Reference Manual
The Geometer’s Sketchpad
Dynamic Geometry Software for Exploring Mathematics
Version 4.0, Fall 2001
Sketchpad Design: Nicholas Jackiw
Software Implementation: Nicholas Jackiw and Scott Steketee
Support: Keith Dean, Jill Binker, Matt Litwin
Reference Manual Authors: Scott Steketee, Nicholas Jackiw,
Steven Chanan
Production: Jill Binker, Deborah Cogan, Diana Jean Parks,
Caroline Ayres
The Geometer’s Sketchpad project began as a collaboration between
the Visual Geometry Project at Swarthmore College and Key
Curriculum Press. The Visual Geometry Project was directed by Drs.
Eugene Klotz and Doris Schattschneider. Portions of this material are
based upon work supported by the National Science Foundation under
awards to KCP Technologies, Inc. Any opinions, findings, and
conclusions or recommendations expressed in this publication are
those of the authors and do not necessarily reflect the views of the
National Science Foundation.
 2001 KCP Technologies, Inc. All rights reserved. No part of this
publication may be reproduced or transmitted in any form or by any means
without permission in writing from the publisher.
 The Geometer’s Sketchpad, Dynamic Geometry, and Key Curriculum Press
are registered trademarks of Key Curriculum Press. Sketchpad and
JavaSketchpad are trademarks of Key Curriculum Press. Fathom Dynamic
Statistics is a trademark of KCP Technologies Inc. All other brand names and
product names are trademarks or registered trademarks of their respective
holders.
Key Curriculum Press
1150 65th Street
Emeryville, CA 94608 USA
http://www.keypress.com/sketchpad
10 9 8 7 6 5
ISBN 1-55953-531-8
04 03
Contents
Introduction
1
Elements
3
Documents.................................................................................... 4
Objects.......................................................................................... 8
Motion Controller ........................................................................ 41
Calculator.................................................................................... 49
Text Palette................................................................................. 55
Script View .................................................................................. 59
Color Picker ................................................................................ 64
Toolbox Reference
67
Overview of the Toolbox ............................................................. 68
Selection Arrow Tools................................................................. 70
Point Tool.................................................................................... 80
Compass Tool............................................................................. 81
Straightedge Tools...................................................................... 83
Text Tool ..................................................................................... 86
Custom Tools.............................................................................. 90
Menu Reference
101
File Menu .................................................................................. 102
Edit Menu.................................................................................. 109
Display Menu ............................................................................ 145
Construct Menu ........................................................................ 156
Transform Menu ....................................................................... 167
Measure Menu.......................................................................... 191
Graph Menu .............................................................................. 201
Window Menu ........................................................................... 211
Help Menu................................................................................. 212
Context Menu ........................................................................... 213
Keyboard Reference
215
Advanced Topics
219
Tips for Experts......................................................................... 220
Advanced Text Topics .............................................................. 226
Advanced Tool Topics .............................................................. 230
Advanced Graphics Export ....................................................... 234
JavaSketchpad and Web-Based Dynamic Geometry .............. 238
Sketchpad’s Internal Mathematics............................................ 246
Index
249
Introduction
The Geometer’s Sketchpad is a software system for creating, exploring, and
analyzing a wide range of mathematics. Using Dynamic Geometry, you
can construct interactive mathematical models ranging from basic
investigations about shape and number to advanced, animated
illustrations of complex systems. If you’re a student, Sketchpad can
help you explore not only the topics from your geometry course, but
mathematical ideas in algebra, trigonometry, calculus, and other areas.
If you’re a teacher, Sketchpad provides a compelling environment with
which to present mathematical concepts, model classroom questions,
and encourage student conjecturing, whether in a hands-on computer
lab or on a demonstration screen before an entire class. Researchers
and other mathematics enthusiasts use Sketchpad to help pose “what
if?” thought experiments, to help probe properties of constructions,
and to help discover new results—as well as to create high-quality
mathematical illustrations for use in activities and assignments, reports
and publications, or simply for their intrinsic visual appeal.
Use the Learning Guide that accompanies this Reference Manual as your
starting point, if you’re just beginning to use Sketchpad. That guide
contains installation instructions and many guided tours to introduce
you to Sketchpad and start you on the path of Dynamic Geometry
discovery. This Reference Manual contains a definitive description of
every Sketchpad tool and command. Consult it after the Learning Guide
when you want to review a particular software function or when you
want to deepen your understanding of how Sketchpad can help you
pursue, discover, and enjoy mathematics.
The Geometer’s Sketchpad Reference Manual
1
Elements
If you’ve worked through the Learning Guide, you already have a good
sense of many of the elements of Sketchpad. Broadly speaking, you use
Sketchpad to create documents containing mathematical diagrams and
geometric figures. Each diagram or figure—each sketch—is
constructed from individual objects that are defined in terms of their
mathematical relationships to each other. You use a combination of
tools and menu commands to interact with documents and the objects
in them. While the next two sections describe each of these tools and
commands in detail, this section introduces the various types of objects
you’ll create and surveys the tools and commands you’ll use to
manipulate them. Refer to this section when you encounter unfamiliar
types of objects in Sketchpad or when you’re looking for properties of
objects you’re already acquainted with.
The Geometer’s Sketchpad Reference Manual
3
Documents
A document in Sketchpad contains one or more sketches—that is, one
or more collections of related mathematical objects. To create these
objects, you’ll use Sketchpad’s Toolbox and menus, described later in
this manual.
A document appears on your screen in a window, and can be saved to
your computer’s hard drive. Once saved, the document can later be
reopened from the hard drive.
The document
window displays one
page at a time.
When a document contains more than a single sketch, each individual
sketch is called a page. A document can also contain any number of
custom tools, which extend the fundamental Point, Compass, and
Straightedge tools in the Toolbox.
Use Document Options in the File menu to manage the pages and tools
contained in a Sketchpad document.
See also: Document Windows (p. 5), Document Pages (p. 6), Document Tools (p. 6),
Document Options (p. 104), Custom Tools (p. 90), Overview of the Toolbox (p. 68),
Menu Reference (p. 101)
Window Border
Page Number
Minimize and
Maximize Boxes
Close Box
Title Bar
Document Window
Sketch
Page Tabs
Tab Divider
Scroll Bars
Resize Area
Document Window (Windows)
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The Geometer’s Sketchpad Reference Manual
Documents
Minimize Button
Zoom Button
Title Bar
Page Number
Close Button
Document Window
Sketch
Page Tabs
Tab Divider
Scroll Bars
Resize Area
Document Window (Macintosh)
Document Windows
A Sketchpad document window is shown above. The window contains
a sketch area within which you construct mathematical figures, and
various controls with which you can manipulate the window and the
sketch.
Title Bar: Drag to reposition the window on the screen.
Close Box: Click to close the window.
Page Tabs: Click to change pages (only present if your document has
more than one page).
Tab Divider: Drag to provide more or less space for page tabs (only
present if your document has more than one page).
Because some sketch
objects, like lines and
rays, extend beyond
the normal scrollable
area, you can always
press a scroll bar’s
buttons even when
the scroll bar itself is
at its limit.
Scroll Bars: Click or drag to scroll the window.
Resize Area: Drag to change the window’s size.
Several of the controls depend on which type of computer you have.
If you’re using a Microsoft Windows computer:
Window Border: Drag any border of the window to change its size.
Maximize Box: Click to expand the window to the largest possible size.
The Geometer’s Sketchpad Reference Manual
5
Documents
Minimize Box: Click to shrink the window to an icon.
If you’re using a Macintosh:
Zoom Box: Click to expand or contract the window.
Collapse Box: Click to shrink the window to its title bar only.
On some Windows and Macintosh computers, you can use System
Preferences to choose whether to display sketches using normal or
high-quality anti-aliased graphics. Sketches will contain smoother lines
and curves when you use anti-aliased graphics, but they will also be
drawn more slowly than sketches using normal-quality graphics.
See also: Document Pages (p. 6), Document Tools (p. 6), System Preferences (p. 141)
Document Pages
A new Sketchpad document always begins with a single page or
sketch—one view of the geometric plane. Over time, you may want to
add additional pages to a document. For example, you may want to
organize a series of sketches that develop an argument; you may want
to present an activity that has several parts; or you may want to explore
a conjecture in more depth than would be possible in a single sketch.
In each of these cases, it’s convenient to store several sketches together
as pages of the same document.
To add pages to a document—and to name, copy, reorder, or remove
existing pages—choose Document Options from the File menu.
Use Document Options
to hide or show the
page tabs.
When a document has more than one page, page tabs normally appear
at the bottom left of the document window, and a page name or
number appears in the window’s title bar. Click on a page tab to switch
from one page of your document to another. You can also create Link
buttons to jump between pages.
See also: Document Windows (p. 5), Document Tools (p. 6), Document Options
(p. 104), Link Buttons (p. 38)
Document Tools
In addition to multiple pages, a Sketchpad document may contain one
or more custom tools—tools that you or someone else has created.
Custom tools extend Sketchpad’s fundamental tools to provide
6
The Geometer’s Sketchpad Reference Manual
Documents
new mathematical objects or to provide new ways to construct
familiar mathematical objects.
Organize custom
tools you use
frequently by storing
each collection of
related tools in a
document of its own.
If you use a
document’s pages to
describe the tools it
contains—and give
examples of their
use—your document
becomes a handy
“toolkit” that you can
share with other
people. For example,
you might want to
make a kit of tools for
constructing different
centers of a triangle
or one for
constructing various
regular polygons.
When you create a new custom tool, it becomes part of your document.
Use Document Options to rename, reorder and remove custom tools from
the active document, and to copy tools from a different open
document into the active document.
When you’re working in a sketch, you can use not only the custom
tools located in the document on which you’re working, but also any
custom tools contained in other open documents. To work with
custom tools contained in a document on your hard drive, open the
document so its custom tools become available. Alternately, if you store
any documents in the Tool Folder located on your hard drive next to
the Sketchpad application, tools from these documents are available
whenever you start Sketchpad—even if the documents that contain
them are not open.
See also: Document Windows (p. 5), Document Pages (p. 6), Custom Tools (p. 90),
Document Options (p. 104), Tool Folder (p. 98)
Script View of a Tool
You can view the details of a tool in written form, using the Script
View. This view shows you a list of each given object and each
constructed object of the tool, allows you to modify the tool in various
ways, and allows you to watch the tool in detail as it operates. Use the
Show Script View command in the Custom Tools menu, or the Show
Script View checkbox in the Tool Options dialog box, to see the Script
View of the active custom tool.
See also: Script View (p. 59), Custom Tools Menu (p. 91), Show Script View (Custom
Tools menu) (p. 96), Show Script View (Tool Options dialog) (p. 106)
The Geometer’s Sketchpad Reference Manual
7
Objects
From one perspective, mathematics is the art of creating new
knowledge by finding new and interesting relationships among existing
mathematical objects. Sketchpad provides you with a rich set of such
mathematical objects and with many ways of connecting them. It’s up
to you to create those objects and their connections, and then to
investigate their behavior, find new relationships, discover symmetry
and patterns, and display and present your results.
The objects you can create in Sketchpad fit into several general
categories. Some of the objects are purely geometric entities—points,
lines, rays, segments, circles, arcs, interiors, loci, and some iterations.
Other objects are either numeric or algebraic entities—measurements,
parameters, coordinate systems, calculations, and functions. And finally,
some objects in Sketchpad—captions and action buttons—are
primarily used in descriptions, explanations, and presentations.
This chapter describes the various kinds of objects with which you can
work in Sketchpad. Refer to the Toolbox section (p. 67) for more
information on how to use Sketchpad’s tools to create and modify
objects, and refer to the Menu section (p. 101) for more details on how
to use Sketchpad’s menus.
See also: Menu Reference (p. 101), Toolbox Reference (p. 67)
Object Attributes
There are several attributes that many or all objects have in common.
Color
Every object in Sketchpad can be colored. To set an object’s color,
select the object and choose a color from the Color submenu of the
Display menu. If an object can have a label, you can set the color for
the label as well, using the Text Palette. Some objects, such as
measurements and functions, display only text; for such objects you can
set the color with either the Color submenu or the Text Palette.
Label
Most geometric objects in Sketchpad can be labeled. To show or hide
an object’s label, use the Text tool or use the Show/Hide Label command
from the Display menu. To change the label, use the Text tool, use
8
The Geometer’s Sketchpad Reference Manual
Objects
Label Properties, or choose the Label command from the Display menu.
To change the font, size, style or color of the label, use the Text Palette.
Visibility
Objects can be hidden from view, although such objects remain present
in the sketch and continue to control or influence other objects. Use
the Hide Objects command in the Display menu to hide an object, and
use Show All Hidden or Object Properties to show hidden objects. You
can also create action buttons to show and hide objects.
Animation
All geometric objects and parameters can be animated so that they
move or change of their own accord. Choose Animate from the Display
menu or create an Animation or Movement action button or use the
Motion Controller to start an animation. Movement and Animation
action buttons allow you to repeat a desired animation conveniently.
Tracing
Any geometric object can be traced so that as it moves it leaves behind
on the screen a trace showing where it’s been. Choose Trace from the
Display menu to trace an object (or to stop tracing an already traced
object). Use Erase Traces to remove collected traces from the screen and
use Color Preferences to determine whether—and how quickly—traces
fade from the screen.
Line Width
Many geometric objects are displayed with straight or curved lines. Use
the Line Width submenu on the Display menu to set the width of such
lines or to display dashed lines.
Properties
You can display and change many of the properties of an object using
the Object Properties dialog box. Select the object and choose Properties
from the Edit menu. You can also right-click (Windows) or Ctrl-click
(Macintosh) on the object to display a Context menu and choose
Properties from the Context menu.
See also: Display Menu (p. 145), Color (p. 8), Text Tool (p. 86), Show/Hide Labels
(p. 149), Label Command (p. 150), Label Properties (p. 122), Text Palette (p. 55), Hide
Objects (p. 148), Show All Hidden (p. 148), Animate (p. 153), Animation Button
The Geometer’s Sketchpad Reference Manual
9
Objects
(p. 112), Trace (p. 151), Line Width (p. 145), Properties (p. 120), Context Menu
(p. 213), Motion Controller (p. 41), Movement Button (p. 113)
Object Relationships: Parents and Children
Without relationships,
this figure could be
dragged apart into six
disconnected points
and six disconnected
segments.
When you create a sketch, the sketch includes
C
not just the geometric objects you’ve
constructed, but also the relationships between
D
those objects. When you construct the triangle
E
shown at right, your sketch includes more than
just six points and six segments; it also includes
A
B
F
the relationships between those 12 objects. For
instance, midpoint E depends on segment BC;
when you choose Midpoint from the Construct menu you create, not just
a point, but also a relationship. Specifically, point E is the midpoint of
segment BC. More generally, we describe that relationship by saying
that point E is the child of segment BC and that segment BC is the parent
of point E. Similarly, segment AB is the child of its two endpoints A
and B, and those endpoints are the parents of segment AB.
These parent-child relationships define the mathematics of your sketch
and are crucial to the way your sketches behave when you explore them
by dragging. Object relationships keep the triangle together as a
triangle, and relationships make the midpoints stay where they belong
when you drag a vertex.
You can think of a sketch as a family tree, defined both by the objects
in the sketch and by their parent-child relationships.
These parent and child relationships are implicit in everything you do in
Sketchpad. For instance, to use a command from the Construct menu,
you must first select certain objects (prerequisites) in your sketch. These
prerequisites become the parents of the newly constructed child.
To explore an object’s mathematical definition—its family tree—use
Select Parents and Select Children from the Edit menu, and use the Parents
and Children pop-up menus in Object Properties. You can even
rearrange your sketch’s family tree using the Split and Merge commands.
See also: Object Properties (p. 121), Select Parents (p. 115), Select Children (p. 115),
Construct Menu (p. 156), Midpoint (p. 157), Arrow Tool (p. 70), Split/Merge (p. 116)
10
The Geometer’s Sketchpad Reference Manual
Objects
Points
Points are the fundamental building blocks of classical geometry, and
geometric figures such as lines and circles are defined in terms of
points. All of Sketchpad’s geometric constructions begin with points.
Points in Sketchpad are of three kinds.
•
An independent point has no parents, and so does not depend on any
other object. An independent point is free to move anywhere on
the sketch plane.
•
A point on path is constructed on a path object such as a line or
circle. A point on path is free to move along its path, but cannot
leave that path.
•
A dependent point—such as a point of intersection—is constructed in
such a way that its position is completely determined by its parents.
A dependent point cannot move by itself. The point can move only
if at least one of its parent objects also moves. Thus, a point at the
intersection of two segments cannot move unless one or both of
the intersecting segments is moved, and the reflected image of a
point cannot move unless either the pre-image or the mirror
moves.
See also: Object Relationships: Parents and Children (p. 10), Point Tool (p. 80),
Straightedge Tools (p. 83), Compass Tool (p. 81), Custom Tools (p. 90), Path Objects
(p. 13)
Moving and Animating Points
Independent points and points on paths can be dragged and animated
(using either Animate from the Display menu or using an Animation
button), and they can be moved with Movement buttons. When you
move or animate any other geometric object, it moves by moving the
parents on which the object depends.
See also: Animate (p. 153), Animation Buttons (p. 38), Animation Button (p. 112)
Movement Button (p. 113), Object Relationships: Parents and Children (p. 10)
Splitting and Merging Points
There may be times when you want to attach an independent point to
another object or make a point that has parents independent of those
parents. The Split and Merge commands in the Edit menu allow you to
make these changes in the family tree of your sketch.
The Geometer’s Sketchpad Reference Manual
11
Objects
Transformed and
plotted points cannot
be split.
Use Split to separate any point on a path, midpoint, or intersection from
its parents, making it into an independent point.
Use Merge to merge an independent point with any other point or with a
path.
See also: Split/Merge (p. 116), Object Relationships: Parents and Children (p. 10)
Constructing Points
There are many different ways to construct points in a sketch.
You can also
construct a point by
using commands
from the Transform
menu to create a
transformed image of
an existing point.
•
Click the Point tool to construct a point.
•
Click the Straightedge tool or the Compass tool in empty space to
construct a point that determines a straight object or circle. Many
Custom tools can also construct points as you use them.
•
Choose Point On Object to construct a point on each selected path
object.
•
Choose Midpoint to construct a midpoint on each selected segment.
•
Choose Intersection to construct the intersection point of two
selected objects. Each object must be a straight object, a circle, or
an arc.
•
Choose the Plot Points command to plot a point with coordinates
defined by two selected measurements or by two numbers you
specify.
See also: Point Tool (p. 80), Straightedge Tools (p. 83), Compass Tool (p. 81), Custom
Tools (p. 90), Path Objects (p. 13), Point On Object (p. 156), Midpoint (p. 157),
Intersection (p. 158), Plot Points (p. 205)
Using Points in Transformations
You can transform points using the Transform menu. You can also use
points in a number of different ways to help specify how other objects
will be transformed.
12
•
Use Mark Center to designate a center point for rotation and dilation.
•
Use Mark Angle to designate the angle formed by three points for
translation and rotation.
•
Use Mark Ratio to designate the ratio formed by three collinear
points for dilation.
The Geometer’s Sketchpad Reference Manual
Objects
See also: Transform Menu (p. 167), Mark Center (p. 169), Mark Angle (p. 169), Mark
Ratio (p. 170)
Measuring Points
Several of the Measure menu’s commands apply to selected points or
to combinations of points.
•
Use Distance to measure the distance between two points.
•
Use Angle to measure the angle formed by three points.
•
Use Ratio to measure the ratio defined by three collinear points.
•
Use Abscissa to measure the x-coordinate of a point.
•
Use Ordinate to measure the y-coordinate of a point.
•
Use Coordinates to measure the coordinates of a point.
See also: Angle (p. 194), Ratio (p. 196), Abscissa (p. 199), Ordinate (p. 199), Coordinates
(p. 198)
Path Objects
There are certain geometric objects in Sketchpad on which you can
construct and animate points. These objects are collectively referred to
as path objects. Path objects include:
For polygons and
other interiors, the
perimeter of the
interior forms that
interior’s path.
•
straight objects (segments, rays, lines, and axes)
•
circles and arcs
•
polygons and other interiors
•
point loci and function plots
You can use the Point On Object command to construct a point on any
path object.
See also: Segments, Rays, and Lines (p. 14), Circles and Arcs (p. 15), Polygons and Other
Interiors (p. 16), Loci (p. 24), Point On Object (p. 156), Animate (p. 153), Principles of
Animation (p. 44)
The Geometer’s Sketchpad Reference Manual
13
Objects
Segments, Rays, and Lines
Segments, rays, and lines are fundamental objects in Euclidean
geometry. In classical geometric constructions, a straightedge is used to
construct these objects.
Constructing Straight Objects
Sketchpad provides several different ways to construct straight objects.
•
Use the Straightedge tool to construct a straight object using two
points.
•
Use the Segment, Ray, or Line command to construct a straight object
using two selected points.
•
Use the Perpendicular Line or Parallel Line command to construct a line
parallel or perpendicular to a selected straight object.
•
Use the Angle Bisector command to construct the ray bisecting the
angle formed by three selected points.
Using Straight Objects
• Use a straight object as a path along which to animate a point by
creating the point with the Point On Object command.
If you use two
selected segments, the
first defines the unit
for x, and the second
for y.
•
Use a straight object as a path along which to animate an
independent point by using the Merge command.
•
Use a straight object as a mirror for reflections by choosing the
Mark Mirror command.
•
Construct a segment’s Midpoint.
•
Construct a straight object’s Intersection with any other straight
object, circle, or arc.
•
Use Define Unit Distance to define a coordinate system with a unit
distance defined by one or two selected segments. (The unit of the
coordinate system is defined by the length of the segment.)
Measuring Straight Objects
You can measure:
•
14
the Length of a segment.
The Geometer’s Sketchpad Reference Manual
Objects
•
the Ratio of two segments’ lengths.
•
the Equation of a line.
•
the Slope of any straight object.
See also: Straightedge Tools (p. 83), Segment (p. 83), Ray (p. 83), Line (p. 83),
Perpendicular Line (p. 160), Parallel Line (p. 159), Angle Bisector (p. 161), Mark Mirror
(p. 169), Point On Object (p. 156), Split/Merge (p. 116), Midpoint (p. 157), Intersection
(p. 158), Length (p. 192), Ratio (p. 196), Equation (p. 200), Slope (p. 200)
Circles and Arcs
Circles and arcs are fundamental objects in Euclidean geometry. In
classical geometric constructions, a compass is used to construct these
objects.
Constructing Circles
Sketchpad provides several different ways to construct a circle.
•
Use the Compass tool to construct a circle using a center point and
another point which defines the radius.
•
Use the Circle By Center+Point command to construct a circle using a
center point and another point which defines the radius.
•
Use the Circle By Center+Radius command to construct a circle using a
center point and either a segment or a distance measurement to
define the radius.
Constructing Arcs
• Use the Arc Through 3 Points command to construct an arc which
passes through three selected points.
•
Use the Arc On Circle command to construct an arc which lies on a
selected circle and is bounded by two selected points on the circle.
Using Circles and Arcs
• Use a circle or an arc as a path along which to animate a point by
creating the point with the Point On Object command.
•
Use a circle or an arc as a path along which to animate an existing
point by using the Merge command.
The Geometer’s Sketchpad Reference Manual
15
Objects
•
Construct the Interior of a circle or the Arc Segment or Arc Sector
interior of an arc.
•
Construct the Intersection of a circle or an arc with a straight object,
a circle, or an arc.
•
Use Define Unit Circle to define a coordinate system that uses the
selected circle as its unit circle.
Measuring Circles and Arcs
You can measure:
•
the Circumference of a circle.
•
the Radius of a circle or an arc.
•
the Area of a circle.
•
the Equation of a circle.
•
the Arc Length of an arc.
•
the Arc Angle of an arc.
See also: Compass Tool (p. 81), Circle By Center+Point (p. 161), Circle By
Center+Radius (p. 162), Arc Through 3 Points (p. 163), Arc On Circle (p. 163), Point
On Object (p. 156), Split/Merge (p. 116), Interior (p. 164), Intersection (p. 158), Define
Unit Circle (p. 201), Circumference (p. 193), Radius (p. 196), Area (p. 194), Equation
(p. 200), Arc Length (p. 196), Arc Angle (p. 195)
Polygons and Other Interiors
There are four kinds of objects in Sketchpad that define a region of a
plane. These include polygons, circle interiors, and two kinds of arc
interiors: arc sectors and arc segments.
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The Geometer’s Sketchpad Reference Manual
Objects
Constructing Interiors
• Use Polygon Interior to construct a polygon defined by three or more
selected vertex points.
•
Use Circle Interior to construct the interior of each selected circle.
•
Use Interior | Arc Sector to construct the sector interior of each
selected arc. An arc sector is bounded by the arc and by the radii to
the two endpoints of the arc.
•
Use Interior | Arc Segment to construct the segment interior of each
selected arc. An arc segment is bounded by the arc and by the
chord connecting the endpoints of the arc.
Using Interiors
• Use an interior as a path along which to animate a point by creating
the point with the Point On Object command. Such a point travels the
perimeter (or circumference) of the interior.
•
Use an interior as a path along which to animate an independent
point by using the Merge command.
Measuring Interiors
You can measure:
•
the Area of any interior.
•
the Perimeter of a polygon interior or an arc interior.
•
the Circumference of a circle interior.
•
the Radius of a circle interior or an arc interior.
•
the Arc Angle or Arc Length of an arc interior.
See also: Interior (p. 164), Point On Object (p. 156), Merge (p. 116), Area (p. 194),
Perimeter (p. 193), Circumference (p. 193), Radius (p. 196), Arc Angle (p. 195), Arc
Length (p. 196)
The Geometer’s Sketchpad Reference Manual
17
Objects
Measurements, Calculations, and Parameters
Numeric values can
be used to control
transformed objects,
plotted points,
calculations,
functions, and
iterations.
Measurements, calculations, and parameters are the three kinds of
Sketchpad objects that display numeric values and so have many
common characteristics. The numeric values generated by
measurements—and by calculations based on measurements—can be
observed in order to discover relationships among objects in your
sketch. And the numeric values of all three kinds of objects can be used
to define or control the behavior of your sketch in powerful and
revealing ways.
Measurements
Measurements quantify the size, orientation, coordinates, and other
m AB = 2.00 cm
characteristics of Sketchpad objects. All measured values update
m∠ABC = 35°
dynamically in Sketchpad when you change the objects they measure.
Often you can gain important mathematical insights by observing how
measurements change and how they relate to each other and to other
objects in the sketch.
All measurements have numeric values of one sort or another. Most
measurements (all except for coordinate pair and equation
measurements) have a single value; these single-valued measurements
can be used for many purposes in your sketch.
To create new measurements, first select objects to measure, then
choose a command from the Measure menu.
See also: Measure Menu (p. 191), Using Values (p. 20)
2⋅AB = 4.00 cm
Calculations
Calculations are mathematical expressions that relate one or more
terms—such as measurement—by arithmetic. For example, where each
of the interior angles of a triangle may be measured by an angle
measurement, the sum of these three angles may be calculated by a
calculation. As you change the measurements on which a calculation
depends, the calculated result changes accordingly.
Use calculations to gain valuable insights into mathematical
relationships or to control sketch objects that you create based on your
calculations. To perform a calculation, choose the Calculate command.
You can define the calculation to use a number of different
mathematical operators, built-in functions, and even functions that
you’ve defined yourself.
18
The Geometer’s Sketchpad Reference Manual
Objects
•
To create a calculation, choose the Calculate command.
•
To edit an existing calculation, double-click it with the Arrow tool,
or choose the Edit Calculation command.
•
To change a calculation into a parameter, edit the calculation so
that it includes only a single numeric value and uses no operators
or functions. It can have no units, or it can have angle or distance
units.
See also: Calculate (p. 198), Calculator (p. 49), Edit Calculation (p. 120), Using Values
(p. 20), Functions (p. 27)
t1 = 1.00
t2 = 1°
t3 = 1.00 cm
Parameters
Parameters are simple given numeric values. Unlike measurements and
calculations, they do not depend on other objects for their value. A
parameter is defined by a single number and an optional unit. Values
with degrees or radian units define an angle parameter; values with cm,
pixel, or inch units define a length parameter; and values with no units
define a scalar parameter. Once defined, parameter values can easily be
changed by typing new values or by animating the parameter so that it
changes value gradually over some numeric domain.
Use parameters to define mathematical constructions when you want to
explore the effects on the construction of varying a numeric quantity.
•
To create a parameter, choose the New Parameter command or
choose New Parameter from within the Calculator.
•
To change the value of a parameter, double-click it with the Arrow
tool or use the Value Properties panel.
•
To change the domain, speed or other aspects of how a parameter
is animated, use the Parameter Properties panel.
•
To increase of decrease the value of a parameter, select it and press
the + or – key on your keyboard.
•
To animate a parameter, select it and choose the Animate command
or click the Animate button of the Motion Controller.
•
To animate a parameter with an action button, select it and choose
Animation from the Action Buttons menu.
The Geometer’s Sketchpad Reference Manual
19
Objects
•
To make a parameter depend on some other value in your sketch,
choose the Edit Parameter command and use the Calculator to
redefine the parameter as a calculation, based on other values.
See also: New Parameter (p. 206), Edit Parameter Definition (p. 120), Value Properties
(p. 124), Parameter Properties (p. 126), Using Values (p. 20), Calculator (p. 49),
Animate (p. 153), Animation Buttons (p. 38) The Motion Controller (p. 41), Animation
Button (p. 112), Keyboard Reference (p. 215)
Using Values
All calculations, all parameters, and most measurements have a single
numeric value. The values of any of these objects can be used in the
same ways.
You can also use Unit
Preferences to set the
precision for all new
measurements,
calculations, and
parameters.
•
Change the value’s precision using the Value Properties panel.
•
Change the distance units or angle units of this value (and of all
values in the sketch) using Unit Preferences.
•
Change how the value is labeled using the Value Properties and
Label Properties panels.
•
Include the value in a calculation or function by clicking the object
while using the Calculator.
•
Choose Mark Distance to use a distance value as the marked distance
for translation.
•
Choose Mark Angle to use an angle value as the marked angle for
translation and rotation.
•
Choose Mark Scale Factor to use a value with no units as the marked
scale factor for dilation.
•
Choose Define Unit Distance to use a distance value as the unit
distance for a new coordinate system.
•
Select two values and choose Plot As (x, y) to plot a point with
coordinates given by the selected values.
•
Use Iterate To Depth to define the depth of an iteration by a selected
value.
See also: Value Properties (p. 124), Label Properties (p. 122), Units Preferences (p. 136),
Calculate (p. 198), New Function (p. 207), Calculator (p. 49), Mark Distance (p. 173),
Mark Angle (p. 169), Mark Scale Factor (p. 170), Define Unit Distance (p. 201), Plot As
(x, y) (p. 205), Iterate (p. 181), Parametric Depth (p. 189)
20
The Geometer’s Sketchpad Reference Manual
Objects
Coordinate Systems and Axes
2
2
-2
-2
The default
coordinate system has
an origin point in the
center of your screen
and a unit point at
(1, 0). Drag the origin
to relocate the
coordinate system
and drag the unit
point to change its
scale.
A coordinate system quantizes the plane and the location of objects on
it. A coordinate system is defined by an origin, a scale, and a grid form
or shape. The origin of a coordinate system is the point or the position
at the center of the coordinate axes. The scale of a coordinate system
determines the size of each unit on an axis. The grid form of a
coordinate system determines how coordinates are measured. In the
most common form—a square coordinate system—coordinates are
indicated as horizontal and vertical distances from the origin, measured
in the same unit scale. Rectangular coordinate systems are like square
coordinate systems in that they measure coordinates horizontally and
vertically from the origin, but they have separate scales for each axis.
Finally, a polar coordinate system measures coordinates by a distance
and a direction from the positive horizontal axis, rather than by
horizontal and vertical distance from the origin.
Many of Sketchpad’s analytic measurements—such as Coordinates,
Equation, and Slope—and all of the plotting commands—such as Plot
Points, Plot As (x, y), or Plot New Function—are defined in reference to a
coordinate system. If you do not create a new coordinate system before
using these commands, they will create a default square coordinate
system for you, and their results will be defined in terms of that
coordinate system.
To create a coordinate system:
•
Choose Define Coordinate System from the Graph menu. Depending
on the objects you’ve selected, there are several different ways this
command constructs the coordinate system.
•
Measure or plot a quantity that requires a coordinate system, such
as Coordinates, Coordinate Distance, Slope, Equation, or Plot Points. If you
don’t already have a coordinate system, any of these commands
will create one.
See also: Graph Menu (p. 201), Define Coordinate System (p. 201), Coordinates (p. 198),
Abscissa (p. 199), Ordinate (p. 199), Coordinate Distance (p. 199), Slope (p. 200),
Equation (p. 200)
The Geometer’s Sketchpad Reference Manual
21
Objects
Modifying a Coordinate System
Depending on how
• To change the form of the grid, choose Polar Grid, Square Grid, or
the coordinate system
Rectangular Grid. The grid form determines whether the Coordinates
was defined, some of
command measures polar or rectangular coordinates and
the Grid Form
determines whether the command to plot measured values is Plot As
commands may not
be available.
(x, y) or Plot As (r, theta).
•
To hide or show the grid lines, choose Hide Grid or Show Grid.
•
To change the appearance of the grid lines or dots, select the grid
by clicking on a grid intersection, then choose Dotted, Thin, or Thick
from the Line Width submenu.
•
To change the color of the grid lines, select the grid by clicking on
a grid intersection, then choose a color from the Color submenu.
•
To make points snap to integer coordinate positions when dragged,
choose Snap Points.
See also: Graph Menu (p. 201), Grid Form (p. 203), Coordinates (p. 198), Plot As
(x, y)/Plot As (r, theta) (p. 205), Show Grid (p. 204), Line Width (p. 145), Color
(p. 146), Snap Points (p. 204), Define Coordinate System (p. 201)
The scale of a
coordinate system
may be defined by an
object in the sketch;
such a coordinate
system doesn’t have a
unit point and cannot
be scaled by dragging
axis tick-mark labels.
Using a Coordinate System
• To change the scale of a coordinate system, drag the unit point.
Square coordinate systems have one unit point, and rectangular
coordinate systems have two.
•
To change the scale of a coordinate system, press and drag any axis
tick-mark label with the Arrow tool.
•
If you have more than one coordinate system, make one of them
the active one by choosing Mark Coordinate System.
•
To plot a point on the coordinate system, choose Plot Points.
•
To plot two measured values on the coordinate system, select them
and choose Plot As (x, y).
•
To plot a new function on the coordinate system, choose Plot New
•
To plot an existing function on the coordinate system, select the
function and choose Plot Function.
Function.
See also: Graph Menu (p. 201), Mark Coordinate System (p. 203), Plot Points/Plot Table
Data/Plot As (x, y) (p. 205), Plot New Function (p. 207), Functions (p. 27)
22
The Geometer’s Sketchpad Reference Manual
Objects
Multiple Coordinate Systems
While many sketches require at most one coordinate system, you can
create more than one coordinate system if you wish. You might want
more than one coordinate system to compare objects—the coordinates
of a point or the plot of a function—in the two systems.
When you have more than one coordinate system, only one of those
systems is the marked or active coordinate system. The marked
coordinate system is the one that’s used for plotting points and
functions and for measurements such as coordinates or slope that
require a coordinate system. To change which coordinate system is
marked in your sketch:
1. Select the coordinate system you wish to mark by clicking at a grid
intersection or on its origin, unit point, or either axis.
2. Choose Mark Coordinate System from the Graph menu.
See also: Mark Coordinate System (p. 203), Plot Points/Plot Table Data/Plot As (x, y)
(p. 205), Plot New Function (p. 207)
How To . . . Construct a Geoboard
D
A
C
B
Area ABCD = 7.00 cm
2
You can use a coordinate system to make a Sketchpad geoboard.
Straight objects you construct on this geoboard will always stick to the
grid dots, just like rubber bands attached to the posts on a physical
geoboard. And with a Sketchpad geoboard you can construct polygons,
measure slopes of lines, measure perimeters and areas of polygons, and
“measure” coordinates of points.
1. Choose Preferences from the Edit menu and click on the Units tab.
2. Set the Distance units to cm and click OK to close the dialog box.
3. Choose Define Coordinate System from the Graph menu.
4. Choose Display | Line Width | Dotted to show dots at the grid
intersections.
5. Hide the origin, unit point, and axes.
6. Choose Graph | Snap Points to make points snap to your geoboard’s
grid intersections.
Your geoboard is ready for use.
The Geometer’s Sketchpad Reference Manual
23
Objects
Loci
In geometry, a locus is the set of all possible
positions of an object that satisfy some
specific condition. For example, you might
investigate the locus of points equidistant
from two fixed points or the locus of circles
that have their center point on a fixed circle
and which pass through a fixed point.
Possible paths include
straight objects,
circles, arcs, polygons,
other interiors, and
point loci themselves.
In Sketchpad, a locus describes the position
of an object while some point (on which the
object depends) travels along a path. More
formally, a Sketchpad locus is the set of positions of a driven object
generated as some driver point on which the object depends moves to a
finite number of positions along a drive path.
A few examples may help: In the first pair of illustrations, point E is
the midpoint of segment CD. The illustration at right shows the locus
of point E as point C travels along segment AB. The locus of point E
forms a smaller segment, parallel to and half the length of segment AB.
In this example, point E is the driven object, point C is the driver
point, and segment AB is the drive path. In the next pair of
illustrations, circle CB is constructed with points C and B defined on
circle AB. The locus of circle CB as point C travels along circle AB is
the cardiod shown at right. In this example, circle CB is the driven
object, point C is the driver point, and circle AB is the drive path.
C
B
A
E
D
Driver: Point C
C
A
B
E
D
Locus of point E
Drive path: Segment AB
Driven object: Point E
24
The Geometer’s Sketchpad Reference Manual
Objects
A
C
B
A
C
B
Driver: Point C
Drive path: Circle AB
Locus of circle CB
Driven object: Circle CB
If you don’t find the terms driver, drive path, and driven object helpful in
understanding the objects that define a locus, consider using other
analogies. Some people prefer to think of a Sketchpad locus as a
visualization of an abstract function. In this analogy, an independent
variable (the driver) is defined over a particular domain (the drive path).
The value of that independent variable (that is, the position of the
driver on the drive path) determines the value of some dependent
variable (that is, the position of the driven object). Each sample of the
locus represents one value of the function, and the entire locus is an
approximation of the range of the function. (It’s an approximation
because Sketchpad uses only a finite number of ordered pairs—or
samples—in constructing the locus.) The abstract function in this
analogy is actually the construction by which the driven object relates to
the driver.
Others prefer to think of a Sketchpad locus as a more durable form of
a traced animation. In that case, you have an animating point (the driver)
moving along the path on which it’s constructed (the drive path) that
defines some traced object (the driven object). As the animating point
moves along its path, the traced object traces out the locus.
The difficulty of naming these dynamic concepts has a long history:
When Johan De Witt and Sir Isaac Newton studied locus constructions
of the conics in the 17th century, they used the term directrix to refer to
what we call the driver. Today, when discussing the same type of locus,
mathematicians use directrix to refer to the drive path instead!
The Geometer’s Sketchpad Reference Manual
25
Objects
Using Plot Properties,
you can set the
number of samples
Sketchpad uses when
calculating and
displaying a locus.
Mathematically, a locus may describe an infinite number of positions of
the driven object. However, to display an infinite number of positions
would require a computer to use an infinite amount of time, so
Sketchpad instead displays a large (but not infinite) number of possible
positions, rather than all possible positions. Each position that
Sketchpad does display is called a sample.
See also: Locus (p. 166), Animate (p. 153), Trace (p. 151), Path Objects (p. 13)
Constructing a Locus
To construct a locus, you must first construct the driven object—the
object whose locus you want to construct— in such a way that it
depends on the driver. Either the driver must be a point on path or you
must construct a separate path along which to move the driver.
1. Select the driver and the driven object.
2. If the driver is an independent point, select the drive path—a path
object that does not depend on the driver.
3. Choose Locus from the Construct menu.
See also: Locus (p. 166), Construct Menu (p. 156), Path Objects (p. 13)
Modifying a Locus
You can change the color and line width of a locus just as you do with
other objects, and you can show the label of a point locus. There also
are some special modifications you can make which apply only to loci.
When a locus is first
constructed, the
number of samples it
uses is determined by
a value on the
Sampling Preferences
panel of Advanced
Preferences.
A point locus is
simply the locus of a
point, as opposed to
the locus of a circle or
line or other object.
26
Determining the Number of Samples
After a locus has been constructed, you can use the Plot Properties
panel of Properties to change the number of samples. In general, the
higher the number of samples, the better the quality of the locus, but
the slower it is to drag.
Continuous and Discrete Point Loci
If the locus is a point locus, you can use the Plot Properties panel of
Properties to determine whether the locus is displayed in continuous
form (with the samples connected to each other) or in discrete form
(with a separate dot for each sample).
The Geometer’s Sketchpad Reference Manual
Objects
Resizing a Point Locus
If a point locus is based on a closed path (such as a circle) or a finite
path (such as a segment or arc), the domain of the driver is fixed. But if
the drive path is infinite and open (such as a ray or line), the domain of
the driver—and therefore, the potential size of the locus—is infinite! If
possible, Sketchpad limits the domain based on the portion of the path
that is visible on the screen. Such a point locus (on an infinite open
domain) displays an arrowhead on the end of the locus.
If you wish to change the displayed size of a point locus, use the Arrow
tool to drag the arrowhead at either end of the locus. Drag in the
direction that the arrowhead points to increase the size of the locus;
drag in the opposite direction to decrease the size of the locus.
See also: Color (p. 146), Line Width (p. 145), Show/Hide Labels (p. 149), Advanced
Preferences (p. 139), Sampling Preferences (p. 140), Properties (p. 120), Plot Properties
(p. 124), Resizing Function Plots and Loci (p. 79), Arrow Tool (p. 70)
Functions and Function Plots
Sketchpad allows you to define functions by their equation and to plot
them on a coordinate system. Throughout Sketchpad and this manual,
the term function refers to a symbolic definition, such as f (x) = 2x, and
the term function plot refers to the graph of a function on a specific
coordinate system.
See also: Coordinate Systems and Axes (p. 21)
Functions
Sketchpad allows you to create functions and families of functions, to
2
f ( x) = a⋅x +b⋅x+c evaluate functions and use them in calculations, to edit functions, to
plot functions and their inverses using either rectangular or polar
coordinates, to combine and compose functions in various ways, and to
differentiate functions.
Creating a New Function
To create a new function, for example, f (x) = sin(x), use New Function.
This command opens Sketchpad’s function calculator to allow you to
define how the function is calculated.
The Geometer’s Sketchpad Reference Manual
27
Objects
To create a new function and plot it immediately, use the Plot New
Function command.
See also: New Function (p. 207), Plot New Function (p. 207), Calculator (p. 49)
Parameters are
particularly useful for
investigating families
of functions because
they can easily be
changed or animated.
Families of Functions
While you’re using the function calculator to enter or edit a function,
you can create a new parameter or use an existing parameter or other
measurement from your sketch. When the value of this parameter or
measurement changes, the definition of the function changes. For
example, if you create parameter a while you’re specifying the function
f (x) = a · sin (x ), you can investigate the behavior and plots of this
entire family of functions, including functions such as f (x) = –1 · sin(x)
and f (x ) = 3 · sin (x ) by varying the parameter a. Similarly, you can use
three parameters as you define the function f (x) = ax 2 + bx + c to
investigate the effect on this family of functions of changing each
parameter’s value.
With the Calculator open add a new parameter to the function
definition by choosing New Parameter from the Calculator’s Values popup menu. Insert an existing parameter by clicking on it in the sketch.
See also: New Function (p. 207), Plot New Function (p. 207), Edit Parameter Definition
(p. 120), New Parameter (p. 206), Parameters (p. 19), Calculator (p. 49)
Evaluating and Using Functions
Functions can be thought of as rules for turning input
f (x) = 2 ⋅ x + 3
values into output values. For example, the function
f (5) = 13.00
f (x ) = 2 · x + 3 can be thought of as the rule which
says “To get an output value, take the input value,
multiply it by 2, then add 3 to the result.” Evaluating a function
means following the rule for a particular value. For example,
f (5) = 2(5) + 3 = 13.
When using the
Calculator, you can
click any function
visible in your sketch
to insert it into the
calculation—or new
function—that you’re
defining in the
Calculator.
Once you’ve defined a function, you can use it in later calculations and
in later function definitions. The Function pop-up menu in the
Calculator includes every selected user-defined function in your sketch.
To insert a user-defined function that isn’t in the list, click the function
in the sketch. (If the Calculator is hiding the function, you may have to
move the Calculator out of the way first.) You can choose any of these
user-defined functions in the definition of the new calculation or
function.
See also: New Function (p. 207), Calculate (p. 198)
28
The Geometer’s Sketchpad Reference Manual
Objects
Plotting a Function
To define a new function and plot it immediately on the marked
coordinate system, choose Plot New Function. Define the function as
described above. When you close the function calculator, the function
is plotted in the form that was set in the function calculator’s Equation
pop-up menu when you defined the function. The plot can take any of
four possible forms.
You can use the
x = f ( y ) form to plot
the inverse of a
function y = f (x ).
•
y = f (x )
•
x = f (y )
•
r = f (θ )
•
θ = f (r )
To plot one or more existing functions on the marked coordinate
system, select every function you want to plot and choose Plot Function.
Each function is plotted in the form that you set in the function
calculator’s Equation pop-up menu at the time you defined that
function.
See also: Plot New Function (p. 207), Function Plots (p. 27)
Editing Functions
You can edit an existing function to change its definition or how it’s
plotted. Editing a function is useful, for example, if you’ve plotted the
graph of y = 2 · sin(x ), and you want to change the function to
y = 3 · sin(x ) in order to see how the two graphs differ. Instead of
changing the constant to 3, you can insert a new parameter, allowing
you to investigate the plots of the family of functions y = a · sin(x ).
To edit a function, select that function and choose Edit Function from the
Edit menu or from the Context menu.
Choose a new
Equation form while
editing a function to
change how that
function is plotted.
For instance, to plot a
function f as a polar
function, change f’s
Equation form from
y = f (x) to r = f (θ).
When you edit a function, you can redefine it in any way you want—
introducing new parameters and existing parameters, measurements
and calculations—provided you don’t create a circular definition. A
circular definition is one that uses a calculation that depends on the
function you’re editing. For instance, if you’ve defined a function f and
used it to calculate f (3) in your sketch, you cannot later edit function f
so that it uses the calculation of f (3) in its definition.
See also: Edit Function (p. 120), Calculator (p. 49)
The Geometer’s Sketchpad Reference Manual
29
Objects
Transforming Functions
Once you’ve defined a function f (x )—for example, f (x ) = x 2— you can
define and plot other functions that are transformations of f (x ).
Transformations of f(x) include such functions as g (x ) = 2 · f (x ),
g (x ) = f (–x ), or even g (x ) = –3 · f (2x + 5) – 1.
To define a function that is a transformation of f (x), choose New Function
from the Graph menu. Define the new function as you normally would,
inserting the original function f wherever you want by choosing it from
the user-defined section of the Calculator’s Function pop-up menu.
See also: New Function (p. 207)
Composite Functions
Composite functions are functions of functions. For
instance, f ( g(3)) is the composition of functions f and g
evaluated at 3 and can be thought of as follows: “First
evaluate function g at 3. Then use this result as the
input for function f. The result of evaluating f is the
value of the expression f ( g(3)).”
f (x) = 2 ⋅ x + 3
g( x ) = x 2
f (g (3)) = 21.00
To compose two functions f and g, you must first define each of the
functions separately. Then you can evaluate the composite function (as
in the example) or define a new function that is the composite of the
two original functions.
To evaluate f (g(3)) as in the example, select functions f and g and choose
the Calculate command. In the Calculator, choose f (x ) from the userdefined section of the Function pop-up menu, then choose g (x )
similarly. And finally, enter the argument (“3” in this example), close
the parentheses, and click OK.
To create the composite function h (x ) = f ( g(x )), choose the New Function
command to define function h (x ), and follow the same process, using x
rather than 3 as the argument.
See also: Calculate (p. 198), New Function (p. 207)
30
The Geometer’s Sketchpad Reference Manual
Objects
Differentiation
To create the derivative of a function with respect to its independent
variable, select the function, then choose Derivative from the Graph
menu. The result is a derivative function that can be plotted or
evaluated like any other function.
See also: Derivative (p. 208)
Function Plots
To plot a new function,
a = 1.00
2
choose the Plot New Function
b
=
2.00
command and use the
c = -1.00
function calculator to define
your function. While you’re
defining the function, use the -5
Equation pop-up menu to
define the form of the plot.
2
(Sketchpad supports several
-2
f ( x ) = a⋅x +b⋅ x+c
Cartesian and polar equation
forms.) When you close the
function calculator, both the function and its plot appear.
To plot one or more existing functions on the marked coordinate
system, select every function you want to plot and choose Plot Function.
Each function is plotted in the form that was set in the function
calculator’s Equation pop-up menu at the time you defined that
function.
See also: Functions (p. 27), Plot New Function (p. 207), Coordinate Systems and Axes
(p. 21)
Iterations and Iterated Images
To iterate an action or an operation is to repeat it some number of
times. Mathematically, iteration refers to the process of repeatedly
applying some mathematical construction, calculation, or other
operation to the previous result of that same operation. The operation
must define an output in terms of some input, and the iteration uses
the output of one step as the input for the next step.
The Geometer’s Sketchpad Reference Manual
31
Objects
Sketchpad allows you
to iterate any of the
mathematical
relationships you use
to construct
relationships in a
sketch. You can use
iterations to create
repeated
transformations (such
as tessellations), to
produce fractals and
other self-similar
objects, or to generate
other sequences and
series.
In algebra, an iteration is the repetition of a calculation that uses an
input value to calculate an output value. The iteration repeatedly applies
the calculation to the value that resulted from the previous
calculation—the output from one step is the input for the next. To
begin the process, there must be a starting value; this value is called the
seed. Consider the calculation “add 2” applied to the seed 5. When you
apply this operation once to 5, the result is 7 (because 5 + 2 = 7). When
you then apply the rule to the first result (7), the second result is 9
(because 7 + 2 = 9). Iterating this operation produces the sequence of
values 7, 9, 11, 13, 15, 17, … .
In geometry, an iteration uses an operation performed on a set of
geometric objects that produces a new set of objects. The original set of
objects is the input, and the new set is the output. To begin the process,
there must be a starting set of objects; these starting objects are the preimage. Consider the transformation “translate to the right by 1 cm.” If
you apply this transformation to an initial pre-image ∆ABC, the first
image result is ∆A'B'C', translated 1 cm to the right. Iterating this
transformation produces a sequence of triangles congruent to the initial
pre-image ∆ABC, each shifted 1 cm to the right of the previous triangle
in the sequence.
In these examples, it’s helpful to think of the operation—“add 2” or
“translate to the right by 1 cm”—as distinct from any individual value
or triangle in the sequence; rather, think of it as mapping each value or
image in the sequence to the next value or image in the sequence. Thus,
one might say that 47 maps to 49 under the operation “add 2.” The
entire iteration is defined by the pre-image (the seed) and the mapping
operation. When you apply the operation to the pre-image once, the
result is the first image of your pre-image according to your mapping
operation. As you iterate the operation, you generate the second, third,
and fourth images, and so on.
Any parameter used
to define an iteration
must have geometric
children.
32
Creating Iterations
Iterated operations and constructions in Sketchpad are always created
by example, and are always defined in terms of points and parameters.
Use the tools and menus to construct a figure in which a set of
independent points or parameters is used to produce (through whatever
mathematical relationships you wish) an equal number of dependent
objects (points or calculated values). The independent objects represent
the pre-images or seeds of your iteration; and the corresponding
dependent objects represent the first iterated images of those initial
The Geometer’s Sketchpad Reference Manual
Objects
objects. Then use Iterate in the Transform menu to indicate the
correspondence between pre-images and images. The Iterate dialog box
allows you to specify the number of times you wish to iterate the
construction. The result is a collection of iterated images of the preimages and of every object that depends on the pre-images.
A
C'
B
More generally, if a
geometric point preimage A is used to
construct a dependent
point A', then the
iterated image of that
point—or the orbit of
that point’s
iteration—is the
sequence of points
A', A'', A''', and so
forth.
A
C'
B'
A'
C
B
B'
A'
C
In the illustration on the left, ∆ABC and its midpoints A'B'C' have
been constructed. In the illustration on the right, the independent
vertices of the triangle have been mapped to their midpoint images using
the Iterate dialog box, and the constructed relationship has been
iterated four times. The result is a series of images of the segments and
points that defined the initial construction as the triangle is iterated
toward its midpoint triangle.
See also: Iterate (p. 181)
Working with Iterations
Once you’ve created an iteration and produced some iterated images,
you can
•
Select, color, hide, or delete the iterated images of individual
objects within the overall iteration. For example, in the illustration
above, you might wish to hide or delete the images of the iterated
triangle’s vertices, so that only the edges of the triangle are visible
in your illustration.
•
Alter the number of times the construction is iterated. Use
Properties to visit the Iteration Properties dialog box of any iterated
image, where you can adjust the number of iterations numerically.
•
Adjust the number of iterations of a construction by first selecting
one or more of its iterated images and then by pressing the + or –
key to increase or decrease the number of iterations by one.
The Geometer’s Sketchpad Reference Manual
33
Objects
•
Use the Iteration Properties dialog box to alter other properties of
the iteration.
When defining new iterations, you can also use the Iterate dialog box to
•
Create iterations in which each iteration step produces more than
one copy of the pre-image. Such iterations allow you to create
tessellations and fractals.
•
Create iterations in which the depth of iteration is controlled by a
parameter or other calculation in your sketch.
See also: Iterate (p. 181), Iteration Properties (p. 134), Multiple Iteration Maps (p. 185),
Parametric Depth (p. 189)
Tables
Use tables to examine how
measured values change over
time. Tables are organized in
rows and columns, with each
column describing a single
measurement, and each row
containing measurement
values at the moment that the
row was added to the table.
Area
AB
(Ra diu s
AB )2
(Area
(Radiu s
AB )
AB )2
cm2
3.1416
14.73 cm2
4.69 cm2
3.1416
32.31 cm2
10.28 cm 2
3.1416
74.43 cm2
23.69 cm 2
3.1416
161.50 cm2
51.41 cm 2
3.1416
5.60
cm2
1.78
Once you create a table, you can add rows to it one at a time, or you
can instruct Sketchpad to collect rows automatically as you drag points
or other objects that cause the tabulated values to change. You can also
remove one or all rows from an existing table.
Working with Tables
• To create a table, select one or more measured values—
measurements, calculations, coordinate pairs, or equations—and
choose Tabulate from the Graph menu. (Sketchpad also creates a
table for you automatically when you create an iteration that causes
one or more values to change.)
•
34
When you want to add—or “capture”—the current values of your
tabulated measurements, double-click the table to add a new row to
it. Or select the table and choose Add Table Data from the Graph
menu.
The Geometer’s Sketchpad Reference Manual
Objects
•
To capture changing values in a table automatically as you drag or
animate measured objects in your sketch, choose Add Table Data
from the Graph menu. Sketchpad displays a dialog box that lets
you specify how many rows to add, and how frequently to add
them.
•
To remove a table’s most recently added row, or to remove all of
the rows from a table, select the table and choose Remove Table Data
from the Graph menu. You can also double-click a table with the
Arrow tool while holding down the Shift key to remove the most
recently added value.
•
To change the label of a table column, change the name of the
original measurement tabulated in that column.
•
To change the precision or units of a table’s data, change the
precision or units of the original measurements tabulated by the
table.
•
Use a table’s Table Properties panel to specify whether the last row
of the table changes dynamically as its measured values change.
•
To change the color of a table’s borders, use the Color submenu of
the Display menu.
•
To change the color of a table’s text, or to change the text’s font,
size, or style, use the Text Palette.
•
To plot the data from a table on a coordinate system, choose Plot
Table Data from the Graph menu.
•
To export the data in a table to another application such as Fathom
Dynamic Statistics™ software or Microsoft Excel®, select the table and
choose Copy from the Edit menu. Switch to the other application
and choose Paste. Sketchpad exports the table data to the other
application as tab-delimited text.
Note that when Sketchpad creates a table for you automatically as the
result of an iteration, the number of rows of data displayed in that table
is controlled by the level of iteration. You can only add and remove
data from these automatic tables by changing the level of iteration.
See also: Tabulate (p. 209), Add Table Data (p. 209), Remove Table Data (p. 210), Table
Properties (p. 127), Tables of Iterated Values (p. 188), Copy (p. 110), Iterate (p. 181),
Text Palette (p. 55), Plot Table Data (p. 205)
The Geometer’s Sketchpad Reference Manual
35
Objects
Captions
Pythagorean Theorem
2
2
a +b =c
2
Chris Williams
Period 4
Use captions in Sketchpad to display text that identifies or explains
your sketch. You can format captions to include a variety of sizes,
styles, colors, and symbolic notation, and you can incorporate values
from measurements, calculations, and labels of objects in captions.
Working with Captions
• Press and drag the Text tool in empty space in your sketch to create
and begin typing a caption.
•
While you’re editing, use the mouse or the arrow keys to move the
insertion point or to select parts of the caption.
•
Use the Text Palette to change the font, size, style, or color of the
selected part of the caption.
•
Click the Symbolic Notation tools in the Text Palette to add
mathematical notation and formatting to your caption.
•
Drag a caption’s resize handle to resize the caption and reflow its
contents.
See also: Creating Captions (p. 88), Editing Captions (p. 88), Text Palette (p. 55)
Composite Captions
In some cases, you may want a caption to contain dynamic text—text
that you do not type, but that Sketchpad provides automatically. A
composite caption combines several different text elements into a
single caption. The elements you can incorporate into composite
captions include:
•
other captions: The caption is incorporated as plain text.
•
measurements and calculations: The current value of the
measurement is incorporated, without the name or label.
•
labeled objects: The label of the object is incorporated.
Using these elements, you could compose a single caption, such as
“A 3 cm by 2 cm rectangle has an area of 6 cm2,” from several other
captions and measurements. As you alter your measured geometry, the
composite caption dynamically updates to reflect the changed values in
your sketch.
36
The Geometer’s Sketchpad Reference Manual
Objects
Creating a Composite Caption
To create a composite caption, select the objects you want to
incorporate. At least one of the objects must be a caption, and each
object must either be displayed as text or have a label. Then choose
Merge Text from the Edit menu to create the caption.
Editing a Composite Caption
To edit a composite caption, separate the composite object into its
parts using Split Text, make your changes to the parts, then recombine
them using Merge Text.
See also: Creating Captions (p. 88), Editing Captions (p. 88), Text Palette (p. 55),
Split/Merge (p. 116)
Action Buttons
Action buttons are objects you create in your sketch which you can
press to perform a variety of actions, including hiding or showing
objects, moving or animating objects, linking to a different page in your
document or to a web site, scrolling the sketch window to a particular
position, or making a presentation. Use action buttons to repeat
frequent actions conveniently or to help explain the mathematics of
your sketch to others who may interact with it.
Using Action Buttons
Create action buttons using the Action Button submenu in the Edit
menu. Once you’ve created an action button, there are several things
you can do with it.
Some buttons stay
down after you press
them, indicating that
their action is still
continuing. You can
click such a button a
second time to stop
its action.
•
Start the button’s action by clicking the
button body (not the handle) with the Arrow
tool.
•
Select the button by clicking the handle (not
Handle
Body
the body) with the Arrow tool. Once the
button is selected, you can hide it, delete it, and perform other
actions on it.
•
Change the button’s font, size, style, and color by first selecting it,
then using the Text Palette.
•
Move the button to a different position by using the Arrow tool to
drag the button’s handle.
The Geometer’s Sketchpad Reference Manual
Hide Point
37
Objects
•
Hide Point A
Change the button’s label by double-clicking it with the Text tool.
Hide/Show Buttons
A Hide/Show button hides or shows a group of objects.
Use Hide and Show buttons when there are details in a sketch which
you sometimes want visible and sometimes want hidden. For example,
your sketch might use a single triangle to show the construction of the
circumcenter, centroid, and orthocenter. If you show all the
construction lines at the same time, the sketch will be very confusing.
You can use Hide/Show buttons to show or hide the construction lines
for each of the three different constructions.
See also: Hide/Show Button (p. 112), Hide/Show Properties (p. 127)
Animate Point
Animation Buttons
An Animation button animates one or more objects. The objects must
be either geometric objects or parameters.
Use Animation buttons to automate motion in your sketch. You can
use an Animation button to move a point along its path, to move an
independent point around randomly in the plane, or to vary a
parameter. Use Animate Properties to set the animation speed and
direction and to set the domain for parameter animation.
See also: Animation Button (p. 112), Animate Properties (p. 128)
Move A -> B
Movement Buttons
A Movement button moves one or more points toward defined
destinations.
Use a Movement button to move an independent point or a point on
path toward a specific destination.
See also: Movement Button (p. 113), Move Properties (p. 130)
Page 2
www.keypress.com
Link Buttons
A Link button links to a different page in the current document, or
links to a web site or other location defined by a URL.
Use a Link button to make it easy to navigate among pages in a
document that contains more than one page, or use a Link button to
open up a web site that’s related to the topic of your sketch.
See also: Link Button (p. 114), Link Properties (p. 133)
38
The Geometer’s Sketchpad Reference Manual
Objects
Scroll
Scroll Buttons
A Scroll button scrolls the sketch window so that a specific point in the
sketch is located either in the center of the window or at the top left
corner of the window.
Use a Scroll button in large sketches to position the window to show a
particular part of your sketch.
See also: Scroll Button (p. 114), Scroll Properties (p. 134)
Present 2 Actions
Presentation Buttons
A Presentation button automatically activates a group of other buttons.
The buttons can be activated either simultaneously or in sequence.
Use a Presentation button to choreograph a complex set of motions or
to present a Sketchpad slide show.
See also: Presentation Button (p. 113), Presentation Properties (p. 131)
Pictures
Pictures are images not created by Sketchpad, which can be used to
enhance or decorate a sketch.
To add a picture to your sketch:
1. Use some other program, such as Paint or a web browser, to create
or locate an image you want to insert into your sketch.
2. Still using the other program, copy the image to the clipboard.
3. Switch back to Sketchpad.
4. Choose Paste Picture from the Edit menu. The picture appears in
your sketch.
If, before pasting, you select one point, the top left corner of the
picture will be attached to the point. If you select two points, the top
left corner will be attached to the first point and the bottom right
corner to the second point.
Hold down the Shift
key while you drag to
maintain the picture’s
original aspect ratio.
If you create the picture without selected points, the selected picture
has resizing handles at the top left and bottom right corners. Drag
these handles with the Arrow tool to resize the picture.
If you create the picture with a single selected point, there’s only one
resizing handle, at the bottom right corner.
The Geometer’s Sketchpad Reference Manual
39
Objects
If you create the picture with two selected points, there are no resizing
handles. The positions of the two points determine how the picture is
resized.
See also: Paste (p. 111), Resizing Pictures (p. 78)
40
The Geometer’s Sketchpad Reference Manual
Motion Controller
Use the Motion Controller to start
objects animating and to control the
motion of objects in your sketch.
Movement is at the heart of dynamic
geometry. By animating and dragging
objects in Sketchpad, you can rapidly
explore many different variants of a
construction, allowing you to make discoveries and investigate
conjectures in a way that would be impossible without motion.
Animation also allows you to demonstrate and present your findings in
a more interesting and effective way than you could ever do with a
static diagram.
Many of the functions of the Motion Controller are available through
Sketchpad’s menus, action buttons, and so forth. By providing a
shortcut to these functions, the Motion Controller makes it quicker and
easier for you to use animation in your sketch.
If you want even
more control of
animation than the
Motion Controller
provides, create
Animation buttons as
described in the Edit
menu chapter
(p. 112).
You use the Motion Controller to start objects moving, stop them,
reverse their direction, and change their speed. It gives you easy access
to the functionality of the Animate, Increase Speed, Decrease Speed, and Stop
Animation commands in the Display menu, while providing better
control of the speed and direction of moving objects. It makes it easy
to control the motion of a specific moving object without affecting
other moving objects.
The Motion Controller appears when you start an animation or when
you choose Show Motion Controller from the Display menu.
When you move objects in Sketchpad, whether by using the Motion
Controller, by choosing Animate, or by pressing an Animation or
Movement action button, different objects move or change in different
ways.
•
Independent points move freely in the plane.
•
Points on paths move along their paths.
•
Parameters change their values.
•
All other objects move by moving their parent objects.
See also: Animate (p. 153), Increase/Decrease Speed (p. 153), Stop Animation (p. 154),
Show Motion Controller (p. 154), Animation Buttons (p. 38), Movement Buttons
(p. 38), Object Relationships: Parents and Children (p. 10)
The Geometer’s Sketchpad Reference Manual
41
Motion Controller
Motion Controller Elements
Each element of the Motion Controller affects a different aspect of
motion.
You can also change
the target by selecting
an object with the
Selection Arrow tool,
although that can be
hard to do if the
object is moving.
Target: This button describes the
objects that will be affected by the
Motion Controller buttons. If at
least one object is moving in your
sketch, you can click to display a
pop-up menu of moving points and changing parameters. Choose an
object from the menu to select it and to make it the new target.
If there are movable selected objects in your sketch, the target matches
the selections. If there are selected objects that cannot be moved, there’s
no target. If the sketch contains moving objects but no selections, the
target is all moving objects.
Animate: Click this button to animate each target object.
This button has the same effect as the Animate command.
Stop: Click this button to stop each moving target object.
This button has the same effect as the Stop Animation command.
Reverse: Click this button to reverse the direction of each
moving target object. This button is enabled when at least one
target object is moving in a fixed direction (rather than in
random directions).
The Pause button
affects all moving
objects, not just the
target.
Pause: Click this button to pause all motion. Unlike the other
elements of the Motion Controller, this button affects every
moving object, not just the target.
Speed: Click and type a new speed here to change
the speed of each moving target object.
Each speed step is a
little more than 25%;
three speed steps in a
row changes the
speed by a factor of
two.
Increase/Decrease Speed Arrows: Click one or
the other of these arrows to increase or decrease by
one step the speed of each moving target. These arrows have the same
effect as the Increase Speed and Decrease Speed commands.
Drag the Motion Controller by the title bar to reposition it. If you’re
using a Microsoft Windows computer, you can also dock it to either the
top or bottom of the Sketchpad window.
See also: Animate (p. 153), Stop Animation (p. 154), Increase/Decrease Speed (p. 153),
Show Motion Controller (p. 154)
42
The Geometer’s Sketchpad Reference Manual
Motion Controller
Using the Motion Controller
There are many different animation-related tasks you can accomplish
using the Motion Controller.
The only objects you
can’t animate are
captions, calculations,
functions, action
buttons,
measurements, and
pictures.
Starting an Animation
1. Select one or more objects you want to animate. The objects must
be geometric objects or parameters.
2. Click the Animate button in the Motion Controller. Each selected
object begins moving.
Selecting a Moving Object
If you want to modify the motion of a particular moving object, you
must select that object to make it the target of the Motion Controller. If
the object were not moving, you would simply click it with the Arrow
tool to select it—but it’s not always so easy to click a moving object.
You may be able to select the desired moving object with a mouse click
if it’s moving slowly or with a selection rectangle if there aren’t any
other objects near it. If not, follow these steps.
1. Press and hold on the Target menu to show the list of target
objects.
2. If the object you want appears on the list, choose it. The object is
selected, and you’re done.
If the object is not an independent point, a point on path, or a
parameter, the object is moved by moving its parents, and it
doesn’t appear on the list. In this case, continue with steps 3
through 5.
3. Click the Pause button in the Motion Controller. Motion stops.
4. Select the object.
5. Click the Pause button again to release and restart the motion. The
object remains selected and is listed as the target object.
Stopping a Moving Object
You can stop the motion of a single object while leaving other objects
in motion.
Clicking Stop with
nothing selected stops
all moving objects.
1. Select the moving object as described in the previous section.
2. Click the Stop button in the Motion Controller. The selected object
stops moving.
The Geometer’s Sketchpad Reference Manual
43
Motion Controller
Reversing the Direction of a Moving Object
You can reverse the direction of any moving object as long as it’s not
animating randomly.
Clicking Reverse with
nothing selected
reverses all moving
objects’ directions.
1. Select the moving object as described in that section.
2. Click the Reverse button in the Motion Controller. The selected
object reverses its direction.
Setting, Increasing, or Decreasing the Speed of a Moving Object
You can change the speed of any moving object.
When you have
several objects
moving at different
speeds and you want
to make all of them
go faster or slower,
use the speed arrows
rather than typing a
specific speed.
1. Select the moving object as described in that section.
2. To set the speed, click in the Motion Controller’s speed control
and type a new speed. The selected object moves at the new speed.
3. Alternatively, click on the up or down arrow in the Motion
Controller’s speed control. The selected object speeds up or slows
down.
See also: Animate (p. 153), Stop Animation (p. 154), Increase/Decrease Speed (p. 153),
The Arrow Tool (p. 70), Selection Rectangle (p. 71), Object Relationships: Parents and
Children (p. 10)
Principles of Animation
The only objects you
can’t animate are
captions, calculations,
functions, action
buttons,
measurements, and
pictures.
The default direction
for points on most
paths is bidirectional.
For circle paths,
however, the default
direction is counterclockwise—the
direction of an
opening angle.
44
Just about every object you can create in Sketchpad can be animated.
To animate an object, you can choose Animate from the Display menu,
use the Motion Controller, or create an Animation action button. Of
these three methods, action buttons give you the most control of the
details of motion.
Different objects in Sketchpad move in different ways.
•
Independent points move freely in the plane.
•
Points on paths move along their paths.
•
Parameters change their values.
•
All other objects move by moving their parent objects.
The Geometer’s Sketchpad Reference Manual
Motion Controller
When you animate a geometric object that isn’t
B
a point, Sketchpad animates that object by
animating its parents. For instance, if you
animate a triangle interior, Sketchpad animates
it by animating the vertices of the triangle. If
A
C
one vertex is a point on path, it’s moved along
its path. If another is an independent point, it
moves randomly on the plane. And if the third
point is an object such as an intersection that
isn’t free to move, it’s moved by moving its parents in turn. Thus, your
ability to animate any conceivable geometric object is ultimately based
on the animation of independent points and of points constructed on
paths.
Most objects that display text (action buttons, measurements,
calculations, functions, and captions) cannot be animated. The one
exception is parameters. A parameter is like an independent point or a
point on path in the sense that the value of a parameter, like the
position of an independent point, does not depend on other objects.
This means Sketchpad can animate the parameter by changing its value.
Because independent points, points on paths, and parameters are the
only objects that can be animated independently of their parents, these
are the only objects that Sketchpad directly animates, and they are the
only objects that appear in the Motion Controller’s Target pop-up
menu. Other objects animate indirectly—by animating their parents.
Other objects can be listed as the Motion Controller target if they are
selected. For example, if you select the interior of ∆ ABC, the triangle
will be listed as the target. Even though the triangle is listed as the
target, any motion changes you make will directly affect points A, B,
and C and will affect the triangle indirectly.
See also: Points (p. 11), Point On Object (p. 156), Parameters (p. 19), Object
Relationships: Parents and Children (p. 10), Animate (p. 153), Motion Controller
(p. 41), Animation Buttons (p. 38), Animate Properties (p. 128)
Animation of an Independent Point
An independent point is animated by moving it about in the plane. The
Merge independent
points to paths if you animation speed determines how far it’s likely to move in each random
want them to move in step. You cannot control the direction for an independent point. To
a specific direction.
specify the starting animation speed explicitly or to make the motion
The Geometer’s Sketchpad Reference Manual
45
Motion Controller
occur one time only, create an Animation action button that animates
the point.
See also: Points (p. 11), Animate (p. 153), Motion Controller (p. 41), Animation Buttons
(p. 38), Animate Properties (p. 128), Once-Only Motion (p. 48), Merging a Point to a
Path (p. 117)
To designate a
specific portion of a
line or ray as the
domain for an
animating point,
construct a segment
collinear with the line
or ray and merge the
point to the segment.
Animation of a Point on Path
A point on path is animated by moving along its path. If the path is
closed (for example, the interior of a circle, a polygon, or an arc) the
animation proceeds around and around the path. If the path is a
segment or an arc, the animation proceeds bidirectionally—back and
forth along the path. If the path is infinite, as with a line or a ray,
Sketchpad animates the point bidirectionally and tries to use the
portion of the path that’s visible in the window. To specify the
direction or speed explicitly or to make the point travel its path only
once, create an Animation action button that animates the point.
See also: Points (p. 11), Point On Object (p. 156), Animate (p. 153), Motion Controller
(p. 41), Animation Buttons (p. 38), Animate Properties (p. 128), Once-Only Motion
(p. 48), Merging a Point to a Path (p. 117)
Animation of a Parameter
A parameter is animated by changing its value within its domain. The
default domain, direction, and speed of a parameter’s variation depends
on the units of the parameter, as shown in this table.
The speed listed here
is a maximum. A
parameter may
change more slowly if
your computer is busy
with many tasks.
Units
None
Domain
Direction
–100 to 100 units Bidirectional
Speed
1 unit/sec
Degrees
0º to 360º
Increasing
45º/sec
Radians
0 to 2π
Increasing
π/4 radians/sec
Inches
0 to 100 inches
Bidirectional
1 inch/sec
Cm
0 to 100 cm
Bidirectional
1 cm/sec
To specify the direction, domain, or speed explicitly, use the Parameter
Properties panel or create an action button that animates the parameter.
See also: Parameters (p. 19), Parameter Properties (p. 126), Animate (p. 153), Motion
Controller (p. 41), Animation Buttons (p. 38), Animate Properties (p. 128), Once-Only
Motion (p. 48)
46
The Geometer’s Sketchpad Reference Manual
Motion Controller
Motion Direction
The possible directions in which you can animate an object depends on
the kind of object. When you start an animation using Animate or the
Motion Controller, Sketchpad uses the most common direction for the
objects you animate. To access more advanced direction choices, create
an Animation action button.
A point on a path can be animated forward, backward, bidirectionally,
or randomly. (If the path is a circle, the choices are counter-clockwise
and clockwise instead of forward and backward.) If you specify random
motion on a path, each time the point is moved it’s given a brand-new
random position somewhere on the path.
An independent point always moves randomly. Each time it moves, its
new position depends both on its previous position and on its location
relative to the window. A point moving slowly takes only small steps
from its previous position, whereas a point moving quickly takes larger
steps. If the point is near an edge of the window or outside the
window, it’s more likely to move toward the center of the window than
away from it. In this way, independent points usually remain visible in
the window while animating.
A parameter has a domain within which it is animated and can increase
in value, decrease in value, change bidirectionally, or change randomly
within that domain.
See also: Parameters (p. 19), Animation Buttons (p. 38), Animate Properties (p. 128)
Motion Speed
You can use the Motion Controller, the Increase/Decrease Speed
commands, or an Animation button to set the speed of an animating
object relative to the normal speed for that object.
You can set the ideal
speed for points using
Advanced Preferences.
The Motion Controller displays normal speed as speed 1.0. The actual
velocity of normal speed depends both on your System Preferences and
on your computer. Sketchpad attempts to keep this speed constant, but
if your sketch is complex or your computer is busy with other tasks,
normal speed may be slightly slower than requested in System
Preferences.
See also: Motion Controller (p. 41), Increase/Decrease Speed (p. 153), Animation
Buttons (p. 38), Animate Properties (p. 128), Parameters (p. 19), System Preferences
(p. 141)
The Geometer’s Sketchpad Reference Manual
47
Motion Controller
Once-Only Motion
When you create an Animation action button, you can specify that an
object moves one time only. If the object is moving randomly, this
means that one press of the button causes the object to move one time
to a new random position. If the object is not moving randomly, onceonly motion means that the object will stop once it returns to its
starting position.
See also: Animation Button (p. 112), Animate Properties (p. 128)
48
The Geometer’s Sketchpad Reference Manual
Calculator
Use the Calculator to create or
edit two kinds of Sketchpad
objects: calculations and
functions.
Calculations are values calculated
using numbers, mathematical
operations, functions, and
measurements or other values
from your sketch. These
calculations can be used for a
wide variety of purposes: to
define distances, angles, and scale
factors by which objects are
transformed; to plot points; and
to determine the results of other
calculations and functions.
When you define a function in Sketchpad, you also use numbers,
mathematical operations, functions, and measurements or other values
from your sketch. Once you’ve defined a function, you can use it for
various purposes: You can plot it, evaluate it, differentiate it, or use it to
define other calculations or functions.
There are several different purposes for which you’ll use the Calculator:
•
To create a new calculation, choose Calculate from the Measure
menu.
You can use the Plot
•
command to create a
new function and plot
it immediately.
To create a new function, choose New Function from the Graph
menu.
•
To edit a calculation, double-click the calculation with the Selection
Arrow tool; or select the calculation and choose Edit Calculation from
the Edit menu.
•
To edit a function definition, double-click the function with the
Selection Arrow tool; or select the function and choose Edit Function
from the Edit menu.
•
To change a parameter into a calculation, select the parameter,
choose Edit Parameter from the Edit menu, and change the
expression to something more complicated than just a number.
New Function
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49
Calculator
•
To change a calculation into a parameter, edit it so that it consists
of one number only, with or without units.
See also: Calculate (p. 198), New Function (p. 207), Plot New Function (p. 207), Edit
Definition (p. 120), Measurements, Calculations, and Parameters (p. 18), Functions
(p. 27), Selection Arrow Tools (p. 70)
Parts of the Calculator
The Calculator has several parts: a keypad, an input line, a preview area,
and a group of buttons for inserting special mathematical elements.
Keypad
Click the buttons in the Calculator’s
keypad to insert numbers, decimal points,
operators, and parentheses into your
calculation or function. Instead of clicking
one of these buttons, you can also type
the corresponding key on your computer’s
keyboard. (Use the / key—the forward
slash—on the keyboard for division.)
The x key is present
only when you’re
defining a function; it
changes to y, θ, or r
depending on the
form of the equation.
Click the left-arrow key on the keypad, or
the Backspace key (Microsoft Windows) or Delete key (Macintosh) on
the keyboard, to delete the last item you typed. If you’re using the
Calculator to define a function, you can click the key labeled x to enter
the value of the independent variable.
Input Line
The input line displays each element you insert
into the calculation or function as you enter it.
Refer to the input line to see exactly what
you’ve entered. You can also click in the input
line or press the right- or left-arrow key on the keyboard to change the
insertion point and add new numbers, operators, and so forth in the
middle of an existing expression.
If the input line does not form a valid mathematical expression, the
portion of the input line up to the first error is shown in black. The
portion following the first error appears in red.
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The Geometer’s Sketchpad Reference Manual
Calculator
Preview Area
The preview area shows a mathematically
formatted preview of the input line when it
contains a valid expression. Use the preview to
be sure that you’ve entered the desired
calculation or function correctly—that you have
the parentheses in the right places and that you have the correct order
of operations.
Pop-up Menus
Depending on whether you’re defining a calculation or a function,
either three or four pop-up menus appear. The last of these, the
Equation pop-up menu, appears only when you’re defining or editing a
function rather than a calculation.
Values: This pop-up menu allows you to
enter the value of any selected measurement in
the sketch, to insert a new parameter, or to
insert the value of the constant π or e. The
values that appear in this menu include
You can click on an
existing measurement measurements or calculations that were
selected in the sketch when the Calculator was
or calculation in the
sketch to insert it into opened. If you want to use a value from the
the Calculator.
sketch that doesn’t appear in this menu, click
on the value in the sketch to insert it into your
expression. (If the value is hidden behind the Calculator, drag the
Calculator by its title bar to move it out of the way and then click on
the value.)
When you edit a calculation or function, you can insert only values that
don’t depend on the object you’re editing. In other words, if you’re
editing a calculated value 2 · AB, and your sketch has another
calculation which uses this result to calculate 2 · AB + 2 · CD, you
cannot insert the value of the second calculation into the first.
If you’re defining a function, you can also use the Values pop-up menu
to insert the value of the independent variable x, y, r, or θ.
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51
Calculator
You can also click on
an existing function
that you’ve defined
previously in the
sketch to insert it into
the Calculator.
The signum function
is especially useful in
creating a calculation
that makes a decision
based on the value of
a variable,
measurement, or
parameter.
Functions: This pop-up menu allows you
to use in your expression any selected
function you’ve already defined in the
sketch, or to use any of Sketchpad’s
standard functions. Sketchpad’s standard
functions include the trigonometric
functions, the inverse trig functions, and
these additional functions:
abs
Absolute value
sqrt
Square root
ln
Natural logarithm (base e)
log
Common logarithm (base 10)
sgn
Signum (Returns +1, 0, or –1,
depending on whether its
argument is positive, zero, or
negative.)
round
Round (Rounds its argument to
the nearest whole number.)
trunc
Truncate (Truncates its argument by removing the fractional
part. For example, trunc (2.6) = 2, and trunc (–7.8) = –7.)
Units: This pop-up menu allows you to insert any
desired angle or distance unit (degrees, radians, inches,
cm, or pixels). A unit must be attached to a number,
so the Units menu is enabled whenever you’ve just
inserted a numeric constant—either an ordinary
number or one of the constants π and e.
See also: Plot Function (p. 207), New Parameter (p. 206)
Equation: This pop-up menu allows you to set the
form in which the function appears. By setting the
form of the function, you determine whether the
independent variable is x, y, θ, or r. If you plot the
function, the form you set here determines how the
function plot appears. Choose y = f (x ) or x = f (y ) for
a rectangular plot; or choose r = f (θ ) or θ = f (r) for a
polar plot.
See also: Plot Function (p. 207), Grid Form (p. 203)
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The Geometer’s Sketchpad Reference Manual
Calculator
How To . . . Use Signum to Construct a Piecewise Function
The signum function
is useful any time you
want a calculation
which makes some
sort of decision—
which performs a
different calculation
when some value
changes.
Sometimes you may want to construct a function that behaves one way
in part of its domain and differently in another part of its domain. Such
functions, called piecewise functions, are very important in interpolation,
in fitting curves to data, and in designing and producing shapes and
surfaces in applications such as automobile design. The signum
function makes it possible to do this in Sketchpad.
For instance, you could construct a function whose plot is in the shape
of a cosine wave when x > 0, but is parabolic when x < 0. Here’s how
to do this with the signum function:
1. Choose Plot New Function from the Graph menu.
2. Enter the cosine-wave part of the
(
)
(
)
function, as shown at right. When x > 0,
1+sgn(x)
cos(x)⋅
the signum function returns +1, and the
2
value of the multiplier is 1. But when
x < 0, the signum function returns –1,
and the value of the multiplier is 0. This first part of the function
will be a cosine wave to the right of the origin, but will always be
zero to the left of the origin.
3. Enter the parabola part of the function
as shown at right. When x > 0, the
1-sgn(x)
(-x2+1)⋅
multiplier will be 0, and when x < 0,
2
the multiplier will be 1.
Here’s the result when you click OK:
1
2
4
6
-1
-2
( 1+sgn2 x )+ -x +1 ( 1-sgn2 x )
f(x) = cos(x)⋅
( )
(
2
)⋅
( )
The functions in this example were chosen in such a way that the two
functions join continuously and smoothly. Can you construct a
different piecewise function in which the join is not continuous, or in
which it’s continuous but not smooth?
The Geometer’s Sketchpad Reference Manual
53
Calculator
Here’s how to use the signum function to solve the general problem of
defining a function h(x) whose value is f(x) for all x < k and is g(x) for
all x > k:
h(x) = f(x)
If a value or function
you want to insert is
hidden behind the
Calculator itself, drag
the Calculator to the
side so you can click
on the object you
want to insert.
(
1-sgn(x-k)
2
)
+ g(x)
(
1+sgn(x-k)
2
)
Inserting Values and Functions from the Sketch
While you’re using the Calculator to define a calculation or a function,
you can click existing values or functions in the sketch to insert these
objects into your new expression. For instance, if you have a sketch in
which you’ve already measured the length of segment AB, you can
insert this length into the Calculator’s expression by clicking on the
existing measurement in the sketch. Similarly, if you have a sketch in
which you’ve defined a function f (x ) = ax2 + bx + c, you can use this
function in other calculations and functions by clicking on it in the
sketch.
Inserting a New Parameter
While you’re using the Calculator to define a calculation or a function,
you can create a new parameter and insert it into your expression.
1. Choose New Parameter from the Values pop-up menu. A New
Parameter dialog box appears.
2. Type the name you want to use for the new parameter. You can
also set the initial value of the parameter.
3. Click OK. The parameter is inserted into your expression and
added to your sketch.
See also: Parameters (p. 19), New Parameter (p. 206)
54
The Geometer’s Sketchpad Reference Manual
Text Palette
Use the Text Palette to format the font, size, style, and color of labels,
captions, measurements, and other text. You can also use the Text
Palette to insert mathematical symbols and formatting into captions.
The Text Palette can be used:
•
when you have selected objects in your sketch and one or more of
those objects either shows text or has a label. Use the Text Palette
to change the appearance of the text or label of each selected
object.
•
when you’re editing a caption and you have selected text in that
caption. Use the Text Palette to change the appearance of the
selected text.
Showing, Hiding, and Moving the Text Palette
Normally the Text Palette appears automatically when you edit a
caption. You can turn this behavior on or off using the Text panel of
Preferences. To show or hide the Text Palette manually, choose Show Text
Palette or Hide Text Palette from the Display menu.
If you’re using a Microsoft Windows computer, the Text Palette
normally appears attached (or docked ) to the bottom of the Sketchpad
application window. You can drag it to a different position and leave it
either floating or docked to the top or bottom of the window. If you’re
using a Macintosh computer, the Text Palette always floats above or
beside your document and can be repositioned by dragging its title bar.
On either kind of computer, when the Text Palette is floating, you can
hide it by clicking its Close box, and you can move it to a different
position on the screen by dragging its title bar.
See also: Show Text Palette (p. 154), Text Preferences (p. 139)
Using the Text Palette
The various parts of the Text Palette allow you to change the font, size,
style, and color of labels and text. You can also use the Text Palette to
add mathematical symbols and symbolic notation to your captions.
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55
Text Palette
Pressing the down
arrow triangles
displays Font, Size,
and Color pop-up
menus.
Text Palette (Windows)
Font
Size
Style
Color Menu
Color Picker
Symbolic Notation
Text Palette (Macintosh)
Font
Size
Color Picker
Color Menu
Style
Symbolic Notation
Font: Change text font by choosing a font from the pop-up menu.
Size: Change text size by typing a size or by choosing a size from the
pop-up menu.
Style: Change bold, italic, or underline text style by clicking on a button.
One click sets the style and depresses the button; a second click removes
the style and releases the button.
When a Text Palette
color is applied to
selected geometric
objects, it affects the
color of those objects’
labels. To change the
color of the objects
themselves, use the
Color command from
the Display menu.
(You can change the
available colors by
clicking Edit Color
Menu in System
Preferences.)
56
Color Menu: Change text color (to one of Sketchpad’s default colors) by
choosing a color from the pop-up menu.
Color Picker: Change text color (to any color your computer can display)
by clicking this button to bring up your system’s Color Picker dialog box.
Symbolic Notation: Display additional tools for mathematical notation
by clicking here when you’re editing a caption. Click a second time to hide
the math formatting buttons.
See also: Captions (p. 36), Show Text Palette (p. 154), Advanced Preferences (p. 139),
Color Picker (p. 64)
The Geometer’s Sketchpad Reference Manual
Text Palette
Using the Text Palette with Selections
To change the font, size, style, or text color of one or more labeled
objects or objects displaying text, select those objects and use the Text
Palette to make your desired modifications.
Using the Text Palette While Editing a Caption
While you’re editing a caption, select any portion of the text and set the
font, size, style, and color using the Text Palette.
You can also enter mathematical symbols and mathematical formatting
in a caption.
Entering Mathematical Symbols and Formatting
Press the Symbolic Notation button in the Text Palette to display
additional notation tools you can use to enter mathematical symbols
and other formatting, like overbars, fractions, exponents, and grouping
symbols.
If you’re using a Microsoft Windows computer, you can drag these
symbolic notation tools so that they’re floating or so that they’re
docked to any side of the Sketchpad window. If you’re using a
Macintosh computer, these symbolic notation tools appear as a second
row of buttons in the Text Palette.
The symbolic notation tools are divided into four groups.
Overbar Buttons: Click one of the overbar buttons to
add a segment, a ray, a line, or an arc overbar to the
selected text. These buttons are available only when
there’s text selected.
AB ⊥ CD
Operator Buttons: Click one of the operator buttons to add
a fraction, square root, superscript (exponent), or subscript to
the selected text. If there is no selected text, the operator is
inserted with ? symbols in the places which you need to fill
in.
Grouping Buttons: Click one of the grouping buttons
to enclose the selected text in parentheses, square
brackets, curly braces, or absolute value symbols.
Unlike parentheses and brackets you type from the
The Geometer’s Sketchpad Reference Manual
3
x
5
( x 2+ y )
a
57
Text Palette
keyboard, mathematical grouping symbols always appear in pairs and
resize automatically to fit whatever expression they enclose.
Symbol Buttons: Click one of the symbol buttons to
α+β ≥ θ-π
insert an angle symbol, pi, the degree symbol, or any of
the additional common symbols that appear in a pop-up menu when
you press and hold on the last of these buttons.
You can combine the symbolic notation tools to add traditional
mathematical notation to your captions or to invent new notations of
your own.
To display symbolic notation, Sketchpad uses a special font containing
mathematical symbols. By default, it uses the Symbol font that comes
installed on every computer. If you wish to use some other font, you
can change the Math Symbol font in System Preferences.
See also: Captions (p. 36), Text Palette (p. 55), Show Text Palette (p. 154)
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The Geometer’s Sketchpad Reference Manual
Script View
The Script View
offers an “inside
look” at the workings
of any custom tool.
You should be
familiar with using
custom tools before
working with the
Script View.
The Script View window displays a
readable description of the
mathematical construction
performed by a custom tool. Use
the Script View to review the given
object requirements of a particular
custom tool, or to investigate how a
tool was originally defined. In the
Script View, you can add or read
comments about the tool written
by the tool’s author; change
properties—such as color or line
weight—of the objects that the tool
creates; and even apply the tool’s
construction in a step-by-step fashion to objects in your sketch.
To display the Script View window, choose Show Script View from the
Custom Tools menu or click the Show Script View checkbox in the
Document Options dialog box’s view of custom tools. The Script View
window sits on top of open document windows and displays the Script
View of the most-recently chosen custom tool. To change the tool
described by the Script View, choose a different custom tool from the
available tools in the Custom Tools menu or in the Document Options
dialog box.
Show Script View is
only available when
one or more custom
tools have been added
to your Custom Tools
menu.
You can reposition the Script View by dragging its title bar and resize it
by dragging its resize area. The Script View remains visible until you
close it by choosing Hide Script View from the Custom Tools menu, or by
clicking the window’s close box.
See also: Custom Tools (p. 90), Document Options (p. 104)
Tool Comment
The Tool Comment appears
at the top of the Script View
window, and contains any
comments about the custom
tool’s purpose or behavior
that its author chose to add.
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59
Script View
Adjust the size of the Tool Comment by dragging the divider bar
beneath it, and add to or change the Tool Comment by typing. If you
create new custom tools you plan to share with other Sketchpad users,
use the Tool Comment to describe the tool’s given objects, to identify
yourself as the tool’s author, and to explain how the tool should be
used.
Any comments you add are saved with the tool, so you—or another
Sketchpad user—can always refer to them when using the tool. When a
Script View construction is printed, the comment is printed with the
rest of the construction.
See also: Printing a Script View Construction (p. 63)
Object List
The Script View’s Object List describes, in a step-by-step fashion, all of
the objects and constructed relationships that make up the tool.
The object list is divided into two major sections: Given and Steps.
Given Objects
The Given section shows all the given objects for the tool—all the
You can change the
order of given objects objects that don’t depend on any other objects in the tool. When using
by dragging individual the tool in a sketch, these are the objects which you “match” in a
given objects up or
sketch by clicking the tool. Given objects are listed in the order in
down in Script View.
which you must match them, which by default is the order chosen by
the tool’s author when the tool was created.
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The Geometer’s Sketchpad Reference Manual
Script View
In some tools, the Given section may be divided into two parts, labeled
Assuming and Given. The Assuming part contains assumed objects—
given objects that are automatically matched to sketch objects having
the same label when the tool is used in a sketch. The Given part
contains the remaining given objects—the given objects that must be
matched explicitly.
Assumed objects don’t need to be matched explicitly when you use the
tool, unless the sketch does not contain objects to match with the same
label. (By default, a tool’s given objects are explicit, as opposed to
assumed. But when a tool is designed to be used repeatedly in the same
sketch, in some situations the tool author finds it convenient to change
some explicitly given objects into assumed objects to make the tool
more convenient to use.) For more information about assumed objects
and how to indicate them, see Advanced Tool Topics (p. 230).
The script step for an
intermediate object
describes it as
“hidden.”
Steps
The Steps section shows all of the objects, and the mathematical
construction, defined by the tool’s given objects. These include the
tool’s intermediate objects (which are not displayed when the tool is
used in a sketch), and the tool’s final results (which are displayed).
Unlike given objects, a tool’s steps cannot be reordered: they are
determined by the tool’s construction.
Working with the Object List
You can double-click any object in the Object List to change its
Use the Hidden
checkbox in Object
properties. When you double-click, the Object Properties dialog box
Properties to
appears and you can use the various panels to make any desired
determine whether an
changes to the properties. You can also use the Parents and Children
object is an
intermediate object or pop-up menus on the Object panel to view and change the properties
a result.
of the parents or children of the object.
You can right-click (Windows) or Ctrl-click (Macintosh) on any object
in the list to display a Context menu for the clicked object. Use this
Context menu to change the color, line width, or tracing for the object
on which you click. The Context menu also allows you to print the
script.
See also: Object Properties (p. 121), Object Relationships: Parents and Children (p. 10),
Automatically Matching a Given Object (p. 230), Printing a Script View Construction
(p. 63)
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61
Script View
Using a Custom Tool with the Script View
Use the Script View
feedback to walk
through a custom tool
the first time you’re
trying a tool created
by someone else.
If the Script View window is showing when you use a custom tool in a
sketch, it provides visual feedback about the tool’s use. Given objects
that you’ve already matched with the tool appear highlighted. Given
objects that you haven’t matched yet appear after those you’ve
matched, and are not highlighted. The object that the tool is currently
matching appears between the two outlined.
Applying the Script View Step-By-Step
Sometimes you may wish to apply a custom tool’s construction to a
sketch in a step-by-step fashion, rather than use it as a tool. If you are
reasoning through the construction described in the Script View, for
example, it’s convenient to watch the construction occur in your sketch
one object at a time. Likewise, if you’re demonstrating a particular
construction to your teacher or students, it may be useful to advance
object by object. When you use a custom tool directly in the sketch, its
constructed steps happen all at once, as soon as you’ve matched given
objects, and only the tool’s final results are displayed. But when you
apply a Script View construction step-by-step, objects appear one at a
time, and the construction’s intermediate objects are temporarily
displayed (so you can see them) until all steps are complete (when
Sketchpad hides them for you).
To apply Script View’s construction step-by-step to a particular sketch:
1. Use the Selection Arrow tool to select sketch objects that match the
Assumed objects
match automatically
only when using
custom tools directly,
not when applying the
Script View step-bystep.
62
given objects displayed in Script View, in the order they appear in
the Script View object list. For example, if the Script View lists
three given points, select three matching points in your sketch.
If the Script View displays both Assumed and Given objects, select
matching objects first for the Assumed objects and then for the
Givens.
As you select matching sketch objects, the Script View displays
feedback about your selections. Given objects that match your
selections appear highlighted; given objects for which you must still
select matching sketch objects appear on a normal background.
The Geometer’s Sketchpad Reference Manual
Script View
2. Once you have selected objects to match all of the Script View
given objects, two buttons appear at the bottom of the Script View
window.
If the tool’s results
match the givens, you
can apply the tool
repeatedly to its own
results simply by
clicking the All Steps
button.
Click Next Step to apply the first step of the construction. Click
this button repeatedly to walk through the entire construction,
step-by-step. As each step is applied, the corresponding object
appears in your sketch. While stepping through the construction,
you can drag objects in your sketch to investigate their
relationships to other objects.
Click All Steps to finish the construction by immediately
constructing all remaining steps.
3. When the last step of the construction has been completed, the
construction’s intermediate objects—which the Script View
temporarily displays in your sketch during stepping—are hidden,
the results are selected, and the Next Step and All Steps buttons
disappear. The Script View returns to its normal appearance.
To apply the Script View construction a second time, select new
matching objects as described in step 1.
During stepping, you can press the mouse on any already-matched
given object or already-constructed step in the Script View to highlight
the corresponding sketch object to which it matches. If you want to
stop step-by-step application of the construction without completing
All Steps, press the Esc key or close the Script View window.
See also: Object List (p. 60)
Printing a Script View Construction
When the Script View is showing, you can print its Tool Comment and
Object List. Right-click (Windows) or Ctrl-click (Macintosh) on any
object in the script, and choose Print... from the Context menu that
appears.
The Geometer’s Sketchpad Reference Manual
63
Color Picker
The Color Picker dialog box allows you to choose a specific color to be
used for displaying a Sketchpad object, for displaying text in Sketchpad,
or for changing any of the colors available on Sketchpad’s Color menu.
The Color Picker appears when you choose the Display | Color | Other
command or when you click the Color Picker swatch on the Text
Palette. It also appears when you click the Edit Color Menu button on
the System panel of the Advanced Preferences dialog box.
The actual Color
Picker dialog box is
provided by the
particular Macintosh
or Windows operating
system installed on
your computer, and
may differ from the
sample dialog boxes
shown below.
The Color Picker dialog box allows you to specify a color in several
different ways. The most common methods involve RGB (Red-GreenBlue) values and HLS (Hue-Luminance-Saturation or Hue-LightnessSaturation) values.
To specify a color using RGB, you set numeric values for the red,
green, and blue components of the color. These numeric values are
most commonly expressed either in percentages or in a range from 0 to
255, with higher numbers for a color corresponding to the presence of
more of that color in the mix. Thus 255 for red and 0 each for green
and blue specifies the purest possible red. Setting all three values to 0
specifies black (no color at all), and setting all three values to 255
specifies white (the brightest color, with the maximum amount possible
of all three components).
The three values of HLS (Hue, Luminance, and Saturation) also allow
you to specify a color numerically. Hue determines the color, such as
red, blue, or green. Luminance determines how light or dark the colors
are, with the maximum value corresponding to white and the minimum
value (zero) corresponding to black. Saturation determines how much
of the color is present; high saturation corresponds to vivid colors, and
low saturation to pale colors. A saturation value of 0 specifies gray, with
the shade of gray determined by the luminance. The most vivid pure
colors correspond to the midpoint of the luminance scale and
maximum saturation.
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The Geometer’s Sketchpad Reference Manual
Color Picker
An alternative system called HSV (Hue-Saturation-Value) is similar to
HLS, with the difference that the most vivid pure colors correspond to
the maximum of the V scale (rather than the midpoint of the L scale) at
maximum saturation. In this system white corresponds to zero
saturation and maximum value.
See also: Other Color (p. 148), Text Palette (p. 55), Color Preferences (p. 138), System
Preferences (p. 141)
Macintosh Color Picker
The Macintosh Color Picker lists on the left several different ways in
which you can pick a color, including crayon colors, RGB, HLS and
HSV.
If you use the HLS method, you can click anywhere in the color wheel
to set the hue and saturation, with minimum saturation corresponding
to the center of the wheel and maximum saturation to the edge of the
wheel. Move the slider on the bottom to set the lightness.
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65
Color Picker
Windows Color Picker
The Windows Color Picker shows at
the top the 16 colors available from
the Display menu’s Color submenu.
The New Color swatch shows the
color that will be chosen if you click
the OK button.
Hue, Saturation, and Luminance allow
you to specify a color numerically.
The Hue value determines the color,
such as red, blue, or green. The
Saturation determines how much of
the color is present; high saturation
corresponds to vivid colors, and low
saturation to pale colors. Luminance
determines how light or dark the
colors are.
Alternatively, you can use the Red,
Green, and Blue values to specify a
color based on how much of each of
these primary colors is used.
Finally, you can use the rectangle and the vertical strip in the lower
portion of the dialog box to specify the hue, saturation, and luminance.
First click in the rectangle to choose the hue and saturation. Change the
hue by moving horizontally in the rectangle, and change the saturation
by moving vertically, with the minimum values at the bottom of the
rectangle and the maximum values at the top. Once you’ve chosen the
hue and saturation, click in the strip on the right to set the luminance.
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The Geometer’s Sketchpad Reference Manual
Toolbox Reference
For thousands of years, since the time of Euclid, the fundamental tools
of geometry have been the compass and the straightedge. Sketchpad’s
Toolbox includes these two tools—to construct circles and straight
objects—and several other tools that allow you to select and drag
objects, to construct points, to create and manipulate text and labels,
and to define and manage custom tools. This section describes how to
use each of the tools in Sketchpad’s Toolbox.
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67
Overview of the Toolbox
The Toolbox appears on the left of the screen when you start
Sketchpad, and includes six tools.
Selection Arrow tools (Translate, Rotate, Dilate)
Point tool
Compass tool
Straightedge tools (Segment, Ray, Line)
Text tool
Custom tools
•
Selection Arrow tools:
•
Point
•
Compass
•
Straightedge
•
Text
•
Use this tool to select and drag objects in your
sketch. The three variations of the tool allow you to drag-translate
(move), drag-rotate (turn), and drag-dilate (shrink or grow) objects.
tool: Use this tool to construct points.
tool: Use this tool to construct circles.
tools: Use this tool to construct straight objects. The
three variations of the tool allow you to construct segments, rays,
and lines.
tool: Use this tool to create and edit text and labels.
Custom
tools icon: Use this icon to define, use, and manage custom
tools.
See also: Selection Arrow Tools (p. 70), Point Tool (p. 80), Compass Tool (p. 81),
Straightedge Tools (p. 83), Text Tool (p. 86), Custom Tools (p. 90)
Choosing and Using Tools
To use any of Sketchpad’s tools, click the desired tool in the Toolbox.
Then move the mouse over the sketch and click or press and drag to
use the tool.
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The Geometer’s Sketchpad Reference Manual
Overview of the Toolbox
The Selection Arrow and Straightedge tools each come in three variations.
To change from one variation to another, press and hold the Selection
Arrow tool or Straightedge tool in the Toolbox until a menu pops out.
Choose a different variation of the tool from this menu to make it
active.
Windows users who
have a mouse wheel
can change the active
tool by turning the
wheel, and can switch
between varieties of
the Selection Arrow or
Straightedge tool by
pressing the wheel.
You can change the active tool by using the keyboard instead of the
mouse. Hold down the Shift key and press the up or down arrow key
to change the active tool. You can also hold down the Shift key and
press the right or left arrow key to switch the Selection Arrow or
Straightedge tool from one variation to another.
Once you’ve chosen a tool, that tool remains active until you choose a
different one, so there’s no need to click the same tool repeatedly to
use it multiple times.
See also: Selection Arrow Tools (p. 70), Point Tool (p. 80), Compass Tool (p. 81),
Straightedge Tools (p. 83), Text Tool (p. 86), Custom Tools (p. 90)
Using Tools to Scroll the Sketch Window
You can use any of Sketchpad’s tools to scroll the sketch window and
see different parts of the sketch. Hold down the Alt key (Windows) or
the Option key (Macintosh) and press and drag in the sketch to scroll
the entire sketch in the direction in which you drag. This feature works
no matter which tool is active in the toolbox.
Hiding and Showing the Toolbox
You may want to hide Hide or show the Toolbox by using the Hide Toolbox and Show Toolbox
the Toolbox when
commands at the bottom of the Display menu.
making presentations.
See also: Hide/Show Toolbox (p. 155)
Moving, Resizing, and Docking the Toolbox
To move the Toolbox, (Windows or Macintosh) grab it by the title bar
or (Windows only) by the gray area surrounding the buttons and drag it
to a different location on the screen.
Microsoft Windows users can dock the Toolbox to
the left, top, right, or bottom edge of the
application window. When the Toolbox is not
docked, Windows users can also resize the Toolbox
by dragging a border of the Toolbox window.
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69
Selection Arrow Tools
The Selection Arrow
tool is sometimes just
called the Arrow tool.
The Selection Arrow tool is at the heart of Sketchpad’s
Dynamic Geometry capabilities—it’s the tool you use to
move (or drag) objects in your sketch. You’ll drag objects to
investigate mathematical relationships, to explore variations,
to test conjectures, and to discover new properties.
You also use this tool to select objects in your sketch. Many of
Sketchpad’s menu commands act on selected objects. By selecting
objects, you focus the software’s attention on one or a handful of the
many objects that make up a sketch. For example, you’ll use selection
to identify the objects to be transformed by the Transform menu or the
objects to be measured by the Measure menu.
In addition to dragging and selecting, you can use the Arrow tool for
several other purposes: constructing points of intersection, pressing
action buttons, and resizing coordinate systems, function plots,
pictures, and loci.
See also: Selecting and Deselecting Objects (p. 70), Dragging Objects (p. 73), Other
Arrow Actions (p. 76)
Selecting and Deselecting Objects
You select objects in your sketch to drag them, to apply menu
commands to them, or to manipulate or modify them with the Motion
Controller or the Text Palette.
When a command is
not available, it is
grayed out in its
menu. Often this
means your current
selection is not
appropriate for that
command.
At any point in time, your selections determine which menu commands
are available at that point. For example, you can use the Midpoint
command in the Construct menu when you have a segment selected,
but not when you have a circle, a point, or a ray selected.
Your selections also determine how dragging works. If you select and
drag both endpoints of a segment, the entire segment moves as a unit,
with no change in its length or direction. But if you select and drag only
one endpoint, the other endpoint remains fixed and the segment’s
length and slope change as you drag.
Selected objects appear outlined, or with special marks, as in the
following illustrations.
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The Geometer’s Sketchpad Reference Manual
Selection Arrow Tools
m AB = 71.00 pixels
A
B
m AB = 71.00 pixels
A
Unselected Objects
B
Selected Objects
To select or deselect objects using a Selection Arrow tool:
•
Select an unselected object by positioning the tip of the Selection
Arrow over the object and clicking it.
•
Select additional objects by clicking each object in turn.
•
Deselect a single selected object the same way, by positioning the
tip of the Selection Arrow over the object and clicking on it.
You can also deselect
all objects by pressing
the Esc key one or
more times.
•
Deselect all objects by clicking in empty space in the sketch.
•
Select one of several overlapping or coincident objects by clicking
repeatedly until the desired object is selected.
Clicking the body of
an action button
doesn’t select it, but
instead performs the
action associated with
the button.
•
Select or deselect an action button by
clicking its handle, not its body.
•
Select multiple objects simultaneously by
enclosing them in a selection rectangle.
The Arrow flips
horizontal when it’s
pointing at an object.
In addition, you can:
Hide Point
Handle
Body
You can also select objects by using Select All, Select Parents, or Select
from the Edit menu or by navigating to them using the Object
Properties dialog box. You can select moving points by choosing them
from the Motion Controller’s Target menu.
Children
See also: Midpoint (p. 157), Selection Rectangle (p. 71), Selecting Overlapping Objects
(p. 72), Action Buttons (p. 37), Select All (p. 115), Select Parents (p. 115), Select
Children (p. 115), Object Properties (p. 121), Motion Controller (p. 41), Esc Key
(p. 215), Object Relationships: Parents and Children (p. 10)
Selecting Objects Using a Selection Rectangle
Use a selection rectangle to select multiple objects located near each
other in a sketch.
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71
Selection Arrow Tools
1. Imagine a rectangle surrounding the objects you wish to select.
2. Position the tip of the Selection Arrow in empty space at one corner
If you want
previously selected
objects to remain
selected, hold down
the Shift key while
starting to drag the
rectangle.
of this rectangle.
3. Press and hold the mouse button and drag diagonally toward the
opposite corner.
A dashed rectangle appears, and every object that the rectangle
touches or encloses is selected.
4. When all desired objects are selected, release the mouse button. If
you started your rectangle in the wrong place to select the desired
objects, just begin again from step 2.
A properly positioned selection rectangle can be very useful for
selecting multiple objects at once, for commands that apply to multiple
objects, such as constructing the three midpoints of a triangle’s sides, or
for constructing a perpendicular.
Select all three sides to construct
midpoints.
Select a point and segment to
construct a perpendicular.
See also: Selecting and Deselecting Objects (p. 70)
Selecting Overlapping Objects
Sometimes objects overlap or may even be geometrically coincident on
the screen. If you want to select one of a set of coincident objects, here
are some useful tips.
72
•
Keep your eye on the status line at the bottom of the Sketchpad
window. It will tell you which object you are about to select. If the
object described isn’t the one you want to select, try moving the
Arrow around until the status line describes the desired object.
•
If possible, point at a portion of your object that doesn’t overlap
other objects. In the following figure, for example, segment CD has
been constructed collinear to line AB. You can select segment CD
by clicking on the segment itself between points C and D; but to
The Geometer’s Sketchpad Reference Manual
Selection Arrow Tools
select line AB, it’s easiest to click on the portion of the line outside C
and D.
A
When you click
overlapping objects,
Sketchpad always
selects points in
preference to any
other objects and
always selects path
objects in preference
to interiors.
C
D
B
•
If clicking selects the wrong object, click again in the same spot. A
second click on similar overlapping objects will deselect the first
object and select the next one. Keep clicking until you get the
object you want.
•
If clicking selects an object which isn’t of interest to you and isn’t
important to the appearance of the sketch, choose Hide from the
Display menu, then click again in the same spot.
•
If all else fails, select a related object, use the Properties command to
view Object Properties, and select the desired object using the
Parents or Children pop-up menu.
To select more than one object of several located near each other on
the screen:
•
Use a selection rectangle to select all the overlapping objects.
•
Select the first object, then hold down the Shift key while clicking
additional objects.
See also: Selecting and Deselecting Objects (p. 70), Hide Objects (p. 148), Properties
(p. 120), Object Properties (p. 121), Object Relationships: Parents and Children (p. 10)
Dragging Objects
Drag objects in your sketch for several purposes: to reposition the
objects, to resize them, to change the shape of a construction, and to
investigate the geometry embedded in the sketch, thereby discovering
and revealing mathematical relationships between them. To drag an
object, position the tip of the Arrow tool over the object, then press and
drag.
The Arrow flips
horizontal when it’s
pointing at an object.
•
If the object was not selected, only that object drags. (Any other
selected objects deselect.)
•
If the object was already selected, it and all other selected objects
drag to follow your mouse.
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73
Selection Arrow Tools
Sketchpad uses the
terms parent and child
to describe geometric
relationships. A
segment is the child
of its endpoints; the
endpoints are the
parents of the
segment.
When you drag an object, other related objects stretch and shrink to
maintain their relationship to the dragged object. For instance, if you
drag one endpoint of a segment, the segment stretches because the
segment depends upon the location of both of its endpoints. Similarly,
if you drag the segment itself, both endpoints move with it because the
segment depends upon these endpoints and cannot move separately
from them.
See also: Selecting and Deselecting Objects (p. 70), Object Relationships: Parents and
Children (p. 10)
Transformations and Dragging
Mathematically, moving an object in your sketch transforms that object,
and Sketchpad’s dragging behavior is based on three geometric
transformations: translation, rotation, and dilation. To allow you to use
each of these transformations, Sketchpad actually has three Arrow tools:
the Translate Arrow tool, the Rotate Arrow tool, and the Dilate Arrow tool.
These three tools behave identically when used to select objects; it’s
only their dragging behavior that differs.
With the Arrow tool
active, you can also
switch transformations by holding
down the Shift key
and pressing the left
or right arrow key on
your keyboard.
When Sketchpad starts, the active Selection Arrow tool is
the Translate Arrow tool. Choose a different Arrow tool,
and a different transformation, by pressing and holding
on the Arrow tool icon in the Toolbox. When you press
and hold, a menu pops out and you can choose one of
the three arrow tools.
The Arrow Tools
Translate Rotate Dilate
•
Use the Translate Arrow tool to translate objects by
any distance in any direction while maintaining their size, angle, and
shape. (This is the default tool.)
•
Use the Rotate Arrow tool to rotate objects about a center point,
changing their angle while maintaining their size and distance from
the center.
•
Use the Dilate Arrow tool to dilate objects about a center point, only
moving them closer to or farther from the center and making them
correspondingly smaller or larger while maintaining their angle and
shape.
See also: Dragging Objects (p. 73), The Translate Arrow Tool (p. 75), The Rotate Arrow
Tool (p. 75), The Dilate Arrow Tool (p. 75)
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The Geometer’s Sketchpad Reference Manual
Selection Arrow Tools
The Translate Arrow Tool
Drag selected objects with this tool to translate them—that is, to slide
them by any distance in any direction without turning or changing size
or shape. This is the default Arrow tool.
You can use the Translate Arrow tool to select and deselect objects, press
action buttons, and construct points of intersection just like the other
Selection Arrow tools.
See also: Transformations and Dragging (p. 74), The Rotate Arrow Tool (p. 75), The
Dilate Arrow Tool (p. 75), Translate (p. 173)
The Rotate Arrow Tool
Drag selected objects with this tool to rotate them—that is, to turn
them around a center point by any desired angle without changing their
distance from the center, their size or their shape.
You can also mark a
point as the center of
rotation by doubleclicking it with an
Arrow tool.
The center point used for rotation is the point you marked most
recently using the Mark Center command. If you haven’t marked a center
point, Sketchpad automatically marks the point nearest to the center of
the screen.
You can use the Rotate Arrow tool to select and deselect objects, press
action buttons, and construct points of intersection just like the other
Selection Arrow tools.
See also: Transformations and Dragging (p. 74), The Translate Arrow Tool (p. 75), The
Dilate Arrow Tool (p. 75), Rotate (p. 177), Mark Center (p. 169)
The Dilate Arrow Tool
Drag selected objects with this tool to dilate them—that is, to stretch
or shrink them farther from or closer to a center point by any desired
amount without changing their direction from the center or their shape.
You can also mark a
point as the center of
dilation by doubleclicking it with an
Arrow tool.
The center point used for dilation is the point you marked most
recently using the Mark Center command. If you haven’t marked a center
point, Sketchpad automatically marks the point nearest to the center of
the screen.
You can use the Dilate Arrow tool to select and deselect objects, press
action buttons, and construct points of intersection just like the other
Selection Arrow tools.
See also: Transformations and Dragging (p. 74), The Translate Arrow Tool (p. 75), The
Rotate Arrow Tool (p. 75), Dilate (p. 179), Mark Center (p. 169)
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75
Selection Arrow Tools
How To . . . Zoom In and Out on a Sketch
Sometimes a sketch may have so much detail
that it becomes hard to select the object you
want. Other times, a sketch may get spread
out so that you can’t see everything in the
sketch window. If you want a Zoom command
at such a time, you won’t find it in Sketchpad’s
Display menu. However, you don’t really need
this command because Sketchpad has the
Dilate Arrow tool, which is the mathematical
equivalent of zooming.
To zoom in or out on a sketch, expanding it or shrinking it:
1. Choose Select All from the Edit menu to select everything in your
sketch.
If you haven’t marked a center point, Sketchpad marks one for you.
2. Use the Dilate Arrow tool to drag any selected object toward or away
from the marked center. As you drag, your sketch zooms in or out.
See also: The Dilate Arrow Tool (p. 75), Select All (p. 115), Mark Center (p. 169)
Other Arrow Actions
Beyond dragging and selecting objects, the Arrow tool has several special
actions when used with specific types of objects.
Arrow Tool Double-Click Shortcuts
Double-clicking the Arrow tool on several different kinds of objects
quickly accomplishes the most common action for that kind of object.
Double-clicking provides shortcuts for these actions that would
otherwise require you to select the object and then choose a menu
command.
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Selection Arrow Tools
Double-Clicked Object
Resulting Action
Point
Mark Center for
rotations and dilations
Straight Object
Mark Mirror for
reflections
Calculation
Edit Calculation
Function
Edit Function
Parameter
Allows you to change the parameter’s value
Label
Caption
Shows Label Properties to edit the label
Begins editing the caption
See also: Mark Center (p. 169), Mark Mirror (p. 169), Label Properties (p. 122), Edit
Calculation (p. 120), Edit Function (p. 120), Editing Captions (p. 88)
Arrow Tool Text Shortcuts
Often, you can use the Arrow tool with labels and captions without
bothering to change to the Text tool. With the Arrow tool, you can:
•
Press and drag a label.
•
Double-click a label to edit the label with Label Properties.
•
Double-click a caption to edit the caption.
See also: Text Tool (p. 86), Label Properties (p. 122), Editing Captions (p. 88)
Constructing a Point of Intersection
Click the Arrow tool on the intersection of two straight objects, circles,
or arcs to construct a point at the intersection.
Clicking the Arrow tool at a potential point of intersection has the same
effect as clicking the Point tool or using the Intersection command. So if
you want to select an intersection that you haven’t yet constructed, use
the Arrow tool to construct and select it in one action.
See also: Point Tool (p. 80), Intersection (p. 158)
Pressing Action Buttons
Click the Arrow tool on the body of an action button to perform that
button’s action. (Different buttons have different actions: Some may
cause an animation to start, others may hide or show objects, and so
forth.)
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77
Selection Arrow Tools
Hide Point
Handle
Body
When you point at the body of an action button, the cursor turns into a
pointing finger indicating you are about to press the button. If you
want to select or drag the button rather than perform its action, click
on the handle of the button rather than its body.
If an action button performs an action that takes time to complete, it
remains pressed until the action completes. Click the Arrow tool on the
body of a pressed button to release it, halting the action in process.
See also: Action Buttons (p. 37)
Rescaling an axis lets
you zoom in or zoom
out on the coordinate
system.
Changing an Axis’ Scale
To rescale an axis or a coordinate system, press and
drag any visible tick-mark number on the horizontal
or vertical axis. When you position the Arrow tool over
an axis’ tick-mark number, the cursor turns into a
bidirectional arrowhead, indicating you can drag to
grow or shrink the scale of that axis.
If the coordinate system has square units, rescaling
one axis will rescale the other. If the coordinate
system has rectangular units, rescaling one axis has no
effect on the other. If the coordinate system was
defined in terms of fixed units, you will not be able to
rescale it by dragging axis tick-mark numbers, as its
scale is fixed by the quantities that define it.
3
2
1
-1
See also: Coordinate Systems and Axes (p. 21), Grid Form (p. 203), Define Coordinate
System (p. 201)
Resizing Pictures
To resize a pasted picture:
1. Select the picture with the Arrow by clicking on it.
A frame appears around the picture, indicating it’s
selected. In the upper left and bottom right corners of
the frame, two small resize handles appear.
2. Press and drag one of the two resize handles.
The picture begins resizing. Holding down the Shift
key while dragging causes the picture to retain its original aspect
ratio.
3. Adjust the picture to the desired size and stop dragging.
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The Geometer’s Sketchpad Reference Manual
Selection Arrow Tools
If the picture was originally pasted onto a point to determine the
location of its upper left corner, only the resize handle in the lower
right of the selection frame will appear and be draggable. If the picture
was originally pasted between two points, neither resize handle will
appear, as the picture’s location and size are determined by the
positions of the points between which it is pasted.
See also: Pictures (p. 39), Paste (p. 111)
Resizing Function Plots and Loci
Function plots and certain loci display an arrowhead at one or both
endpoints, indicating that the function plot or locus extends farther in
the indicated direction than is presently displayed.
You can also set or change the domain of a function plot numerically in
its Plot Properties panel.
You can resize such a function plot or locus—to extend farther or less
far in the direction of an endpoint arrowhead—by dragging that
endpoint arrowhead with the Arrow tool. As you point at the arrowhead,
the cursor becomes a multidirectional arrow. Press and drag in the
direction in which the arrowhead points to extend the locus or function
plot; drag in the opposite direction to contract it.
See also: Functions and Function Plots (p. 27), Loci (p. 24), Plot Properties (p. 124)
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79
Point Tool
Use the Point tool to draw or construct independent points, points on
paths, and points at intersections.
•
Click in an empty area of your sketch to create an
independent point.
A point constructed
on a path can move
anywhere along the
path, but nowhere
else.
•
Click on a path object—such as a segment, a
circle, or the edge of a polygon interior—to
construct a point on the path. When the Point
tool is in the right place to construct a point on
path, the path is highlighted, appearing thicker
and in a special color.
In addition to
highlighting your
targeted objects,
Sketchpad displays a
message in the status
line (at the bottom of
the window) telling
you when you can
construct a point on a
path or at an
intersection.
•
Click at the intersection of two path
objects—such as a segment and a circle, two
lines or two circles—to construct a point of
intersection. When the Point tool is in the
right place to construct an intersection, both
paths are highlighted, appearing thicker and
in a special color.
There are several ways to create points without using the Point tool. For
example, clicking with the Arrow tool on an intersection constructs a
point of intersection there in the same way the Point tool does. The
Compass tool, Straightedge tools, and most Custom tools sometimes
construct their own points as part of their operation. The Point On Object,
Midpoint, and other menu commands also construct points.
See also: The Toolbox (p. 68), Points (p. 11), Path Objects (p. 13), Selection Arrow
Tools (p. 70), Compass Tool (p. 81), Straightedge Tools (p. 83), Custom Tools (p. 90),
Point On Object (p. 156), Midpoint (p. 157), Intersection (p. 158)
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The Geometer’s Sketchpad Reference Manual
Compass Tool
Use the Compass tool to construct circles determined by two points—
the center point and another point through which the circle passes.
This second point is sometimes called the radius point, because it
determines the radius of the circle.
Another way to use
the Compass tool is to
press the mouse
button at the center
point, drag, and
release the button at
the radius point.
Constructing a Circle
1. Choose the Compass tool if it’s not already active.
2. Click to locate the center of your circle. (You can click in empty
space, on an existing point, on a path object such as a segment or
another circle, or on an intersection.)
3. Click again to locate the radius point.
Attaching a Circle to an Existing Object
You can attach either the center point or the radius point to an existing
object. To do so, you can click on
•
an existing point
•
a path object (such as a segment, a line, a circle or an arc)
•
an intersection of two path objects.
When the Compass tool is in the right place to click on an existing
object, that object is highlighted, appearing thicker and in a special
color.
Be careful when you want to attach the radius point of a circle to an
existing object. It’s not enough to position the circle so that the circle
appears to go through the desired point. You must actually position the
tool itself over the point or object to which you want to attach before
clicking or releasing to locate the radius point.
In the left box of the following illustration, even though the circle
appears to pass through the vertices of the triangle, the radius point will
be located in empty space. The result is that the radius point will be
independent, and dragging either the radius point or any part of the
triangle will show that the circle isn’t attached to the vertices.
In the right box of the same illustration, the tool is positioned at a
vertex, and the vertex is highlighted. Thus, that vertex will be the radius
point of the circle, and the size of the circle will be linked to the
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81
Compass Tool
position of the vertex no matter how various parts of the figure are
dragged.
Circle not attached to vertex
Circle attached to vertex
Related Commands
There are two Construct menu commands that construct circles
without using the Compass tool.
The selected distance
can be either a
segment or a distance
measurement.
•
Circle By Center+Point
•
Circle By Center+Radius
constructs a circle determined by two selected
points. The first point is the center point, and the second is the
radius point.
constructs a circle determined by a selected
point and a selected distance. The selected point is the center point,
and the selected distance determines the radius.
See also: The Toolbox (p. 68), Circles (p. 15), Path Objects (p. 13), Circle By
Center+Point (p. 161), Circle By Center+Radius (p. 162)
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Straightedge Tools
Use the Straightedge tools to construct straight objects: segments, rays,
and lines. Each straight object constructed by one of these tools is
determined by two points.
A straightedge is a
ruler without marks.
It can be used for
drawing straight lines
but not for
measuring.
Use the Segment tool to construct a segment between its two
endpoints.
With any Straightedge
tool active, you can
switch to other
Straightedge tools by
holding down the
Shift key and pressing
the left or right arrow
key on your keyboard.
When Sketchpad starts, the active Straightedge tool is the Segment tool.
Choose a different Straightedge tool by pressing and holding on the
Straightedge tool icon in the Toolbox. When you press and hold, a menu
pops out and you can choose any of the three tools.
If you prefer, you can
use any of the
Straightedge tools by
pressing the mouse
button at the location
of the first point,
dragging the mouse,
then releasing the
mouse button at the
location of the second
point.
Use the Ray tool to construct a ray from one endpoint through
another point.
Use the Line tool to construct a line through two points.
To Construct a Segment:
1. Choose the Segment tool if it’s not already active.
2. Click to locate the first endpoint of the segment. (You can click in
empty space, on an existing point, on a path object such as another
segment or a circle, or on an intersection.)
3. Click again to locate the second endpoint.
To Construct a Ray:
1. Choose the Ray tool if it’s not already active.
2. Click to locate the endpoint of the ray. (You can click in empty
space, on an existing point, on a path object such as a segment or
another circle, or on an intersection.)
3. Click again to locate the point through which it travels.
To Construct a Line:
1. Choose the Line tool if it’s not already active.
2. Click to locate a first point through which the line travels. (You can
click in empty space, on an existing point, on a path object such as
a segment or another circle, or on an intersection.)
3. Click again to locate a second point through which the line travels.
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83
Straightedge Tools
Attaching a Straight Object to an Existing Object
You can attach either the first or second determining point of a straight
object to an existing object. To do so, you can click
•
an existing point
•
a path object such as a segment, a line, a circle, or an arc
•
an intersection of two path objects.
When the Straightedge tool is in the right place to click on an existing
object, that object is highlighted, appearing thicker and in a special
color.
Be careful when you want to attach a defining point to an existing
object. It’s not enough to position the tool so that the ray or line
appears to go through the desired location. You must actually position
the tool itself over the object to which you want to attach before
clicking or releasing the mouse to locate the defining point.
For example, a student wants to draw a diagonal line passing through
the intersection of the horizontal and vertical lines. In the following
illustration, even though the line in the left box appears to pass through
the intersection, the cursor is positioned to click in empty space, so the
second defining point of the line won’t be attached. But the cursor in
the right box is over the intersection, so the intersecting lines are
highlighted, and when clicked on, the point will be defined at the
intersection.
Line not attached to intersection
Line attached to intersection
In another example, a student wants to attach a ray’s through point to
the circle so the through point can be animated around the circle. The
next illustration shows how to accomplish this. Positioning the cursor
to click where shown in the left box won’t work because the through
point will not be attached to the circle. Clicking where shown in the
right box, with the cursor over the circle and the circle highlighted,
defines the point on the circle.
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Straightedge Tools
Ray not attached to circle
Ray attached to circle
Constructing Straight Objects at Specific Angles
While you’re constructing a straight object, you can hold down the
Shift key to make the object horizontal, vertical, or at an angle of 15º,
30º, 45º, 60º, or 75º. Construct the second point before you release the
Shift key.
Related Commands
There are several commands in the Construct menu that construct
straight objects without using the Straightedge tool.
•
Segment, Ray, and Line
•
Perpendicular Line and Parallel Line construct a line through a selected
point that is perpendicular or parallel to a selected straight object.
•
Angle Bisector
commands construct straight objects
determined by two or more selected points.
constructs a ray that bisects the angle formed by three
selected points.
See also: The Toolbox (p. 68), Construct Segment, Ray and Line Commands (p. 158),
Path Objects (p. 13), Segment (p. 83), Ray (p. 83), Line (p. 83), Perpendicular Line
(p. 160), Parallel Line (p. 159), Angle Bisector (p. 161)
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Text Tool
Use the Text tool to perform a variety of operations on labels and on
other objects that display text.
Text plays an important role in Sketchpad. Labels allow you to
communicate with others about specific objects in your sketch, and
provide a correspondence between measurements and the objects they
measure. Captions allow you to identify and describe both your sketch
as a whole and individual parts of your sketch. Measurements,
calculations, parameters and functions display important mathematical
information in text form. The Text tool allows you to show, hide, edit
and reposition labels of objects, to edit captions, and to modify the text
displayed with measurements, calculations, parameters, and functions.
See also: Label (p. 8), Captions (p. 36), Measurements (p. 18), Functions (p. 27),
Parameters (p. 19)
Using the Text Tool
Use the Text tool to create, show, hide, and edit labels; to create, edit,
and resize captions; and to change the text displayed with
measurements, calculations, and parameters.
The Text tool has five possible appearances when you move the mouse
over the sketch. These appearances depend on what it’s pointing at and
indicate the effect of using the tool.
Objects like points
and circles can either
show or not show a
label. Some other
objects—like action
buttons or
parameters—always
show their label. And
some other objects—
like captions and
pictures—never show
a label.
Filled hand: Click to show or hide the label of the object at which
you’re pointing. This cursor appears only when you’re pointing at
an object that has a label to show or hide.
Labeled hand: Press and drag to reposition the label of a
geometric object or double-click to edit the label of the object
you’re pointing at. This cursor appears when you’re pointing at a
label or at a text object (such as a parameter or a measurement)
that can be labeled.
Open hand: Press and drag to create a new caption. This cursor
appears only when you’re pointing at empty space or at an object
that never displays text.
I-beam: Click, press and drag, or double-click to edit a caption.
This cursor appears when you’re pointing at a caption.
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Text Tool
Resize arrow: Press and drag to resize the caption and reflow the
text. This cursor appears when you point at the resize handle of a
selected caption.
See also: Show/Hide Labels (p. 149), Positioning and Changing Labels (p. 88), Label
Command (p. 150), Creating Captions (p. 88), Editing Captions (p. 88), Resizing
Captions (p. 89), Text Preferences (p. 139)
Showing and Hiding Labels
When you point the Text tool at a geometric object, the tool appears as
a filled hand.
By default, objects are
labeled automatically
when you measure
them. (Use Text
Preferences to turn
off this feature.) You
can also change Text
Preferences so that all
new points are labeled
when they’re created.
•
Click to show the object’s label.
•
Click again to hide the object’s label.
Here are the default labels Sketchpad uses for various kinds of objects
(before you change them):
Object
Default Label
Point
Straight Object
Circle
Circle Interior
Arc
Arc Interiors
Polygon*
Point Locus
Non-point Locus
Function
Function Plot
Measurement
Parameter
Caption
Pictures
Action Buttons
A, B, C, …
j, k, l, …
c 1, c2, c3, …
C1, C2, C3, …
a1, a2, a3, …
A 1 , A2 , A3 , …
P1, P2, P3, …
L1, L2, L3, …
Cannot show a label
f, g, h, …
y=f(x), …
m1, m2, m3, …
t1, t2, t3, …
Cannot show a label
Cannot show a label
Label depends on action
* When possible, a polygon with six or fewer vertices is labeled
according to its vertices.
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Text Tool
If Sketchpad runs out of letters with which to label points or straight
objects, it starts over with A1 and j1.
See also: Show/Hide Label (p. 149)
Positioning and Changing Labels
When you point the Text tool at an object’s label or at a measurement,
parameter, calculation, or function, the tool appears as a labeled hand.
Double-clicking is the
same as choosing
Label from the
Display menu.
•
Double-click to change the label.
•
Press and drag to reposition the label of a geometric object.
You can also change the font, size, style, and color of selected objects’
labels with the Text Palette or with text commands in the Display
menu.
You can use the Label command from the Display menu to change the
labels of several selected objects at once.
See also: Label Properties (p. 122), Label Command (p. 150), Text Palette (p. 55),
Showing and Hiding Labels (p. 87)
Creating Captions
When you point the Text tool at an empty place in the sketch, the tool
appears as an open hand.
Dragging determines
the initial width of
your new caption.
•
Press and drag to create and begin typing and editing a new
caption.
To finish editing your caption, press the Esc key or click anywhere
outside the caption.
See also: Captions (p. 36), Editing Captions (p. 88), Show/Hide Text Palette (p. 154),
Text Palette (p. 55)
Editing Captions
When you point the Text tool at a caption, the tool appears as an
I-beam.
88
•
Click to place an insertion point and begin editing.
•
Press and drag to select a block of text to replace.
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Text Tool
You can turn off this
default behavior using
Text Preferences, and
instead use Show Text
Palette to summon the
Text palette as
needed.
•
Double-click to select the word under the cursor.
•
While editing, drag the caption’s resize handle to resize the caption
and reflow its contents.
When you create or edit a caption, by default Sketchpad displays the
Text Palette. You can change the font, size, style, or color of selected
caption text by using this palette or by using text commands in the
Display menu. The Text Palette also contains symbolic notation tools
for adding mathematical text to your caption.
To finish editing, press the Esc key or click anywhere outside the
caption.
See also: Captions (p. 36), Text Palette (p. 55), Display Menu (p. 145), Show/Hide Text
Palette (p. 154)
Resizing Captions
When you point the Text tool at a caption’s resize handle, the tool
appears as an arrow. A caption’s resize handle is displayed at the lower
right corner of the caption whenever the caption is selected or being
edited.
•
Press and drag the handle to resize the caption and reflow the text.
See also: Captions (p. 36)
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Custom Tools
The Custom tool icon allows you to define and use custom tools.
Overview of Custom Tools
Custom tools are tools that you create yourself or that other Sketchpad
users create for you. In just the way that Sketchpad’s Compass tool
constructs a circle given its center and radius point, custom tools that
you create can construct figures of arbitrary complexity. For example,
you can make a custom tool that constructs the perpendicular bisector
of a given segment, or one that constructs the circumcircle of a given
triangle, or one that constructs a square given two adjacent vertices. A
more advanced custom tool might create a fractal, or a tangent to an
arbitrary point on a function plot, or a complex tessellation. Any
custom tool you define can be used an unlimited number of times in an
unlimited number of sketches. By defining new custom tools, you
extend the built-in tools available to you in Sketchpad. Since you can
create any type or number of custom tools you wish, the possibilities
for extending Sketchpad’s tools are limitless.
Tools that you create reside inside the document in which you create
them. You can always use them in that document (unless you remove
them from that document) just as if you were using a built-in tool like
the Compass or Straightedge. When a document containing custom tools
is open, you can also use those tools in any other open document.
Finally, you can place documents containing frequently used tools
inside a special Tool Folder on your hard drive. Any tools in your Tool
Folder will be available all the time, even when the documents that
contain them are not open in Sketchpad.
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Custom Tools Menu
When you press the Custom tool icon in the Toolbox, Sketchpad
displays the Custom Tools menu. This menu has several parts.
Create New Tool. This command defines a
new custom tool based on your selections in
the sketch.
Tool Options. This command allows you to
organize, rename, copy, or remove the custom
tools contained in your sketch.
Show Script View. This command shows or
hides the most recently chosen custom tool’s
script—a step-by-step description of what the
tool constructs. The command is Hide Script
View if the script is already showing.
This Document. This part of the menu lists
all of the custom tools defined in the current
(active) document. If the active document does not yet contain any
tools, this part of the menu does not appear. When you define a new
custom tool, it appears first in this part of the menu.
Other Documents. This part of the menu lists all of the other open
documents that contain custom tools. Each entry in this part of the
menu lists an open document that contains tools and displays a
submenu of each of the tools in that document. If no other open
documents contain tools, this part of the menu does not appear.
Tool Folder. This part of the menu lists any tools from documents
that were stored in the Tool Folder when Sketchpad started. This
folder is a special folder (directory) named Tool Folder that’s stored next
to the Sketchpad application on your hard disk. If no documents were
stored in this folder when Sketchpad started (or if the documents in
this folder didn’t contain any tools), this part of the menu does not
appear.
When you want to define a new tool or reorganize your tools, use the
commands at the top of this menu. When you want to use a tool to
construct objects in your sketch, choose it from one of the lower parts
of the menu. The chosen custom tool then becomes active until you
choose a different tool, just as if you’d chosen the Compass or
Straightedge tool. (In other words, if you want to use the same custom
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Custom Tools
tool several times in a row, you don’t need to choose it again from the
Custom Tools menu.) If you’ve switched from a custom tool to some
other tool, such as the Arrow, you can switch back to the last custom
tool you used by clicking—rather than pressing—on the Custom tool
icon. This activates the most recently chosen, checkmarked tool. If you
want to switch from one custom tool to another or from a built-in tool,
such as the Arrow, to a new custom tool, choose the new tool from the
Custom Tools menu.
See also: Using a Custom Tool (p. 92), Making a Custom Tool (p. 94), Advanced Tool
Topics (p. 230), Document Tools (p. 6), Show Script View (p. 106), Script View (p. 59)
Using a Custom Tool
Custom tools are easy to use, even before you learn to make one
yourself. In this example, you’ll use custom tools that are already
defined in one of the sample documents that comes installed with
Sketchpad. The document Sample Tools.gsp contains several custom
tools. Just by opening that document, you can use its tools in your own
sketch.
1. Open the document Sketchpad | Samples |
Custom Tools | Sample Tools.gsp.
2. Press and hold on the Custom tools icon in
You can also press,
drag, and release to
get the same effect as
two clicks (one where
you pressed, and one
where you released).
92
the Toolbox. The Custom Tools menu
appears.
3. Choose Circumcircle from the menu. This
custom tool constructs a triangle and its
circumcircle given three points.
4. Move your mouse over the sketch and click in three different
places. A triangle appears with its circumcircle—the circle that
passes through all three vertices.
5. Continue using the Circumcircle tool. Click three more times in the
sketch for each triangle you want to construct. You can click in
empty space, on existing points, on paths, or on intersections just
as you can with the Point, Compass, and Straightedge tools.
6. When you’re finished making triangles with circumcircles, click on
any other tool in the Toolbox or press the Esc key.
To resume making circumcircles, click the Custom tool icon again. You
don’t need to press and hold to choose from the Custom Tools menu
the next time—just click the Custom tool icon to activate it. Press and
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Custom Tools
hold the Custom tool icon in the Toolbox only when you want to switch
to a different custom tool from the menu.
See also: The Givens and Results of a Tool (p. 93), Matching Given Objects (p. 93), The
Tool Folder (p. 98), Custom Tools Menu (p. 91)
The Circumcircle tool
has three givens—the
vertices of the
triangle. The three
sides of the triangle
and the circumcircle
are the tool’s results.
The Givens and Results of a Tool
Each time you click as you use a tool, you are specifying a given object—
an object that is used to determine the rest of the objects the tool
produces. The objects produced by the tool (that depend on the given
objects) are the results of the tool.
You can think of the various objects and relationships that make up a
tool as a family tree. In this family tree, the givens are the ultimate
ancestors—objects that have children but no parents. The other
objects—those that have parents—are the results of the tool.
See also: Object Relationships: Parents and Children (p. 10)
The status line can
help you figure out
what to match next.
Matching Given Objects
Tools can use as their givens various kinds of objects: points, straight
objects, circles, measurements, functions, and so forth. When you use a
tool, a message appears on Sketchpad’s status line at the bottom of the
window describing what kind of given object you need to match next.
For example, if a tool uses a point, a segment, and a distance
measurement as its three givens, the status line first says “1. Match
Point…” to indicate what object you must match first. After you match
the point, the status line says “2. Match Segment…” to indicate that
you must next specify a segment. And finally, it says “3. Match
Distance Measurement…” when it’s time for you to click on a distance
measurement as the last given object.
You can always match
a given by clicking on
a sketch object of the
correct kind. For
some givens—points,
straight objects, and
circles—you can also
match the given by
constructing it.
If a given object for a tool is a point, you can specify it in either of two
ways.
•
Specify an existing point: Click on an existing point in the sketch.
The point you click on is used as the given.
•
Construct a new point: Click somewhere else in the sketch—in
empty space, on a path, or at an intersection—to construct a new
point. The point you construct is used as the given.
If a given object for a tool is a circle or straight object, you can specify
it in either of two ways.
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Custom Tools
•
Specify an existing object: Click on an existing object of the correct
kind. The object you click on is used as the given.
•
Construct a new object: Click twice (or press and drag) in the
sketch to specify the two points which determine the given. If
you’re constructing a circle given, the first point is the center and
the second point is the radius point. If you’re constructing a
straight object, the two points are the two determining points of
the straight object.
If the given for a tool is any other kind of object, you must click on a
matching object in the sketch. If no object of the correct kind exists in
the sketch, you must create such an object before you can use the tool.
Making a Custom Tool
You define new custom tools by example: you create a construction you
wish to “turn into a tool,” then define a tool based on that example.
Turn any construction into a custom tool by selecting sketch objects
that define a construction you wish to turn into a tool. The objects you
select must be related to each other in such a way that at least one
selected object is completely determined by other selected objects.
Then choose Create New Tool from the Custom Tools menu.
The selected objects to be produced by a tool are called results. The
selected objects that don’t depend on any others, but upon which the
results depend, are called givens. Any unselected objects that relate the
selected givens to the selected results are intermediate objects.
(Intermediate objects are not reproduced when you use the tool; only
the objects that were selected when you defined the tool are
reproduced as results when you use the tool.)
Follow these steps to make a new custom tool:
1. Make a construction to serve as an example of the construction
you want the tool to produce. You can use any of Sketchpad’s tools
or menus to create this exemplar.
2. Select both the given objects (usually, independent points) and the
desired resulting objects you’d like the tool to produce. The order
in which you select the givens determines the order in which you’ll
match givens when using the tool.
3. If there are other objects that relate the givens to the final results,
you can select them or not. If you do select them, they will be
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Custom Tools
shown when you use the tool. If you don’t select them, they won’t
appear when you use the tool.
4. Choose Create New Tool from the Custom Tools menu that appears.
5. A dialog box appears in which you can type a name for the tool.
Type a name and click OK.
Your tool is added to the Custom Tools menu, and is ready to use.
See also: Custom Tools Menu (p. 91)
Specifying the Results of a Tool
When you make a tool, any selected objects that depend on the givens
become results of the tool, and will be shown when you use the tool.
Any unselected objects that depend on the givens will not be shown
when you use it.
For example, if you make a tool that constructs the perpendicular
bisector of a segment, the segment is the given object and the
perpendicular bisector is a result. If you select the midpoint of the
segment when you make the tool, the midpoint is also a result, and is
shown when you use the tool. If you don’t select the midpoint, it’s an
intermediate object and is hidden when you use the tool.
B
C
D
A
Midpoint selected when making tool
B
A
Midpoint unselected when making tool
Midpoint shown when using tool
C
D
Midpoint hidden when using tool
See also: Custom Tools Menu (p. 91)
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Custom Tools
Managing Custom Tools
After you’ve created one or more custom tools in a document, you can
rearrange them, copy them to other documents, rename them, or
remove them by choosing Tool Options from the Custom Tools menu.
This command opens the Document Options dialog box to a view of
your document’s tools. See Document Options for more information.
See also: The Givens and Results of a Tool (p. 93), Using a Custom Tool (p. 92),
Advanced Tool Topics (p. 230), Document Options (p. 104), Custom Tools Menu
(p. 91)
Showing or Hiding the Script View
Choose Show Script View or Hide Script View from the Custom Tools menu
to show or hide the script view of the active tool. This view allows you
to see the given objects and the steps the tool takes to construct its
results, as well as, to change the properties of the steps, and to observe
and control the tool as it functions.
See also: Script View (p. 59), Custom Tools Menu (p. 91)
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Custom Tools
How To . . . Make a Perpendicular Bisector Tool
If you’re doing an investigation in which you need to construct several
perpendicular bisectors, you can make a perpendicular bisector tool to
simplify your work. Suppose you want to construct the perpendicular
bisectors to all three sides of a triangle. Rather than do the
perpendicular bisector construction three times, you can do it once and
make it into a tool. Then you can use the tool for the other two
perpendicular bisectors. This example shows you how.
1. In a new document, use the Segment
Segment AB is the
given object, and the
midpoint and
perpendicular are
results.
If you had also
selected points A and
B, they would have
been the givens, and
the segment would
have been an
intermediate result.
tool to construct ∆ ABC.
2. On segment AB, construct the
B
C
midpoint and the perpendicular to
segment AB through the midpoint.
3. Select segment AB, the midpoint, and
the perpendicular line.
A
4. Press and hold the Custom tool icon
and choose Create New Tool from the
Custom Tools menu that appears.
5. Type Perpendicular Bisector to name your tool and click OK.
6. Click the Custom tool icon to choose your new tool.
7. Click on each of the other two sides of the triangle. The tool
constructs the perpendicular bisectors on those two sides.
8. Press the Esc key or choose a different tool from the Toolbox to
stop using your custom tool.
Because the given of this tool is a segment, you can match it either by
clicking on a segment, as you did above, or by clicking twice (or
pressing and dragging) to construct a new segment to match the given.
This means you can use the tool to construct a triangle from scratch
with the perpendicular bisectors constructed automatically.
1. Click the Custom tool icon to choose your Perpendicular Bisector tool.
2. Press in empty space in the sketch and drag to construct one side
of a triangle.
3. Press and drag two more times to construct the remaining two
sides of the triangle.
You’ve just constructed a triangle, as easily as you’d have done it with
the segment tool, but with perpendicular bisectors on all three sides.
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Custom Tools
Use your new tool to construct a quadrilateral with the perpendicular
bisectors of all four sides. Then construct one diagonal, along with its
perpendicular bisector. What do you notice about the perpendicular
bisector of the diagonal as you drag the quadrilateral into different
configurations?
The Tool Folder
The Tool Folder is a special folder in which you can store frequently
used tools. When Sketchpad starts, it checks the Tool Folder and puts
every tool in this folder into the Custom Tools menu that appears
when you press and hold the Custom tools icon.
If you do not have
permission to write to
the folder containing
Sketchpad, it’s
possible to create a
tool folder in an
alternate location. See
Alternate Tool
Folders (p. 232) for
details.
Creating a Tool Folder
You can create a folder in which you store documents containing
frequently used tools. These tools will be available each time you use
Sketchpad.
1. Locate the folder the Sketchpad program is located in.
2. If there is no folder named Tool Folder within the folder containing
Sketchpad, create a new folder.
3. Name this new folder Tool Folder.
You can now use this folder to store Sketchpad documents containing
frequently used tools. The tools will be available the next time you start
Sketchpad.
Storing a Tool in the Tool Folder
To store a tool in the Tool Folder:
You can copy tools
between open
documents with Tool
Options.
1. Create or copy the tool(s) you want into a new document.
2. Choose Save As from the File menu.
3. Use the Save As dialog box to navigate to the folder named Tool
Folder located within the same folder which contains the Sketchpad
program.
4. Save the document in the Tool Folder.
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Alternately, you can drag Sketchpad documents into this folder from
your computer’s desktop to make their tools available the next time you
start Sketchpad.
The new tool will be available the next time you start Sketchpad.
See also: Overview of Custom Tools (p. 90), Using a Custom Tool (p. 92), Custom
Tools Menu (p. 91), Alternate Tool Folders (p. 232)
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99
Menu Reference
This section describes the commands in Sketchpad’s menus.
The Mac OS X
edition of Sketchpad
contains a Sketchpad
application menu in
addition to the menus
listed here. The
application menu
contains standard
commands used by all
applications, and isn’t
discussed here.
•
The File menu allows you to create, save, and print entire
documents.
•
The Edit and Display menus contain commands that alter the
appearance, format, or definition of existing objects in your
active sketch.
•
The Construct, Transform, Measure, and Graph menus allow
you to define new mathematical content in the active sketch,
most often by expressing new objects’ relationships to existing,
selected objects.
•
The Window menu (Microsoft Windows and Mac OS X only)
lets you manage open document windows on your desktop.
•
The Help menu allows you to consult an electronic version of
this Reference Manual for help with specific Sketchpad
commands and tools.
•
The Context menu appears when you right-click (Windows) or
Ctrl+click (Macintosh) in the sketch, and presents options
relevant to the object that was clicked.
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101
File Menu
This menu contains commands for opening,
saving, printing, and otherwise working with
Sketchpad documents. Many of these
commands are standard commands that appear
in most software applications. This chapter
briefly describes the familiar aspects of these
commands and provides more detail on ways
in which Sketchpad treats them differently. If
you are unfamiliar with any of the basic
commands, you’ll probably want to start by
looking in the manual that came with your
computer.
New Sketch
The keyboard
shortcut for New
Sketch is Ctrl+N
(Windows) or
a+N (Mac).
Opens a new, blank document. A new document window appears on
top of all other windows and becomes the active window. The new
document is untitled until you name it by saving it.
Open
The keyboard
shortcut for Open is
Ctrl+O (Windows) or
a+O (Mac).
In Windows, you can
use a command-line
flag to determine
what folder appears in
the dialog box the
first time you choose
the Open command.
See p. 224 for details.
102
Opens one or more previously
saved documents.
When you choose Open a dialog
appears showing a view of your
disk.
1. Navigate to the folder
containing your document(s).
2. Click on a name to highlight
the document you wish to open.
3. Hold down the Shift key (Macintosh) or Ctrl key (Windows) while
clicking additional document names to open more than one
document at a time.
4. Click Open or double-click the name of the document.
The chosen documents open on your desktop.
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File Menu
Save
The keyboard
shortcut for Save is
Ctrl+S (Windows) or
a+S (Mac).
Saves whatever changes have been made to the current document since
the last time it was saved. If the document is being saved for the first
time, the Save command prompts you for the location in which to save
(see Save As below). This command is enabled only if you’ve made
changes in the document since the last time you saved it.
Save As
Mac users: When first
saving a document,
Sketchpad suggests a
filename ending with
the extension .gsp.
While Macintosh
doesn’t require file
extensions, naming
Sketchpad documents
with this extension
makes it easier to
share them with
Microsoft Windows
users or users on the
Internet.
Names and saves the active document in a location that you specify.
When you choose Save
As, a dialog box like the
one at right appears.
1. Navigate your
folders to locate the
folder in which you
want to save your
document, or create
a new folder, if
needed.
2. Type a name for the
document.
3. Click Save.
Saving Different Copies of a Document
You can use Save As to save an additional copy of a previously saved
document, with or without changes. For example, imagine you have a
document called Pythag.gsp to which you want to add action buttons,
but you also want to keep a copy of the original version. Make sure
you’ve saved the sketch in the original form, then make the changes
and choose Save As. Enter a different name for the modified version,
such as Pythag2.gsp. Now you’ll have both versions saved as separate
documents.
Saving in Cassiopeia™ Sketchpad Format
You can use Save As to save a copy of your document in a form that can
be used with the version of Sketchpad available for Casio’s Cassiopeia
Computer Extender handheld computer. Use the Format (Macintosh)
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103
File Menu
or Save As Type (Microsoft Windows) menu to choose Cassiopeia
Sketchpad Document before you click the Save button.
Hold the Shift key
while pulling down
the File menu to
change the Save As…
command to Save As
HTML.
Saving in HTML/JavaSketchpad Format
You can use Save As to save a copy of your document in a form that can
be used on the World Wide Web with JavaSketchpad. Use the Format
(Macintosh) or Save As Type (Microsoft Windows) menu to choose
HTML/Java Sketchpad Document before you click the Save button.
See also: JavaSketchpad (p. 238)
Saving in Metafile Format (Microsoft Windows only)
You can use Save As to save a graphics file that can be used by many
other Windows programs. The graphics file includes all objects visible
in the sketch window. Use the Save As Type menu to choose Enhanced
Metafile (*.emf) or Windows Metafile (*.wmf) before you click the Save button.
See also: Metafile Export (p. 236)
The keyboard
shortcut for Close is
Ctrl+W (Windows) or
a+W (Mac). You
can also use Ctrl+F4
in Windows.
Close
Closes the current document window. You’ll be prompted to save any
changes you’ve made since you last saved. The Close command does the
same thing as clicking the Close box in the upper-left-hand (Mac) or
upper-right-hand (Windows) corner of the window’s title bar. Close
does not quit Sketchpad itself.
Document Options
Manages the pages and
custom tools contained in a
document. A Sketchpad
document can contain
multiple pages; use this
command to add, remove,
rename, and reorder the
pages. Similarly, a Sketchpad
document can contain
multiple custom tools; use
the Document Options
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File Menu
command to copy, remove, rename, and reorder them.
See also: Document Pages (p. 6), Document Tools (p. 6)
View Pages/Tools
Click Pages to manage the pages in your document, or click Tools to
manage the custom tools.
When you’re viewing
pages, click the name
of a page in the list to
display that page in
the document.
List of Pages/Tools
This list shows all the pages or tools in the current document. You can
perform the following actions directly on the list items:
•
Click on a page or tool in the list, then change its name. (See Page
Name/Tool Name.)
•
Press and drag a page or tool in the list to change the position of
that page or tool in your document.
•
Double-click on a page name to display that page and close the
dialog box.
Page Name/Tool Name
To rename a page or tool, type a new name here for the page or tool
that is chosen in the list box.
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File Menu
Add Page
When you’re viewing pages, you can use this pop-up menu to add new
pages to your document. Choose Blank Page to add a new blank page to
your document. Choose from the Duplicate submenu to add a duplicate
copy of a page from any open document. The top part of the Duplicate
submenu lists pages from the current document. Listed below a divider
are any other open documents; these serve as submenus from which
you can choose pages to duplicate and add to the current document.
When page tabs aren’t
showing, Link
buttons and the
Document Options
dialog box are the
only ways to move
from page to page in
your document.
Show Page Tabs
This checkbox appears only when you view page options, not when
you view tool options. When Show Page Tabs is checked, tabs appear
along the bottom left of the document, showing the names or numbers
of the pages. You can use these tabs to navigate among the pages of a
document.
Moving the divider at the left edge of a document’s horizontal scroll
bar all the way to the left has the same effect as unchecking Show Page
Tabs. Moving the divider away from the left edge of the window has
the same effect as checking Show Page Tabs.
See also: Document Pages (p. 6), Link Buttons (p. 38)
Show Script View
This checkbox appears only when you view tool options, not when you
view page options. Check or uncheck this box to show or hide the
script view of the most-recently chosen custom tool. The script view
allows you to see the given objects and the steps of the tool, to change
the properties of the steps, and to observe and control the tool as it
functions.
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See also: Script View (p. 59)
Tools that you use
frequently and not
just in one document
can be stored in the
Tool Folder. These
tools are always
available whenever
you’re using
Sketchpad. See p. 98
for information on
setting up and using a
Tool Folder.
Before removing a
tool or page from
your document, you
may want to save the
document with a
different name in
order to preserve a
copy of that page or
tool in case you want
it later.
Copy Tool
When you’re viewing tools, you can use this pop-up menu to copy tools
into the active document from other open documents or from
documents in your Tool Folder. The top part of the Copy Tool menu
lists any other open documents that contain tools; each item provides a
submenu from which you can choose a tool to copy and add to the
current document. The bottom part of the Copy Tool menu allows you
to copy tools from Sketchpad documents into your Tool Folder.
Remove Page/Tool
Use this button to permanently remove a page or tool from a
document. First choose from the list box the page or tool you wish to
remove, then click Remove Page or Remove Tool. Every document
must have at least one page, so you cannot remove the only page from
a one-page document.
Once you remove a tool or page and click OK, it’s gone for good—you
cannot get it back. If you decide you don’t really want to remove that
tool or page, you must click Cancel rather than OK in order to leave
your document unchanged.
Page Setup
Sets up the page size, orientation, and other printing options for your
document. This dialog box differs depending on the printer you’ve
chosen as your default printer. To set your default printer, use the
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107
File Menu
Chooser from the Apple menu (Macintosh) or Settings | Printers from the
Start menu (Windows).
Print Preview
Displays a preview of your document as it will appear when printed.
This dialog box allows you to change the scale of your printout. If the
printout will be more than a single sheet of paper, you can view the
different sheets to decide which ones to print.
Print
Prints the current page of the active document on the default printer.
The Print dialog box allows you to specify which sheets to print and
allows you to print multiple copies. Depending on your printer and
operating system, it may also allow you to change printers, decide
between color and black-and-white, save output as Postscript, and
make other adjustments.
Quit
The keyboard
shortcut for Quit is
Ctrl+Q (Windows) or
a+Q (Mac).
108
Closes all open documents and exits Sketchpad. You’ll be prompted to
save any unsaved work in open documents before Sketchpad quits.
The Geometer’s Sketchpad Reference Manual
Edit Menu
The Edit menu contains commands for
undoing and redoing recent operations, for
managing the clipboard, for creating action
buttons, for selecting objects in your sketch,
and for modifying various elements and
properties of your sketch and of Sketchpad
itself.
Undo
The keyboard
shortcut for Undo is
Ctrl+Z (Windows) or
a+Z (Mac).
This command undoes the most recently
performed action.
Use this command in combination with Redo to
move backward and forward through your
recent Sketchpad actions. Sketchpad’s capability
to undo/redo is unlimited: you can use it to
undo your actions, one at a time, all the way back to the point at which
you created or opened the sketch. Similarly, after undoing, you can redo
those actions to restore your sketch to the state it was in before you
started undoing.
Unlimited undo/redo is helpful for correcting mistakes—undoing
something you didn’t mean to do—or for going back and trying a
different approach to a construction—testing a different hypothesis.
Unlimited undo is also a way to review your work or someone else’s
work step-by-step: Undo back to the beginning, then Redo one step at a
time to review each action.
You can undo many steps quickly by repeatedly pressing the keyboard
shortcut for Undo, or you can undo back to the beginning with only one
action by holding down the Shift key, then choosing Undo. When you
hold down the Shift key, the command becomes Undo All.
Undo and Redo only apply to geometrically significant actions: They
don’t apply to formatting changes such as changing labels, text styles,
fonts, colors, and line widths.
See also: Redo (p. 110)
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Edit Menu
Redo
The keyboard
shortcut for Redo is
Ctrl+R (Windows) or
a+R (Mac).
This command redoes an action you have undone. If you’ve undone
several steps, you can redo each of those steps.
Redo is available only immediately after using Undo. If you take any other
action after undoing operations, you can no longer redo the original
operations.
Use Redo in combination with Undo to move backward and forward
through your recent Sketchpad actions.
When you hold down the Shift key, Redo becomes Redo All and redoes
all previously undone actions.
Cut
The keyboard
shortcut for Cut is
Ctrl+X (Windows) or
a+X (Mac).
This command removes from the sketch any object that is selected,
along with any objects that depend on it. Each removed object is
placed on the clipboard and can be pasted into the same or a different
sketch, or as a picture into another application.
See also: Copy (p. 110), Paste (p. 111), Clear (p. 111), Advanced Graphics Export
(p. 234)
Copy
The keyboard
shortcut for Copy is
Ctrl+C (Windows) or
a+C (Mac).
This command places a copy of each selected object on the clipboard.
The contents of the clipboard can then be pasted into the same or a
different sketch, or into another application. If you copy a single table,
you can paste the table’s data into another application such as Fathom or
Microsoft Excel.
If you’re editing a caption or other text, Copy puts the selected text on
the clipboard. It can then be pasted back into the same or a different
sketch, or into another application.
See also: Cut (p. 110), Paste (p. 111), Advanced Graphics Export (p. 234)
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Edit Menu
Paste
The keyboard
shortcut for Paste is
Ctrl+V (Windows) or
a+V (Mac).
This command pastes the contents of the clipboard into the active
sketch. If the clipboard contains sketch objects, these objects are
inserted into the sketch.
If the clipboard contains a picture, the picture is inserted into the
sketch. When you paste a picture, if you have either one or two points
selected, the picture’s corners will be attached to those points.
If the clipboard contains text and you’re editing a caption or other text,
the text from the clipboard is inserted into the text you’re editing.
See also: Cut (p. 110), Copy (p. 110)
Clear
Pressing the Delete or
Backspace key
performs the same
action as choosing
Clear from the Edit
menu.
This command removes from the sketch any selected object, as well as
any objects which depend on it. The removed objects are not placed on
the clipboard.
Removing objects from a sketch using Cut or Clear is very different from
hiding them using the Hide command. The Hide command hides objects
from view, but they still exist as part of the sketch. Hidden objects can
later be shown, and they continue to affect the behavior of objects that
depend on them. Use Hide if you want to remove objects from view
while maintaining their geometric role in your document; use Clear, Cut,
or Undo if you want to remove them permanently from your
construction.
See also: Cut (p. 110), Hide Objects (p. 148)
Action Buttons
Action buttons are sketch objects
that, when pressed, perform a
previously defined action, such as
starting an animation or hiding a
group of objects.
Each command on the Action
Buttons submenu creates a
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111
Edit Menu
particular kind of action button. Most of these commands display a
Properties dialog box panel that allows you to specify how the button
you’ve just created works.
Most action buttons provide a simple way of performing or repeating a
common Sketchpad action or activity. For example, you can hide a
group of objects by selecting each object in the group and then
choosing Hide Objects from the Display menu—or you can create a
Hide/Show button that hides or shows the entire group of objects with
a single click. You can drag point A toward point B using the Arrow
tool—or you can create a Movement button to do it for you. In
general, you’ll create action buttons for commonly repeated actions for
your own convenience and to help communicate about—and
present—your sketch to others who may work with it later.
See also: Action Buttons (p. 37), Hide/Show Button (p. 112), Animation Button (p. 112),
Movement Button (p. 113), Presentation Button (p. 113), Link Button (p. 114), Scroll
Button (p. 114), Arrow Tool (p. 70), Text Tool (p. 86)
Hide/Show Button
This command is available only when at least
one object is selected.
Hide Point
Show Objects
It creates a button that hides or shows each
selected object. Normally the button’s label
changes from Hide to Show depending on whether the objects it
controls are visible or not. (When some objects are visible and some are
not, the button is labeled Hide.)
You can use the Hide/Show panel of Properties to change various
aspects of the button’s behavior. For example, you can make the
button always show its objects or always hide its objects, instead of
toggling between hiding and showing. You can also use this panel to
prevent the button from selecting its objects after showing them or to
make it show or hide its objects instantly instead of fading them in or
out.
See also: Action Buttons (p. 37), Hide/Show Properties (p. 127)
Animation Button
This command is available only when at least
one selected object can be animated. (Only
geometric objects and parameters can be
animated.)
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Animate Point
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Edit Menu
It creates a button that animates each selected object. Sketchpad
animates independent points freely in the plane. Points constructed on
objects are animated along the objects on which they’re constructed;
other geometric objects are animated by animating each point on which
they depend. And finally, numeric parameters are animated by changing
their numeric value.
You may want to
familiarize yourself
with the basic Animate
command from the
Display menu before
creating action
buttons to perform
animations for you.
When you create an Animation button, the Animate panel of the
Properties dialog box appears, allowing you to set the speed and
direction for each animated point and for each animated parameter.
Click the button once to start the animation; the button remains
pressed until the animation is finished. You can click the button a
second time, while the button is still pressed, to stop the animation.
See also: Animate (p. 153), Action Buttons (p. 37), Animate Properties (p. 128)
Movement Button
This command is available only when the
selection contains at least one pair of points. (You
must have selected an even number of points.)
After creating a
Movement button,
you may want to hide
the destination points
so only the moving
points are visible.
Move A -> B
It creates a button that moves the first point of each selected pair
toward the second. You can select as many pairs of points as you want;
the first point of each pair (the moving point) always moves toward the
second (the destination point).
When you create a Movement button, the Move panel of the Properties
dialog box appears, allowing you to set the speed and characteristics of
the motion.
Click the button once to start the movement; the button remains
pressed until all moving points reach their destinations. You can click
the button a second time, while the button is still pressed, to stop the
movement.
See also: Animate (p. 153), Action Buttons (p. 37), Move Properties (p. 130)
Presentation Button
This command is available only when at
least one action button is selected.
Present 2 Actions
It creates a button that activates the selected
buttons’ actions either simultaneously or sequentially to form a
presentation. Pressing a Presentation button has the same effect as
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113
Edit Menu
pressing each of its selected parent actions either all at once or one after
the other. Use Presentation buttons when you want to combine several
related actions into a single button for ease of use.
When you create a Presentation button, the Presentation panel of the
Properties dialog box appears, allowing you to specify whether to
present the selected actions simultaneously or sequentially, as well as to
set other properties of the Presentation.
See also: Action Buttons (p. 37), Properties (p. 120), Presentation Properties (p. 131),
Object Relationships: Parents and Children (p. 10)
Link Button
Use Document Options This command creates a button that links to a different
from the File menu to page of the document or that links to an Internet URL
add multiple pages to such as a remote web site.
your document.
Use URLs to connect
your sketch to related
mathematical,
historic, or reference
material available on
the Internet.
Link
When you create a Link button, the Link panel of the Properties dialog
box appears, allowing you to determine whether the button links to a
different document page or to a URL. If the button links to a
document page, you can also specify an action button on that page that
will be activated when the link occurs.
See also: Action Buttons (p. 37), Properties (p. 120), Link Properties (p. 133), Document
Options (p. 104)
Scroll Button
This command is available only when a single point is
selected.
Scroll
It creates a button that scrolls the window based on the
position of the selected point.
Use Scroll buttons
when you want to be
able to “jump” to a
point anywhere in the
scrollable plane.
When you create a Scroll button, the Scroll panel of the Properties
dialog box appears to allow you to determine whether the button
scrolls the window to put the selected point at the top left corner of the
window, or to put the selected point in the center of the window.
See also: Action Buttons (p. 37), Properties (p. 120), Scroll Properties (p. 134)
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Edit Menu
Select All
The keyboard
shortcut for Select All
is Ctrl+A (Windows)
or a+A (Mac).
This command selects all objects that match the active tool in the
Toolbox.
If a Selection Arrow or Custom tool is active, this command selects all
objects in the sketch.
If another tool from the Toolbox is active (Point, Compass, Segment, Ray,
or Text), all matching objects are selected. For example, if the Ray
tool is active, the command becomes Select All Rays.
Line,
See also: Select Parents (p. 115), Select Children (p. 115), Object Relationships: Parents
and Children (p. 10)
Select Parents
The keyboard
shortcut for Select
Parents is Ctrl+U
(Windows) or a+U
(Mac). (Think “Up
the family tree.”)
This command selects the parents of each selected object.
The parents of an object are those objects upon which the object directly
depends. For example, a segment’s parents are the endpoints used to
define it; a midpoint’s parent is the segment on which it’s constructed.
If the selected object has no parents (in other words, if it’s an independent
object), it remains selected. If the parents of the selected object are
hidden, the selected object is deselected (leaving nothing selected).
See also: Select All (p. 115), Select Children (p. 115), Object Relationships: Parents and
Children (p. 10)
Select Children
The keyboard
shortcut for Select
Children is Ctrl+D
(Windows) or a+D
(Mac). (Think “Down
the family tree.”)
This command selects the children of each selected object.
The children of an object are those objects that directly depend on the
object. For example, a circle constructed by the Circle By Center+Radius
command is a child of both the point and the segment used to define it.
If the selected object has no children, it remains selected. If the
children of the selected object are hidden, the selected object is
deselected (leaving nothing selected).
See also: Select All (p. 115), Select Parents (p. 115), Object Relationships: Parents and
Children (p. 10)
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115
Edit Menu
Split/Merge
The Split and Merge commands allow you to alter the relationships of
existing objects by splitting points from their parents, by merging
points either with other points or onto paths, and by merging several
text objects into one. These commands allow you to fix construction
mistakes, to make significant changes in a sketch without starting over,
and to modify geometric and mathematical investigations in flexible
and powerful ways.
Splitting a Point from Its Parent
When you select a single midpoint, a point on a path, or a point of
intersection, the command becomes Split Midpoint From Segment, Split
Intersection From Path Objects, or something similar. If the selected point is
a point on path, it’s removed from its path. If the selected point is an
intersection point, it’s split from the intersecting objects. In either case,
it becomes an independent point and can be dragged anywhere.
Split Point From Circle
Split Intersection From Path Objects
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Splitting a Point Apart
In this example, if you An independent point with more than one child can be split into
want to split the circle multiple points, one for each child. For example, a point that is the
center from the
center of a circle and is also the endpoint of two segments can be split
segment endpoint
while leaving the two so that the circle center and segment endpoints are now three separate,
unrelated points. Select the point you wish to split and choose Split Point
segments with a
common endpoint,
from the Edit menu. The point splits into two or more separate points
first split the point
a small distance from each other.
apart, then select the
two segment
endpoints and choose
Merge Points.
To split a point apart, you must select an independent point with two
or more children.
Split Point
Merging Two Points
Merging two points is Two separate points can be combined into a single point using the
a handy way to fix any Merge command. Select an independent point and the point to which
mistakes you make
you want to merge it. One of the points must be independent so that
while using drawing
it’s free to merge with the other point. The other point doesn’t have to
tools.
be independent, but it must not depend on the first. (If the second
point could depend on the first, after merging, it would be defined in
terms of itself!)
Merge Points
Merging a Point to a Path
A point can be merged to a path (straight object, circle, arc, interior,
point locus, or function plot) using the Merge command. Select an
independent point and a path that doesn’t depend on that point. The
point must be independent, and the path must not be dependent on the
point. In this way, for example, an endpoint of a segment can be joined
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117
Edit Menu
or merged to another segment. After it’s merged, the point is attached
to the segment and can move along the segment but cannot leave it
(unless you Split it from that segment).
Merge Point To Segment
Merging Text
You can merge separate text elements—where at least one is a
caption—into a single “sentence.” For example, you can merge a
caption and two measurements together to read “The model is 5.2 cm
tall and 3.1 cm wide.” Select the text you wish to merge (captions,
measurements, or labeled objects) in the order you wish it read and
choose Merge Text from the Edit menu. Sketchpad consolidates the
separate pieces of text into a single caption. The resulting composite
caption includes the text of selected captions, the labels of selected
labeled objects, and the values of selected measurements and
calculations. The order in which the objects were selected determines
the order in which they appear in the final composite caption.
Splitting Merged Text
To return a single merged caption to its component parts, select the
merged caption and choose Split Merged Text from the Edit menu.
Sketchpad splits the caption back into the parts you originally merged.
2 . m AB = 5 .2 c m
mAB = 5.2 cm
4 . m AC = 3 .1 c m
1 . T he m odel is
mAC = 3.1 cm
→
The m odel i s 5 .2 c m tall and 3.1 cm wi de!
→
The m odel is
3 . ta ll a nd
tal l and
5 . wi de!
wide!
Select the text objects in the order shown at left and choose Merge Text to
create the center caption. Select the merged text and choose Split Merged Text
to return the individual elements to their original positions (right).
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See also: Path Objects (p. 13), How to Use Split and Merge to Explore Constructions
(p. 119), Object Relationships: Parents and Children (p. 10), Composite Captions
(p. 36), Advanced Text Topics (p. 226)
How To . . . Use Split and Merge to Explore Constructions
The Split and Merge commands are often useful when you’re
investigating the behavior of a particular construction and you want to
see what happens in the case of a slightly different construction. In this
extended example, you’ll construct the diagonals of a general
quadrilateral, then investigate what happens when you turn the
quadrilateral into a parallelogram, a rectangle, or a rhombus.
To begin, construct quadrilateral ABCD, its two
diagonals, and the point of intersection E of the
diagonals. Then measure the distances and angle
shown in the figure at right. Drag the vertices
around to see if there’s any relationship between
the various measurements.
Now turn the quadrilateral into a parallelogram.
Construct two parallel lines as shown at right:
one through D parallel to the segment from A
to B and another through B parallel to the
segment from A to D. Also construct the
intersection of the parallels.
C
D
E
B
A
EA = 2.25 cm EC = 1.52 cm
EB = 2.40 cm ED = 0.81 cm
m∠AEB = 100°
C
D
E
A
B
Select both point C and the intersection, and choose Merge Points to
merge the vertex with the intersection point.
Hide the parallel lines. Then drag vertices A, B, and D. How are the
measurements now related?
Next turn the quadrilateral into a rectangle by
making ∠BAD into a right angle. Construct a
perpendicular as shown at right and Merge
point D to the perpendicular. Again, drag the
vertices and watch the measurements. Do you
notice anything different?
The Geometer’s Sketchpad Reference Manual
D
C
E
A
B
119
Edit Menu
Finally, change your quadrilateral into a
rhombus. First Split D from the perpendicular.
Then construct a circle centered at A and
passing through B, and Merge D to the circle.
Drag the vertices once more to see what
relationships the measurements now reveal.
D
C
E
A
B
Edit Definition
Editing definitions
allows you not only to
correct any mistakes
you might make, but
also to explore a
mathematical model
over an unlimited
range of cases.
This command allows you to edit the definition of a selected
calculation, function, numeric parameter, or plotted point. If you select
a parameter, a calculation or a function, the Calculator appears,
allowing you to modify the value or expression. If you select a plotted
point, the Plot Points dialog box appears, allowing you to change the
point’s coordinates.
This command’s name changes to match the selected object.
Select:
For this command:
One calculation
Edit Calculation
One function
Edit Function
One parameter
Edit Parameter
One plotted point
Edit Plotted Point
You can also double-click any of these objects with the Arrow tool as a
shortcut for Edit Definition.
See also: Calculator (p. 49), New Parameter (p. 206), Calculate (p. 198), New Function
(p. 207), Plot Points (p. 205)
Properties
The keyboard
shortcut for Properties
is Alt+? (Windows) or
a+? (Mac).
120
This command allows you to change a variety of properties of a single
selected object. Select one object and choose Properties from the Edit
menu to display a dialog box that allows you to alter the object’s
properties.
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Edit Menu
As an alternative, you
can use the Context
menu to access
Properties. In
Windows, right-click
on the object. On a
Mac, hold down the
Ctrl key while you
click on the object.
Properties is the
command that
appears directly under
your mouse in the
resulting Context
menu.
Keep in mind that for
any specific object,
only a few panels
appear—not all of
them!
The Properties dialog box is arranged into separate panels of related
properties. Switch from panel to panel by clicking on the tabs near the
top of the dialog box. Which panels are available depends on the type
of object selected.
While you’re modifying one object’s properties, you can switch to a
different object by clicking on that object in the sketch. (If the object to
which you want to switch is obscured by the Properties dialog box,
you’ll first have to move the Properties dialog box aside to reveal the
object.)
When you finish modifying an object’s properties, click OK to make
the changes permanent or click Cancel to leave the object with its
original properties. When you click in the sketch to switch to a different
object, your changes for the original object are made permanent before
Sketchpad switches to the new object, just as if you’d clicked OK for
that original object.
The following sections describe all of the possible Property panels, as
well as the types of object for which each panel appears.
See also: Objects (p. 8), Context Menu (p. 213)
Object Properties
All objects have an
Object Properties panel.
In the panel, the
object’s geometric
definition is described,
usually in terms of its
relation to its parents
(the objects that
geometrically define the
object).
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Use the Parents and
Children menus to
navigate the family
tree—that is, to find
an object’s ancestors
or descendants. This
is a good way to learn
how a particular
object was
constructed, to locate
a particular object in a
complicated sketch,
or to display one
particular hidden
object.
Q: If an object is no
longer arrowselectable, how do
you select it again to
change its properties
back to being arrowselectable?
A: Click it with the
right mouse button
(in Windows), or hold
down the Ctrl key
(Mac) to access
Properties via the
Context menu.
Parents, Children: Click on either the Parents or Children pop-up menu
to see the parents or children of the current object. As you move through
either of these lists, the corresponding object in the sketch is highlighted.
Choose an object from either the Parent or Child pop-up list to switch to
showing the properties of this related object. When you switch to a
different object in this way, any changes you’ve made in the properties of
the original object become permanent.
Hidden: Use this checkbox to determine whether the object is hidden or
visible.
Arrow Selectable: Use this checkbox to determine whether the object
can be selected by clicking on it with the Arrow tool. Normally, you’ll leave
this property checked. If you clear the checkbox, the object will no longer
be selected when clicked by the Arrow tool or when you use the selection
rectangle. This can be handy when you’re working with something like a
pasted picture that you want to use as the backdrop of a geometric
measurement activity. You don’t want to accidentally select and drag this
picture while working “on top” of it, so you may wish to make it not
arrow-selectable.
See also: Hide Objects (p. 148), Show All Hidden (p. 148), Object Relationships: Parents
and Children (p. 10), Selecting Objects Using a Selection Rectangle (p. 71)
Label Properties
All points, straight objects,
arcs, interiors, point loci,
function plots, and
measurements have a Label
panel.
Use the Label panel to
change an object’s label, to
change whether and how
the label is displayed, and to
change whether the label is
used in custom tools.
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If you enter a label
that ends in
brackets—such as
A[1]—the bracketed
quantity will display as
a subscript in your
sketch. In other
words, type A[1] in
Label Properties to
display A1 in your
sketch.
Label: This box displays the label of the selected object. It will be blank if
a label has not yet been assigned to the object. Type a new label to change
the object’s label.
Style: Click on this button to display a dialog box in which you can adjust
the font, size, and typeface in which the label is displayed.
Show Label: Use this checkbox to hide or show the object’s label. (This
checkbox is unavailable for objects that cannot display separate labels,
such as measures.)
The final checkbox in Label Properties differs depending on the
context. It appears as Use Label in Custom Tools for a sketch object,
but for a custom tool object appears as either Automatically Match
Sketch Object (for a given) or Use Label in Sketches (for a step).
Use Label in Custom Tools: Use this checkbox to determine whether
the label of a sketch object will be used in custom tools. Normally,
custom tools assign new and unique labels to the objects they create. So, if
you are defining a custom tool that includes an object with a special
label—such as “hypotenuse” or “orthocenter”—and you want to
duplicate that label whenever the tool is used, you should check this
option. This box appears only when viewing properties for a sketch
object.
Use Label in Sketches: Use this checkbox to determine whether or not
the label of a custom tool step object will be used in sketches. With this
box unchecked, the custom tool assigns a new and unique label when it
constructs this step. When the box is checked, the step’s label is used
whenever a sketch object is constructed from this step. This box appears
only when using the Script View to view properties of a custom tool step
object.
To view the
properties of a
custom tool object,
you must make that
tool the chosen tool,
and choose Show
Script View from the
Custom Tools menu.
Automatically Match Sketch Object: Use this checkbox to determine
whether or not a custom tool given object will be automatically matched
to an object with the same label in the sketch. This box appears only
when using the Script View to view properties of a custom tool given
object. When this box is checked, the object appears in Script View as an
assumed given. When not checked, the object appears in Script View as a
normal given.
See also: Text Tool (p. 86), Text Palette (p. 55), Custom Tools (p. 90), Show Script View
(p. 96), Advanced Tool Topics (p. 230), Custom Tools Menu (p. 91)
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Value Properties
Measurements,
calculations, and
parameters have a
Value panel. Use this
panel to set the
precision and the
display name of these
objects and to set the
value of parameter
objects.
Use Units Preferences
to set the default
precision used for
newly created values.
Precision: In this
box you can set the
precision with which the measurement is displayed. Choices range from
units to hundred-thousandths. This setting determines only how
Sketchpad rounds the value, when the value is displayed on-screen.
(Sketchpad stores the actual value with considerably more accuracy, and
this setting doesn’t cause any loss of accuracy in the value Sketchpad uses
internally.)
Display With: Use this to set the on-screen name that will appear before
the value. For instance, if you’ve measured the length of the segment
from A to B and changed the measurement’s label to Length, the three
possible ways to display the value are illustrated here.
Original Name: m AB = 4.00 cm
No Name: 4.00 cm
Current Label: Length = 4.00 cm
You can also choose
Edit Parameter from
the Edit menu or
double-click with the
Arrow tool to change a
parameter’s value.
Parameter: In this box you can set the value of a parameter. (This is
available only for parameters, not for measurements and other
calculations.)
See also: Measurements, Calculations, and Parameters (p. 18), Edit Parameter Definition
(p. 120),Units Preferences (p. 136) Accuracy vs. Precision (p. 137)
Plot Properties
Only loci and function plots have a Plot panel. Both panels allow you
The default number
of samples assigned
to set the number of samples used to plot the locus or function. The
to newly constructed
greater the number of samples, the smoother and more accurate the
loci and function
plots is determined by plot is. However, the greater the number of samples, the slower
Sampling Preferences. Sketchpad can be in calculating and displaying the plot.
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Plot Properties for a Locus
Use this panel to set the
You can adjust the
number of samples in number of samples in a
a locus or function
locus. If the locus is a point
plot without going to
locus, you can also use this
Properties by
selecting the sampled panel to determine whether
object and pressing
the locus is displayed in
the + or – key while
continuous or discrete
your sketch window is
form. The illustration that
active.
follows shows an elliptical
point locus displayed in continuous form on the left and in discrete
form on the right.
You can also set the
domain of a plotted
function by dragging
the arrowheads that
appear at the
endpoints of the
function plot.
When entering limits
for the domain, you
can use mathematical
expressions like 2*3
and π/4. (Windows
users: Press p for π.
Mac users: Press
Option+p.)
Plot Properties for a Function Plot
Use this panel to set the
domain of a plotted
function, to set the
number of samples used
to display the function,
and to determine
whether the function
plot is displayed in
continuous or discrete
form. By default,
Sketchpad plots new
functions over a domain
corresponding to the
width of the screen.
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The illustration that follows shows a function plot displayed in
continuous form on the left and in discrete form on the right.
1
-2
1
2
-1
-2
2
-1
See also: Loci (p. 24), Functions and Function Plots (p. 27), Sampling Preferences
(p. 140)
Parameter Properties
Use this panel to
change the default
animation behavior of a
parameter. The settings
on this panel determine
how the parameter’s
value changes when you
select the parameter and
either press the + or –
key to change the value
manually or choose
Animate from the
Display menu. These
settings are also used as the initial settings when you create an
Animation button that animates the parameter.
Continuously/Discretely: Check the Discretely radio button to make
the parameter’s value jump by the amount in the units box each time it
changes. Check Continuously to make the value change gradually instead
of by jumps.
units per sec: The numbers here determine how quickly the parameter’s
value changes. This rate is not exact; if your computer is very busy doing
other tasks, the parameter may change more slowly than the rate you
specify here.
Domain: These numbers determine the minimum and maximum values
of the parameter during animation. You can still change the value directly
(by double-clicking with the Arrow tool or by choosing Edit Parameter from
the Display menu) to any value you like, no matter what these domain
limits are.
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Keyboard (+/–) Adjustments: This number specifies the amount by
which the parameter changes when you select it and press the + or – key
on the keyboard.
See also: Measurements, Calculations, and Parameters (p. 18), Animation Button
(p. 112), New Parameter (p. 206), Edit Parameter Definition (p. 120), Animate (p. 153)
Table Properties
This panel appears only for tables created using the Tabulate command.
Use it to determine whether the last row of the table dynamically
changes values as the tabulated measurements change values.
See also: Tables (p. 34), Tabulate (p. 209)
Hide/Show Properties
This panel appears only for Hide/Show action buttons. Use it to
determine the type of action the Hide/Show button performs as well as
various aspects of how it works.
If you don’t change
the default label of a
toggling Hide/Show
button, the label
changes between
Hide and Show to
describe the action it
will perform next.
Action: Choose Always
Show Objects to make
this button into a Show
button, which always
shows the objects to
which it applies. Choose
Always Hide Objects to
make this button into a
Hide button, which
always hides the objects
to which it applies.
Choose Toggle Between
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Hide and Show to make this a toggling Hide/Show button, which hides
objects when clicked on if one or more of its objects are showing and
shows its objects when clicked on if they are all hidden.
Effects: If Select Objects After Showing is checked, when you click on a
Show button all its parental objects are selected. (Even if some of the
button’s objects are already showing, they are still selected.) If unchecked,
the objects are left unselected after they’re shown.
If Fade Objects In or Out is checked, objects fade in or out of visibility
gradually when you click on the Hide/Show button. If unchecked, objects
appear or disappear immediately.
See also: Action Buttons (p. 37), Hide/Show Button (p. 112), Hide Objects (p. 148),
Show All Hidden (p. 148), Object Relationships: Parents and Children (p. 10)
Animate Properties
This panel appears only for Animation action buttons. Use it to
determine how each animated object moves.
The list at the top describes the motion of each animated object. If the
button animates an object that can move only by moving its parents,
the list shows the parents rather than the object itself. (For instance, if
you animate the segment from point A to point B, the list shows A and
B rather than the segment.) When you highlight an object in the list, the
details of that object’s animation settings appear in the lower part of the
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panel, and you can modify the settings to change how the animation
works.
Animating points or
parameters randomly
is particularly useful
for investigations in
probability, statistics
and chaos.
Direction: For parameters and for points that are animated on paths, you
can set the direction of the motion. The available choices depend on the
kind of path and the kind of objects. For instance, if the path is a circle or
circle interior, the first two choices are counter-clockwise and clockwise.
If the path is a straight object, arc, or polygon interior, the first two
choices are forward and backward. If the animated object is a parameter,
the first two choices are increasing and decreasing.
If you choose random direction for a point on a path, the point moves to
a new randomly chosen position on its path each time it moves. Similarly,
if you choose random direction for a parameter, the parameter takes on a
new random value within its domain each time it moves.
The Once Only check
box is not available
when a point or
parameter is moving
bidirectionally.
Once Only: For a parameter or a point on a path which is moving
randomly, you can check Once Only to stop the animation after it
generates a single new random position or value. For points on paths that
aren’t moving randomly, you can check Once Only to stop the animation
when the point returns to its starting position. For parameters which
aren’t moving randomly, you can check Once Only to stop the animation
when the parameter returns to its starting value.
Speed: This portion of the dialog box appears only for points, and allows
you to set the speed of an animated point to slow, medium, fast, or some
other desired value.
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The rate at which a
parameter changes its
value is not exact.
Sketchpad tries to
change the value at
the designated rate,
but if your computer
is busy with other
tasks, the parameter
may change more
slowly than the rate
set here.
Change Value: This portion of the dialog box appears only for
parameters, and allows you to determine how a parameter’s value changes.
You can determine whether the value changes continuously or discretely
(jumping by an increment you set each time the value changes). You can
determine how quickly the value changes, and the domain within which
the value can vary.
See also: Animation Button (p. 112), Animate (p. 153), Measurements, Calculations, and
Parameters (p. 18), New Parameter (p. 206), Object Relationships: Parents and Children
(p. 10)
Move Properties
This panel appears only
for Movement action
buttons, and allows you
to set their speed and
behavior.
You can set the
movement speed to
slow, medium, fast, or
instant.
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If the destination
point is moving, it’s
possible that the
moving point will
never reach the
destination and will
keep moving forever.
You can take
advantage of this
“perpetual motion” to
model kinematic
systems.
If the destination point moves while the Movement button is active,
you can decide how the moving point will travel. Choose Follow
Moving Destination to have the moving point alter direction as the
destination point moves, always continuing to move toward that point.
Choose Move Toward Initial Destination to have the moving point
travel in a straight line to the location of the destination point at the
instant the Movement button was clicked, stopping when it reaches
that initial destination.
See also: Action Buttons (p. 37), Movement Button (p. 113)
Presentation Properties
This panel appears only for
Presentation action buttons.
A Presentation button
presents the actions of a set
of other parental buttons.
Present Actions: This
choice is available only
when the Presentation
button presents more than
one other action. (In other
words, it’s available when
the Presentation has more
than one parental action
button.)
Choose Simultaneously to
activate all actions of the presentation at the same time. Pressing a
simultaneous Presentation button has the same effect as activating all of
its parental action buttons at once. If you choose Simultaneously, you can
also set stopping conditions for the Presentation (see Stop After, below).
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When you present a
parental Animation
button sequentially,
Sketchpad waits for
the animation to
complete before
continuing the
sequenced
presentation. If the
animation doesn’t
complete on its own,
you can stop it
yourself either by
releasing the pressed
Animation button, or
by choosing Stop
Animation from the
Display menu while
the animation is
continuing. Once the
animation stops, the
sequenced
presentation resumes.
Choose Sequentially to activate the actions of the presentation one after
another in the order in which they were selected when the button was
created. Pressing a sequential Presentation button has the same effect as
pressing the parental action buttons one at a time, waiting for each
activated action to be completed before proceeding to the next parental
action. If you choose Sequentially, you can also specify a pause between
presented actions (see Pause Between Actions, below).
Before Starting: Check these options to specify any additional effects
you’d like to occur at the moment the Presentation button is pressed.
Based on your choices, Sketchpad will deselect any previously selected
objects, stop any previously started animations, and erase any previously
displayed traces before commencing the presentation.
Stop After: This choice is available only when the presentation presents
actions simultaneously. When Stop After Last Action Stops is chosen,
each of the presented actions is allowed to proceed independently, and
the presentation completes only when the last presented action has
finished. If you choose Stop After First Action Stops, all of the presented
actions stop as soon as the first presented action stops. (This choice can
be useful for coordinating two or more animation or movement actions.)
When you choose Stop After Elapsed Time, the dialog box allows you to
enter an overall duration in seconds for the presented actions. (This
choice is useful when you want to stop an animation after a fixed amount
of time.)
Pause Between Actions: This choice is available only when the
presentation presents actions sequentially. Enter the amount of time (in
seconds) you want to pause between each step of the sequence. If you
enter zero, each step of the presentation commences as soon as the
previous step completes. If you enter a nonzero pause, Sketchpad waits
between steps.
See also: Action Buttons (p. 37), Presentation Button (p. 113), Object Relationships:
Parents and Children (p. 10)
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Link Properties
This panel appears only for
Link buttons. Use the Link
To radio buttons to choose
whether the button links to
another page in your
document or to an Internet
URL such as a web page.
Use the Document
Options command
from the File menu to
add or copy new
pages into your
document.
Page: When this is
chosen, the Link button
will link to a different
page of the current
document. Use the Page
pop-up menu to decide
which page to link to. If that page contains any action buttons of its own
(such as an Animation or Movement button), you can choose one of
those buttons in the Button On Page pop-up menu to cause the Link
button to automatically activate the specified button on the linked page.
The default URL
points to the
Sketchpad Resource
Center, a web site
with additional
resources and
information about
using Sketchpad.
Copy web page URLs
from your browser
and paste them here.
URL: When this is chosen, the Link button will link to a URL using your
web browser. You must enter the actual URL, which can be a web site (if
the URL starts with “http://”) or a local file (if the URL starts with
“file://”).
URL links can also use these three special URL forms:
sketchdoc://: Start the URL with “sketchdoc://” to link to a document
in the same folder as the current document. For example, if you want to
link from a document named Demo1.gsp to a document called Demo2.gsp in
the same folder, you could use the URL “sketchdoc://Demo2.gsp”.
sketchapp://: Start the URL with “sketchapp://” to link to a document
in the same folder as the Sketchpad application.
help://: Start the URL with “help://” to link to a document in the
Sketchpad Help folder.
Internet URLs such as http:// web sites are only accessible if you’re
connected to the Internet and have a web browser installed on your
computer, of course. Local file:/// URLs are useful for accessing files or
other resources on your specific computer, but these buttons may not
function if you open your document on another computer (which may
well be missing the linked-to files). The relative URLs—sketchdoc://,
sketchapp://, and help://—let you refer to resources in relation to
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“known locations” on any computer running Sketchpad. They can be
handy if you’re creating a folder of linked documents you’d like to share
with others. As long as you use sketchdoc:// URLs, the documents
within that folder will stay linked together no matter where you move or
copy that folder.
See also: Action Buttons (p. 37), Link Button (p. 114), Document Options (p. 104)
Scroll Properties
This panel appears only
for Scroll action
buttons. A Scroll button
scrolls the window in
which it’s located to
show a specific portion
of that window. The
Scroll button is based
on a point, and the
scrolling action can
scroll in one of two
ways: so that the
parental point is at the
top left corner of the window or so that the parental point is centered
in the window.
See also: Action Buttons (p. 37), Scroll Button (p. 114), Object Relationships: Parents
and Children (p. 10)
Iteration Properties
Only iterated images
and iteration rules have
an Iteration panel. Use
this panel to set the
number of iterations, to
specify whether the
iterations display all
levels or only the final
iteration, and to
determine how random
points in the iteration
behave.
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You can adjust the
number of iterations
without going to
Properties by
selecting an iterated
image and pressing
the + or – key while
your sketch window is
active.
Number of Iterations: This number determines how many times the
iteration is repeated. The minimum value you can use is 1; the maximum
value depends on the iteration and is smaller for complex iterations that
involve more than a single map. If the number of iterations—the iteration
depth—was defined by a measurement or calculated value when the
iteration was first created, the current depth is displayed here but cannot
be edited.
You can randomize
an iteration without
going to Properties by
selecting an iterated
image and pressing
the ! key while your
sketch window is
active.
Move Iterated Points on Objects: This choice appears only for
iterations in which one or more of the pre-images are mapped to a point
on the path. It determines how iterated images of such points behave. Set
this to Same Locations in order to have each iterated image appear in the
same relative location on its path as the first image does on its original
path. Set this to Random Locations to have each iterated image appear at
a random location on its path. When you’ve set this to Random
Locations, the Randomize Now button is enabled; you can click this
button to assign each image point on the path a new random position.
Display As: Set this to Full Orbit to display all the iterated images (the
images for every level of the iteration). Set this to Final Iteration Only to
display only the images at the final level, as set by the Number of
Iterations.
See also: Iterations and Iterated Images (p. 31), Iterate (p. 181), Keyboard Reference
(p. 215), Parametric Depth (p. 189)
Preferences
Use the Context
menu as a shortcut to
Preferences. Rightclick (Windows) or
Ctrl+click
(Macintosh) in a
blank area to invoke
the Context menu.
This command allows you to change a variety of settings that determine
how Sketchpad works.
If you want some
changes to apply only
to the current sketch
and some to apply
only to new sketches,
you’ll need to open
the Preferences dialog
box twice.
When you make changes in
Sketchpad’s Preferences, you can
use the checkboxes at the bottom
of the dialog box to decide whether your changes will apply to the
current sketch only, to new sketches only, or to both the current sketch
and new sketches. Choose Apply To: This Sketch to have your changes
affect newly constructed objects in the current sketch. Click Apply To:
The Preferences dialog box normally includes three panels: Units,
Color, and Text.
An Advanced Preferences command allows you to control additional
aspects of Sketchpad’s operation that need to be changed rarely, if ever.
That command is described in the next section.
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New Sketches to have your changes affect all new sketches (including
new blank pages you add to the current document).
See also: Units Preferences (p. 136), Color Preferences (p. 138), Text Preferences
(p. 139), Advanced Preferences (p. 139), Context Menu (p. 213)
The unit settings on
this panel also affect
the units in which you
specify angles and
distances for
translations and
rotations in other
dialog boxes.
Units Preferences
The settings on this
panel control the units
and precision
Sketchpad uses to
display measurements
and calculations. For
example, depending
on these settings, a
segment’s length
measurement may
appear as “2.54 cm”
or “1.0 in.”
The units set on this panel apply to all measurements and calculations
in the sketch. The precision applies only to newly created
measurements and calculations. Use Value Properties to change the
precision of an existing measurement or calculation.
The choices for Angle Units are degrees, directed degrees, or radians.
With directed degrees
or radians, ∠ABC has
the opposite measure
as ∠CBA. With
“plain” degrees, both
of these angles have
the same measure.
Directed degrees are
useful in
transformational
geometry, where the
direction as well as the
magnitude of an angle
is important.
•
•
•
Measurements in degrees are always
positive and range from 0° to 180°.
Measurements in directed degrees are
positive for counter-clockwise angles and
negative for clockwise angles. Directed
degrees range from –180° to 180°.
Measures of angles in radians are
always directed and range from –π
to π.
B
C
A
m∠ABC = 20°
B
C
A
m∠ABC = 20°
B
m∠CBA = 20°
m∠CBA = -20°
C m∠ABC = 0.11π radians
A m∠CBA = -0.11π radians
Set the Distance Units to cm, inches, or pixels. A pixel is a single dot on
the computer screen and usually corresponds to about 1/72 inch on
Macintosh and 1/96 inch in Windows.
Sketchpad’s distance measurements are accurate when printed, but may
not be exactly accurate on the screen, depending on the size of your
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monitor and on the resolution you’ve set in your computer’s control
panel. In the event that you require exactly accurate on-screen distance
measurements, use the System Preferences panel to set the precise
number of pixels per inch or per centimeter on your screen. This panel
is available only in Advanced Preferences.
The choices for each of the three Precision settings are units, tenths,
hundredths, thousandths, ten-thousandths, and hundred-thousandths (0, 1, 2, 3, 4,
and 5 decimal places, respectively). These settings only affect how
numbers are displayed, not how they are represented internally.
See also: System Preferences (p. 141), Accuracy vs. Precision (p. 137), Value Properties
(p. 124)
User Tip
Accuracy vs. Precision
Be careful not to
confuse accuracy with
precision. If you are
displaying distances
precise to the nearest
tenth, Sketchpad can
accurately represent
the sum of two
lengths measuring 1.4
as
1.4 + 1.4 = 2.8.
If your precision is set
to round displayed
values to the nearest
unit, however, the
sum may seem
nonsensical:
1 + 1 = 3.
The accuracy of a measurement in Sketchpad refers to how close the
measured value is to the ideal, “correct” value. The precision of a
measurement in Sketchpad refers to the number of decimal places used
when the value is displayed on the screen.
The accuracy of Sketchpad’s measurements and calculations is
determined by the full computational power of your computer. Initial
computations are usually accurate to about 15 significant digits. For
instance, the square root of 2 is stored internally as 1.41421356237310.
Calculations based on these initial values may be less accurate, because
errors in the accuracy compound across calculations.
The precision of displayed measurements is determined by your choice
in Units Preferences. Sketchpad rounds calculated values to your
chosen precision when displaying them.
See also: Sketchpad’s Internal Mathematics (p. 246)
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Edit Menu
Color Preferences
The settings on this
panel control the
updating of color
preferences, the
default colors of new
objects you create, the
background and
selection colors used
in your sketch, and the
fading behavior of
traces.
Even if Update
Automatically When
Applying New Colors
is checked, you can
prevent the
preferences from
being changed by
holding down the
Shift key while you
change an object’s
color.
Update Automatically When Applying New Colors: When this box is
checked, the preferences for your sketch are automatically changed
whenever you color an object in the sketch. For example, if you color a
line green and then construct a new segment, the new segment will also be
colored green. Remove the check to have the preferences remain
unchanged no matter how you modify the colors of objects in your
sketch.
Object and Sketch Colors: The first six color rectangles show the
default colors for various types of objects in Sketchpad: points, lines and
other straight objects (segments, axes, and rays), circles and other curved
objects (arcs), interiors, loci, and plots. New objects are assigned colors
according to these settings. The last two color rectangles show the colors
for selection markers and for the background of the sketch itself.
To change any of these colors, click the color rectangle you want to
change. Your system’s Color Picker dialog box appears, allowing you to
choose a new color.
Fading traces look
best if your monitor is
set to display more
than 256 colors.
Fade Traces Over Time: Use this checkbox and speed control to
determine whether or not traces fade and how fast they fade. When the
box is checked, traces fade gradually over time, so recent traces are more
vivid and older traces are fainter. If you remove this check, traces will
never fade, and you’ll have to remove them by using the Erase Traces
command (or by pressing the Esc key). When fading is turned on, the
speed control determines how quickly the traces fade.
See also: Trace (p. 151), Erase Traces (p. 152), Color (p. 8) Color Picker (p. 64)
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Text Preferences
These settings control when objects are labeled, and whether or not the
Text Palette appears automatically.
Even if you don’t
show labels
automatically, you can
show (or hide)
individual objects’
labels by using the
Text tool or Show
Labels.
Show Labels Automatically: Check For All New Points to show the
labels of new points when they’re created. Check As Objects Are
Measured to show labels when you measure an object. As you make a
measurement, the labels needed to name the measurement will also
appear. For example, if you measure the length of a segment between two
unlabeled points, the segment endpoints’ labels will also appear.
Show Text Palette When Editing Captions: If this box is checked, the
Text Palette appears whenever you edit a caption and disappears when
you finish editing. You can also display and hide the Text Palette by using
the Show Text Palette command.
See also: Text Tool (p. 86), Show/Hide Labels (p. 149), Show/Hide Text Palette
(p. 154), Text Palette (p. 55)
Advanced Preferences
This command allows you to change a variety of advanced settings that
determine how Sketchpad works. Change these settings only after
careful consideration; most users never need to modify them or need to
modify them only once.
To use this command, hold down the Shift key before activating the
Edit menu, causing the Preferences command to become Advanced
Preferences.
Export Preferences
These settings control how sketches are printed and how Sketchpad
objects are copied onto the clipboard.
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Edit Menu
Include Arrowheads: If this box is checked, printouts and pictures
copied to the clipboard include arrowheads at the ends of lines, rays, and
infinite point loci. If this box is not checked, the ends of these objects
appear on printouts and on the clipboard just as they do in the sketch—
with no special markings.
Locus/Plot Export Quality: The value displayed here determines how
loci and function plots appear in printouts and on the clipboard. If the
setting is 1x, such objects appear just as they do on the screen. Set the
value to 5x to use five times as many samples or to 10x to use ten times as
many samples. The default setting of 5x works well on most printers; you
may want to change this to 10x for a very high-resolution printer. The
higher the number of samples, the smoother the appearance of the curve.
Clipboard Image Scale: This value determines the magnification used
for clipboard pictures. Set this to more magnification if you’re copying
pictures from Sketchpad to paste into a document in a word processor or
page layout program that you intend to use for high-quality production. If
you set the scale to more than 100%, the pasted picture will be large, and
you’ll have to scale it down in the word processing or page layout
application into which you paste it. The result will be a higher-quality
printout from the other application.
See also: Advanced Graphics Export (p. 234), Cut (p. 110), Copy (p. 110)
Sampling Preferences
These settings control the number of samples used for loci, function
plots, and iterations. In general, the more samples, the more accurate or
detailed the object appears, but the slower it is for Sketchpad to
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Edit Menu
compute and draw. If your computer is faster than most, you may want
to increase the values of these settings; if it’s slow to draw and
recalculate loci, plots, and iterations you may want to decrease these
values.
You can adjust the
number of samples
used in any specific
locus or function plot
by visiting its Plot
Properties.
Number of Samples for New Point Loci: This value determines the
number of samples used in a newly created point locus. (The number of
samples in a new locus of a nonpoint object is proportionate to, but
smaller than, this number.)
Number of Samples for New Function Plots: This value determines
the number of samples used in a newly created function plot.
Maximum Number of Locus/Plot Samples: After you’ve created a
locus or function plot, you can use Properties to change the number of
samples. This value determines the largest number you can use in the Plot
Properties panel when you’re changing the number of samples for such an
object.
Maximum Number of Iteration Samples: This value limits the number
of samples allowed in an iterated image. For an iteration using a single
map, this is the maximum number of iterations. For an iteration using
more than one map, this value limits the depth in such a way that the total
number of iterated images of a single object never exceeds this number.
See also: Loci (p. 24), Functions and Function Plots (p. 27), Locus (p. 166), Iterate
(p. 181), Advanced Preferences (p. 139), Plot Properties (p. 124), Iteration Properties
(p. 134)
System Preferences
These settings correlate Sketchpad’s behavior with your particular
computer, allow you to edit Sketchpad’s color menu, and allow you to
change all preferences back to their original values.
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Edit Menu
This value is “ideal”
in the sense that
Sketchpad attempts to
reach it. If you’re
animating a
particularly complex
sketch or have a
slower computer,
animations may be
slower than this.
Ideal Normal Speed: This value sets the approximate speed represented
by a value of 1.0 in the Motion Controller, or “normal” speed for a
Movement button. You should modify the speed only if your computer is
consistently too fast or too slow for your tastes when animating objects at
a speed of 1.0 in the Motion Controller.
The Symbol font
comes pre-installed
on your computer.
Math Symbol Font: Sketchpad needs a symbol font so that it can display
certain mathematical symbols such as π and θ. These symbols are
normally found in the font called Symbol. If the Symbol font is not present
on your computer, you can enter the name of a different font that
contains the required symbols. If the Symbol font is missing or the
alternative font you specify doesn’t contain the needed symbols,
Sketchpad won’t be able to display those symbols correctly.
142
Screen Resolution: This value sets the correspondence between the
pixels on your computer screen and real-world length units. The normal
value is 96 pixels/inch for Windows computers and 72 pixels/inch for
Macintosh. You should modify this value only if it’s absolutely necessary
that on-screen distances match Sketchpad’s measured distances. Changing
this value may result in discrepancies when you paste Sketchpad pictures
into other applications, which assume your screen resolution is fixed at 96
or 72 pixels/inch.
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On Macintosh
computers, antialiased graphics are
available only with
OS X. When
Sketchpad is installed
normally on a
Windows computer,
anti-aliased graphics
are available on
Windows XP, 2000,
98, and Me, but not
on Windows 95 or
NT4.
Anti-Aliased Graphics: (This option is available only on newer
computers). This setting determines the quality of the graphics displayed
in your sketches. Sketchpad’s normal graphics quality balances the need
for both mathematical accuracy and drawing speed. On newer computers,
however, you may choose high-quality anti-aliased graphics, which display
smoother lines and curves but take considerably longer to produce. For
simple drawings or on very fast computers, the higher quality is usually
worth the speed tradeoff. But on more complicated sketches or on slower
computers, anti-aliased graphics can cause Sketchpad to feel less
responsive, especially when you are animating or dragging objects.
Depending on your visual tastes and the speed of your computer,
therefore, you can choose any of three settings for anti-aliased graphics:
Never, When Speed Permits, or Always. Choose Never for fastest
performance. Choose Always for highest-quality, at the expense of speed.
Choose When Speed Permits to have Sketchpad switch back and forth
between anti-aliased and normal graphics depending on the complexity of
the sketch and on your computer’s speed.
Edit Color Menu: This button displays your system’s Color Picker dialog
box, which allows you to change the colors in the Color menu. You can
use this feature to add your favorite shades of magenta, teal, or chartreuse.
If you create a sketch with a dark or black background, you may want to
add white to the Color menu. (To edit the Color menu in Windows, see
p. 144 for step-by-step instructions.)
Reset All Preferences: This button allows you to reset all Sketchpad
preferences to their original values. Use it with care: you’ll lose all the
preferences values you’ve set in any other way.
See also: Motion Controller (p. 41), Animation Buttons (p. 38) Animate (p. 153),
Animation Button (p. 112), Movement Button (p. 113), Color (p. 146), Color Picker
(p. 64)
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Edit Menu
Editing the Color Menu in Windows
When you click the Edit Color Menu button on the Advanced
Preferences System panel, the Edit Color Menu form of the Color
Picker dialog box appears. Use this dialog box to set the colors that
appear in Sketchpad’s Color menu. The colors currently in the Color
menu appear under Menu Colors.
1. Click one of the Menu Colors boxes to choose which menu color
2.
3.
4.
5.
to change.
Use the HSL controls, the RGB controls, or the color box and
slider on the right to set a new color.
Click Set New Color to change the menu color to your new color.
Click another of the Menu Colors to choose a different menu color
to change.
Click OK when you’re done, or Cancel to leave the Color menu
unchanged.
See also: System Preferences (p. 141)
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Display Menu
Display menu commands allow you
to control the appearance of objects
in your sketch and of the tools you
use to work with them.
With these commands you can greatly
enhance the visual appeal of a sketch
and its effectiveness in
communicating the mathematics it
embodies. Using appropriate line
widths and colors and hiding some
objects while showing others help
focus attention on the important parts
of a sketch. Properly styled labels and
captions help describe the purpose of
the sketch and the mathematics
behind it. Tracing and animation
create dynamic visualizations of your
sketch’s underlying principles.
Line Width
These commands set the line width of each
selected object to dashed, thin, or thick. Line
width applies to straight objects, circles, arcs, loci,
plots, and grids. To change an object’s line width:
To change an object’s
line width without
changing the setting
for future objects,
hold down the Shift
key while choosing
the command.
When a grid is
selected, the name of
the command is
Dotted rather than
Dashed.
1. Select the object(s) whose line width you wish to change.
2. Choose the desired line width from the Line Width submenu of the
Display menu.
When you change the selected object’s line width, Sketchpad
remembers your chosen width for future straight objects and curves in
the current sketch.
The Line Width setting has special effects on two types of objects—
coordinate system grids and function and locus plots.
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Display Menu
Line Width
Dotted
(grids) or
Dashed
(plots)
Appearance of Locus
and Function Plots
Appearance of Grids
1
0.5
-1
1
-0.5
-1
Thin
1
0.5
-1
1
-0.5
-1
Thick
1
0.5
-1
1
-0.5
-1
See also: Coordinate Systems (p. 21), Functions (p. 27), Loci (p. 24)
Color
These commands set the color of each selected object.
To change an object’s
color without
changing the setting
for future objects,
hold down the Shift
key while choosing
the command.
146
1. Select every object whose color you wish to change.
2. Choose the desired color from the Color submenu of the Display
menu.
When you set the color of an object, Sketchpad remembers your
chosen color for new objects of the same kind.
If you set the color of a caption that has different colors for different
characters, choosing this command sets all of the characters to the new
color you choose.
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Display Menu
To change the color
of an object’s label,
select the object and
then choose the new
color from the Text
Palette.
Parametric Color lets
you “color by
numbers.” The
number is provided
by a Sketchpad value
(such as a
measurement). When
the value changes, the
color changes, too.
While you cannot
color a locus or an
iterated image
parametrically, the
locus or iterated
image of a
parametrically colored
object will display the
range of that object’s
colors.
If you set the color of a labeled geometric object, note that the color
applies only to the object, not to its label.
You can change the color choices available in the Color submenu by
using the System panel of the Advance Preferences dialog box.
See also: Color Preferences (p. 138), System Preferences (p. 141), Text Palette (p. 55)
Parametric Color
Choose Parametric from the
Color submenu to set the color
of selected objects based on the
numeric value of either one or
three selected values.
This command is available only
when you have selected one or
three numeric values—
measurements, calculations, or
parameters—as well as one or
more objects that can be
colored parametrically—points,
circles, arcs, straight objects, or
interiors.
If you have selected one
measurement, that
measurement is used to set a color from the spectrum (ranging from
violet to deep red) or a grayscale color. You can set the Parametric
Domain—that is, the numeric interval that corresponds to one
complete cycle of the available colors or shades. You can also set the
Color Range so that the cycle doesn’t repeat, repeats one-way, or
repeats bidirectionally.
If you have selected three measurements, all three measurements are
used to determine the object’s color. As with a single color, you can set
the domain and color range. You can also decide whether the three
measurements are interpreted as RGB (red, green, blue) values or as
HSV (hue, saturation, value) settings.
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Display Menu
If your computer is
set to display 256 or
fewer colors, colors
may not appear
accurately, and you
may be limited in
your ability to select
the color you want.
Other Color
Choose Other from the Color submenu to set the color of the selected
objects to a color other than those that appear in the Color submenu.
Your system’s Color Picker dialog box appears, allowing you to specify
any color your computer can display.
See also: Color Picker (p. 64)
Text
The keyboard
shortcut for Increase
Size is Alt+>
(Windows) or a+>
(Mac). The shortcut
for Decrease Size is
Alt+< (Windows) or
a+< (Mac).
This submenu sets the font used for the selected objects and can
increase or decrease the font size for the selected objects.
Choose Increase Size to increase the font size for each selected object to
the next-larger standard font size. Choose Decrease Size to reduce the
font size for each selected object to the next-smaller standard font size.
If you want to set text styles such as bold, italic, or underline or if you
want to set text size to a specific value, use the Text Palette.
See also: Text Palette (p. 55), Show Text Palette (p. 154)
Hide Objects
The keyboard
shortcut for Hide is
Ctrl+H (Windows) or
a+H (Mac).
This command hides each selected object from view, without changing
its geometric role in the sketch. Use this command to hide objects that
shouldn’t be seen, but that are still required for geometric purposes in
the sketch. When an object is hidden, it remains present in the sketch
and can continue to define the position and behavior of other objects
that remain visible.
If you have objects in your sketch that you don’t want to see and that
are not required for geometric reasons, it’s better to delete those objects
using the Backspace key or the Clear command than to hide them.
See also: Clear (p. 111), Show All Hidden (p. 148)
Show All Hidden
This command displays and selects all objects previously hidden in a
sketch.
See also: Object Properties (p. 121), How to Show Just One Hidden Object (p. 149)
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Display Menu
How To . . . Show Just One Hidden Object
In a sketch containing many hidden objects, you may want to show just
one or two of those hidden objects. The Show All Hidden command
shows all the hidden objects, and once you’ve shown them it can be a
lot of trouble to reselect all of them in order to hide them again.
Here’s a convenient way to show one particular object among many
that have been hidden:
1. Choose Show All Hidden from the Display menu.
All previously hidden objects appear and are selected.
2. Using a Selection Arrow tool, click on the object you wish to remain
visible, deselecting it.
The object you click is deselected; the other just-shown objects
remain selected.
3. Choose Hide from the Display menu.
The remaining selected objects are hidden, and the deselected
object remains visible.
Another way to hide or show objects is by using the Hidden checkbox
on the Object panel of the Properties dialog box. To display properties
for a hidden object, you must first display properties for a parent or
child of the hidden object, then choose the desired object from the
Parents or Children pop-up menu.
See also: Object Properties (p. 121), Show All Hidden (p. 148), Hide Objects (p. 148),
Object Relationships: Parents and Children (p. 10)
Show/Hide Labels
The keyboard
shortcut for Show/Hide
Labels is Ctrl+K
(Windows) or a+K
(Mac).
This command shows or hides the label of each selected object. Most
Sketchpad objects can show their labels, with a few exceptions such as
grids, measurements and calculations, captions, pictures, and loci of
objects other than points.
To show or hide labels:
1. Select every object whose label you wish to show or hide.
2. Choose Show Labels or Hide Labels from the Display menu.
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149
Display Menu
Using this command is equivalent to clicking on an object with the Text
tool.
When you measure a quantity that depends on a particular object, the
object’s label may automatically be shown if it’s needed to display the
measurement. For instance, if you measure the angle defined by three
points, the labels of those points may be shown. Use the Text panel of
Preferences to turn this automatic showing of labels on or off.
An object is labeled automatically when the label is first needed for
some purpose—either because you’ve chosen the Show Label command,
because you’ve clicked the Text tool on the object, or because you’ve
measured some quantity that depends on the object. Thus the first
point you label will become point A, even if it’s the tenth point you’ve
actually created in your sketch.
Although you can’t show or hide the labels of measurements and
calculations, you can control how they are named when they’re
displayed, using the Value panel of the Properties dialog box.
See also: Text Tool (p. 86), Text Preferences (p. 139), Label Properties (p. 122), Value
Properties (p. 124)
Label
This command allows you to change the label of one or more selected
objects. If a single object is selected, choosing the command opens the
Properties dialog box for that object and displays the Label panel,
where you can type a new label. If more than one object is selected,
choosing the command displays the Label Multiple Objects dialog box.
If an object in your
sketch already has the
desired first label,
select that object first,
and then select the
other objects to be
relabeled. When you
choose the Label
command you won’t
have to type anything
into the dialog.
150
In the Label Multiple
Objects dialog box, type the
label you want to assign to
the first of the selected
objects. Sketchpad will
create a sequence of labels
to use for the remaining
selected objects.
If some of the selected
objects aren’t already showing their labels, you can check or uncheck
the Show Labels for All Selected Objects checkbox to determine
whether or not the labels will be shown when you dismiss the dialog.
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Display Menu
Here are some examples of the labels Sketchpad will create, depending
on what you type as the first label:
Type This:
To Produce This Sequence:
P
P, Q, R, S, .T, …
d
d, e, f, g, h, …
P1
P1, P2, P3, P4, P5, …
P[1]
P1, P2, P3, P4, P5)
A98
A98, A99, A100, A101, A102, …
W
W, X, Y, Z1, Z2, …
1-a
1-a, 1-b, 1-c, 1-d, 1-e, …
See also: Label Properties (p. 122), Labeling Objects with a Custom Sequence (p. 229)
Trace
The keyboard
shortcut for Trace is
Ctrl+T (Windows) or
a+T (Mac).
The faster a traced
object moves, the
more spread out its
trail; the slower the
object moves, the
denser its trail.
Think of a trace as a
temporary display of
an object’s locus. To
construct a
permanent locus, use
the Locus command
from the Construct
menu.
This command turns tracing on or off for each selected object. If all the
selected objects are currently being traced, a check mark appears next
to the command. Choosing the command turns tracing off for each
selected object. If no check mark appears, choosing the command turns
tracing on for each selected object.
When an object is traced, it leaves behind a trail as it moves, no matter
how the motion is caused—whether by dragging the traced object, by
dragging some object on which the traced object depends, or by
animating.
Tracing can be useful for exploring how an object’s movement is
constrained or for creating interesting mathematical art.
You can use Color Preferences to make traces gradually fade over time
and eventually disappear. You can also control how fast traces fade—or
turn off the fading of traces altogether. If fading is disabled, traces will
remain on the screen indefinitely. To remove traces (whether fading or
not) from the screen, choose Erase Traces from the Display menu.
There’s an important difference between erasing traces and deactivating
tracing for an object or objects. Erasing traces (using the Erase Traces
command) removes all existing traces from the screen but has no effect
on whether objects will leave traces in the future. Turning tracing off
for an object (using the Trace command) prevents that object from
leaving traces as it moves in the future, but has no effect on any traces
the object has already left on the screen. If you want to remove all
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151
Display Menu
existing traces an object has left behind, and also to prevent that object
from leaving traces in the future, you must use both commands:
1. Select the object and choose the Trace command to turn off future
tracing for the object.
2. Choose Erase Traces to erase any existing traces from the screen.
See also: Erase Traces (p. 152), Color Preferences (p. 138), Loci (p. 24)
The perpendicular bisector of a segment attached to a circle is selected and
traced (left). As the segment’s endpoint is dragged around the circle, the
bisector traces a hyperbolic envelope (center). Animating the endpoint
around the circle results in smoother traces (right).
Erase Traces
The keyboard
shortcut for Erase
Traces is Ctrl+B
(Windows) or a+B
(Mac).
This command removes all visible traces from the screen.
This command erases all traces immediately. If you want to control
whether and how quickly traces fade, use the Color panel of the
Preferences dialog box.
This command does not prevent traced objects from leaving new traces
on the screen as they continue to move. To prevent an object from
leaving new traces, you must select that object and turn tracing off for
the object by choosing Trace from the Display menu.
You can also use the Esc key to erase traces.
See also: Trace (p. 151), Color Preferences (p. 138), Esc Key (p. 215)
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Display Menu
Animate
The keyboard
shortcut for Animate is
Ctrl+` (Windows) or
a+` (Mac).
This command puts each selected geometric object into motion.
•
Independent points move freely in the plane, in random directions.
•
Points on paths move along their paths. Points on straight objects
and arcs move bidirectionally. Points on circles and interiors move
around their paths.
•
Other objects move by moving the objects upon which they
depend (that is, their parental objects).
This command has the same effect as pressing the Animate button on
the Motion Controller.
Pause Animation and
Resume Animation have
the same effect as the
Pause button on the
Motion Controller.
When animations are running and nothing is selected, the Animate
command becomes Pause Animation. When animation is paused, the
command becomes Resume Animation.
You can also create Animation action buttons to start and stop specific
animations.
See also: Stop Animation (p. 154), Increase Speed/Decrease Speed (p. 153), Motion
Controller (p. 41), Animation Buttons (p. 38), Object Relationships: Parents and
Children (p. 10)
Increase Speed/Decrease Speed
The keyboard
shortcut for Increase
Speed is Alt+]
(Windows) or a+]
(Mac); for Decrease
Speed it’s Alt+[
(Windows) or a+[
(Mac).
This command increases or decreases by about 25% the animation
speed of each object that is both selected and animating, or of all
animating objects if nothing is selected.
This command has the same effect as clicking on the speed control
buttons of the Motion Controller.
If it’s difficult to select the object whose speed you want to change,
there are two methods you can use:
•
Select any moving point by choosing from the Motion Controller’s
Target menu.
•
Pause animation, select the desired object, and resume animation.
If nothing is selected, these commands change to Increase All Speeds and
Decrease All Speeds.
See also: Animate (p. 153), Stop Animation (p. 154), Motion Controller (p. 41)
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Display Menu
Stop Animation
If at least one selected object is moving, this command stops the
motion of each selected moving object.
If nothing is selected but at least one object in the sketch is moving,
this command changes to Stop All Motions and stops the motion of every
moving object in the sketch.
This command has the same effect as pressing the Stop button on the
Motion Controller.
You can also use the Esc key to stop all motion.
See also: Animate (p. 153), Motion Controller (p. 41), Esc Key (p. 215)
Show/Hide Text Palette
The keyboard
shortcut for Show/Hide
Text Palette is
Shift+Ctrl+T
(Windows) or
Shift+a+T (Mac).
This command shows or hides the Text Palette. The Text Palette allows
you to change the font, font size, and style for the label or text of each
selected object.
If the palette is not currently showing, the command is Show Text Palette;
if the palette is currently showing, the command is Hide Text Palette.
Use the Text panel of the Preferences dialog box to determine whether
or not the Text Palette appears automatically when you create or edit a
caption.
See also: Text Palette (p. 55), Text Preferences (p. 139)
Show/Hide Motion Controller
This command shows or hides the Motion
Controller.
Use the Motion Controller to control the
motion of the selected objects in your
sketch. The Motion Controller controls the
motion both of animated objects and of
objects set in motion by pressing a Movement button.
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If the Motion Controller is not currently showing, the command is Show
Motion Controller; if it is currently showing, the command is Hide Motion
Controller.
The Motion Controller also appears automatically when you choose
from the Display menu.
Animate
See also: Motion Controller (p. 41)
Show/Hide Toolbox
This command shows or hides Sketchpad’s Toolbox.
If the Toolbox is not currently showing, the command is Show Toolbox;
if it is currently showing, the command is Hide Toolbox.
Hide the Toolbox to give you more space when you’re working on a
large sketch or to remove the distraction of visible tools when you’ve
finished a sketch that should be manipulated only by using the Selection
Arrow tool to drag objects or to click on action buttons.
See also: Toolbox (p. 68)
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155
Construct Menu
This menu provides commands for
accomplishing many important geometric
constructions. Most of these are
constructions that could also be
accomplished by using Sketchpad’s Compass
and Straightedge tools, but the Construct
menu provides simpler and quicker ways of
completing the construction.
Using the Construct Menu
If a command you
wish to use is
unavailable, it’s
possible you have too
few or too many objects
selected. Deselect
unwanted objects or
deselect all objects
and try again.
Each command on the Construct menu
requires you first to select one or more
objects in the sketch. These selected
objects are called the selection prerequisites of
the construction: they are the objects to be used to define the
construction itself. (For instance, to use the Midpoint command you must
first select one or more segments.) Commands in the Construct menu
are enabled only when you’ve selected appropriate prerequisites. If you
want to use a command but find that it’s disabled, check the objects
you’ve selected in the sketch and compare them with the prerequisites
listed in the description of that command.
See also: Compass Tool (p. 81), Straightedge Tools (p. 83)
Point On Object
Selection prerequisites: One or more path objects.
You can also
construct a point on
an object by clicking
on it with the Point
tool, or with any
other tool that
constructs points as
part of its operation
(such as the Compass
tool, Straightedge
tools, or most custom
tools).
156
Constructs a point on each selected path object. The point is randomly
placed on the object. It can later be animated or dragged anywhere on
the object, but it will not leave the path.
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Construct Menu
Path objects on which you can construct points include straight objects
(segments, rays, and lines), curved objects (circles and arcs), interior
objects (polygons and circle and arc interiors), point loci, and function
plots. (When you use an interior as a path, the actual path is the
perimeter of that interior. A point constructed on a polygon interior is
free to move around the entire perimeter of the polygon.)
See also: Path Objects (p. 13), Objects (p. 8)
Midpoint
The keyboard
shortcut for Midpoint
is Ctrl+M (Windows)
or a+M (Mac).
Selection prerequisites: One or more segments.
Constructs a point at the midpoint of each selected segment. As a
segment gets longer or shorter (through dragging an endpoint, for
example), the midpoint moves accordingly.
See also: Points (p. 11)
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Construct Menu
Intersection
The keyboard
shortcut for
Intersection is Ctrl+I
(Windows) or a+I
(Mac).
Selection prerequisites: Two intersecting objects, each of which is a straight object,
circle, or arc.
Constructs a point at each intersection of the two selected objects. If
the objects actually intersect in two places, two points are constructed;
otherwise one point is constructed.
You can also
construct a point of
intersection by
clicking with the
Selection Arrow tool or
with any tool that
constructs points as
part of its operation
(such as the Point,
Compass, or
Straightedge tools).
Both intersections constructed
One intersection constructed
If you later drag the intersecting objects apart, the constructed
intersection point disappears, and reappears when you drag the objects
so that they’re again touching.
See also: Points (p. 11)
Segment, Ray, and Line
Selection prerequisites: Two or more points.
The keyboard
shortcut for Segment
is Ctrl+L (Windows)
or a+L (Mac).
158
Constructs a segment, ray, or line through the selected points. ( The Ray
command constructs a ray from the first point through the second.) If
more than two points are selected, this command constructs the same
number of segments, rays, or lines as the number of selected points.
The Geometer’s Sketchpad Reference Manual
Construct Menu
(For instance, choosing Line with the four points A, B, C, and D
selected will construct the four lines shown.)
To quickly construct a
pentagon, hold down
the Shift key while
you click the Point
tool five times. Then
choose the Segment
command to
construct the five
sides of the pentagon.
A
B
A
D
B
C
Four points selected
D
C
Four lines constructed
The results of these commands are the same as the results of using the
Straightedge tools on the selected points. The commands are particularly
useful for constructing multiple straight objects and for making sure
that your straight objects go through the proper points.
See also: Straightedge Tools (p. 83), Segments, Rays, and Lines (p. 14)
Parallel Line
Selection prerequisites: A straight object and one or more points; or a point and one
or more straight objects.
Constructs a line through each selected point parallel to each selected
straight object.
One point, one straight object
One parallel line
Two points, one straight object
Parallel lines through both points
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Construct Menu
One point, two straight objects
Lines parallel to both straight objects
Perpendicular Line
Selection prerequisites: A straight object and one or more points; or a point and one
or more straight objects.
Constructs a line through each selected point perpendicular to each
selected straight object.
160
One point, one straight object
One perpendicular line
Two points, one straight object
Perpendicular lines through both
points
One point, two straight objects
Lines perpendicular to both straight
objects
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Construct Menu
Angle Bisector
Selection prerequisites: Three points, with the vertex point selected second.
Constructs a ray that bisects the minor angle formed by the three
selected points. The second selected point designates the vertex of the
angle. For instance, to bisect ∠ABC, select points in the order A, then
B, and finally C.
A
B
C
A
B
C
See also: Segments, Rays, and Lines (p. 14)
Circle By Center+Point
Selection prerequisites: Two points.
Constructs a circle with its center at the first selected point and with its
circumference passing through the second selected point. The result of
this command is the same as the result of using the Compass tool on the
two selected points.
A
A
B
B
See also: Circles and Arcs (p. 15), Compass Tool (p. 81), Circle By Center+Radius
(p. 162)
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Construct Menu
Circle By Center+Radius
Selection prerequisites: A point and one or more segments and/or distance
measurements; or a segment or distance measurement and one or more points.
This command
duplicates the
function of a
noncollapsible
compass, which
allows you to set a
length, then draw
circles centered
anywhere with that
length as the radius.
Constructs one or more circles centered at each selected point and with
the radius determined by each selected segment or distance
measurement.
One point, one segment
m EF = 0.29 in.
One circle
m EF = 0.29 in.
Two points, one distance
measurement
Two circles centered at the points with
radius determined by the measurement
One point, two segments
Two circles centered at the point with
radii determined by the segments
See also: Circles and Arcs (p. 15), Compass Tool (p. 81), Circle By Center+Point (p. 161)
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Construct Menu
Arc On Circle
Selection prerequisites: A circle and two points on that circle; or a center point and
two other points equally distant from the center point.
The arc is constructed
counter-clockwise
from the first selected
circumference point
to the second. The
order in which you
select the points
determines whether
the constructed arc is
a minor or major arc.
Constructs an arc on the given circle or with the given center, bounded
by the selected circumference points.
B
B
A
A
Circle and two points
Arc on circle
B
B
A
A
C
Center point and two circumference
points
C
Arc with given center
Arc Through 3 Points
Selection prerequisites: Three noncollinear points; that is, three points that do not lie
on the same line.
Constructs an arc through the three selected points. The arc starts at
the first selected point, passes through the second, and ends at the
third.
B
B
A
C
A
C
If subsequently you drag the three points collinear, the second point
determines the arc’s display. If the second point falls between the first
and third, the arc has zero measure but nonzero arc length and displays
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163
Construct Menu
as a straight segment from the first point through the second to the
third. If the second point is collinear to, but outside of, the first and
third, the arc disappears.
See also: Circles and Arcs (p. 15)
Interior
The keyboard
shortcut for Interior is
Ctrl+P (Windows) or
a+P (Mac).
Constructs the interior defined by the selected object or objects.
Depending on your selection, one of four separate commands appears
(Polygon Interior, Circle Interior, Interior | Arc Sector, and Interior | Arc Segment). A
description of each follows.
Constructed interiors give your sketches substance and color and allow
you to measure the areas and perimeters of figures. You can also use
the perimeter or circumference of an interior as a path on which to
construct or animate points.
See also: Polygons and Other Interiors (p. 16)
Polygon Interior
Selection prerequisites: Three or more points.
Constructs a polygon interior using the selected points as vertices. The
order in which you select the points determines the order of the
vertices of the polygon.
B
A
C
E
D
B
A
C
E
D
See also: Polygons and Other Interiors (p. 16)
Circle Interior
Selection prerequisites: One or more circles.
Constructs a circle interior for each selected circle.
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Construct Menu
See also: Circles and Arcs (p. 15), Polygons and Other Interiors (p. 16)
Arc Sector Interior
Selection prerequisites: One or more arcs.
Constructs an arc sector interior for each selected arc. An arc sector is
bounded by the arc and by the radii to the two endpoints of the arc.
See also: Arc Segment Interior (p. 165)
Arc Segment Interior
Selection prerequisites: One or more arcs.
Constructs an arc segment interior for each selected arc. An arc
segment is bounded by the arc and by the chord connecting the two
endpoints of the arc.
See also: Arc Sector Interior (p. 165)
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165
Construct Menu
Locus
Selection prerequisites: The object whose locus you want to construct (the driven
object) and a point (the driver) that determines the position of that object. The driver
must be constructed on a path; or you can select an independent point as the driver
and a separate unrelated path object.
You can use Plot
Properties to change
the number of
samples in a locus and
to determine whether
a point locus is
displayed as discrete
or continuous.
Constructs the locus of the selected object as the driver point moves
along its path. Generally, a locus is the set of points (or other objects)
that satisfies some mathematical condition. In Sketchpad, a locus is
defined as the set of locations of an object—the driven object—whose
position or shape is determined by the motion of a driver point as it
moves along a path. You can construct loci of points, straight objects,
circles, arcs, or interiors. The selected driver must be constructed as a
Point On Object that controls the position of the driven object.
Alternately, you may select as the driver an independent point that
determines the position of the driven object, in which case you must
also select an unrelated path along which the driver can travel.
Driver: Point on circle
Driven object: Point on segment
Result: Point locus
Driver: Independent point
Path: Arc
Driven object: Segment
Result: Segment locus
See also: Loci (p. 24), Path Objects (p. 13), Properties (p. 120), Point On Object (p. 156)
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Transform Menu
The Transform menu commands apply geometric transformations to
figures in your sketches, allowing you to create translations, rotations,
dilations, reflections, tessellations, scale models, kaleidoscopes, fractals,
and much more. These commands include four basic transformations:
Translate (to slide), Rotate (to turn), Dilate (to shrink or enlarge), and Reflect
(to flip).
In addition to the object or objects that they transform—the
transformational pre-image—each type of transformation involves
certain parameters: objects or values upon which the transformation is
based. For instance, a rotation requires both an angle and a center
about which to rotate. The Mark commands—Mark Center, Mark Mirror, Mark
Angle, Mark Ratio, Mark Vector, and Mark Distance—let you specify objects in
your sketch to act as dynamic parameters for your transformations.
These parameters may be either geometric objects or geometric
quantities. For example, a rotation is defined by two parameters: a
center and an angle. The center is a geometric point about which
objects rotate. The angle is a geometric quantity that determines how
far objects rotate about the center.
The Transform menu provides a flexible set of commands that allow
you to specify a rich variety of transformations using either fixed or
dynamic transformational parameters. For instance, you can rotate a
triangle by a fixed angle of 45° about some center point, or you might
rotate by a dynamic parameter such as ∠ABC. (This is a “dynamic”
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167
Transform Menu
parameter in the sense that if you then drag A, B, or C, the rotated
image changes dynamically to the new ∠ABC.)
In general, follow these steps to construct the transformed image of
one or more objects:
Once you mark a
parameter, Sketchpad
remembers that mark
even after you change
the selections. If
you’ve already marked
a parameter for one
transformation, you
don’t need to mark it
again to use the same
parameter for a
second
transformation.
1. Mark any parameters that determine dynamic aspects of your
transformation by selecting objects to define those parameters and
choosing the appropriate Mark commands (for example, Mark Center
to mark a selected point as the center of rotation).
2. Select the object(s) you wish to transform. (Mathematically, this is
the pre-image of your transformation.)
3. Choose the appropriate transformation command from the
Transform menu (for example, Rotate).
4. In the resulting dialog box, enter any fixed parameters you wish to
use in your transformation (for example, type 45°) then leave the
dialog box.
Sketchpad constructs the image of your selection according to the
transformation and parameters you’ve specified.
Finally, the Iterate command allows you to create in a single step
multiple transformed images such as the spiral shown above, or any
figure resulting from a tranformation or construction repeated many
times. Iteration can even be used to create complex images such as
tessellations and fractals.
Mark Commands
Any marked object
remains marked until
you mark a new
object of the same
type. For example,
you don’t need to
mark the same center
more than once, no
matter how many
times you use it as a
marked center.
The first six commands on the Transform menu allow you to mark the
parameters to be used in subsequent transformations.
You can use most of the Mark commands whenever your selections
include the required object(s) for that mark, even if you have other
objects selected as well. Thus, if you’ve selected five polygons you want
to dilate, then realize you forgot to mark the center, you can select your
desired center point, without deselecting the polygons, and choose Mark
Center. The most recently selected point is marked as the center and
removed from your selections, and you can go ahead and choose Dilate
for your polygons.
See also: Mark Center (p. 169), Mark Mirror (p. 169), Mark Angle (p. 169), Mark Ratio
(p. 170), Mark Vector (p. 172), Mark Distance (p. 173)
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Transform Menu
Mark Center
If you haven’t yet
marked a center when
you choose Rotate or
Dilate, Sketchpad will
automatically choose
a point and mark it
for you.
Marks the most recently selected point as the center point about which
future rotations and dilations will take place. To mark a center:
•
Select a point and choose Mark Center from the Transform menu, or
•
Double-click a point with the Selection Arrow tool.
The marked point will flash briefly to indicate it’s been marked as the
center for future rotations and dilations. Once marked, this center
point will be used for all future rotations and dilations until you mark a
different center point.
You can also change the marked center while the Rotate or Dilate
dialog box is open by clicking on the desired point in your sketch.
See also: Mark Commands (p. 168), Rotate (p. 177), Dilate (p. 179)
Mark Mirror
If you haven’t yet
marked a mirror when
you choose Reflect,
Sketchpad will
automatically choose
a straight object and
mark it for you.
Marks the most recently selected straight object as the mirror across
which future reflections will take place. To mark a mirror:
•
Select a straight object and choose Mark Mirror from the Transform
menu, or
•
Double-click on a straight object with the Selection Arrow tool.
The marked straight object will flash briefly to indicate it has been
marked as the mirror for future reflections. Once marked, this mirror
will be used for all future reflections until you mark a different mirror.
See also: Mark Commands (p. 168), Reflect (p. 181)
Mark Angle
Use a marked angle
for a transformation
when you want your
transformed image to
move as the marked
angle changes.
Marks the most recently selected angle or angle measurement as the
angle to be used for future rotations and polar translations. To enable
this command, your selections must include either three points or an
angle measurement.
Rotations and polar translations can use either a fixed angle (such as
45º) or a marked angle (such as ∠ ABC ). When you rotate by 45º, the
angle of rotation never changes, but when you rotate by ∠ ABC, the
rotation changes as the positions of points A, B, and C change.
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Transform Menu
You can mark an angle defined by three points, or you can mark an
angle measurement or calculation.
Marking an Angle Using Three Points
1. Select the three points defining your angle. First,
C
select a point on the initial side of the angle, then
select the vertex, and finally, select a point on the
terminal side of the angle.
For instance, if you select three points arranged
B
A
as shown above, in the order A, B, C, the result
will be a marked angle that specifies the counterclockwise rotation shown by the arrow.
2. Choose Mark Angle from the Transform menu.
A brief animation from the initial side to the terminal side confirms
that your angle has been marked.
Marking an Angle Measurement
1. Select the desired angle measurement, parameter, or calculation.
The units of the value must be either degrees or radians.
2. Choose Mark Angle from the Transform menu.
The selection will flash briefly to confirm that your angle measurement,
parameter, or calculation has been marked.
You can change the marked angle measurement while the Translate or
Rotate dialog box is open by clicking on the desired measurement,
parameter, or calculation in your sketch.
See also: Mark Commands (p. 168), Polar Translation Vector (p. 174), Rotate (p. 177),
Angle (p. 194)
Mark Ratio/Mark Segment Ratio/Mark
Scale Factor
Use a dynamic
marked ratio or scale
factor when you want
your dilated image to
change as the marked
item changes.
170
Marks the most recently selected ratio or scale factor as the ratio for
future dilations. This command displays one of three related
appearances, depending on your selections. If you select:
• two segments, the command becomes Mark Segment Ratio. Dilation
will use the ratio of the length of the first segment to the length of
the second segment.
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Transform Menu
•
a measured or calculated scale factor, the command becomes Mark
Scale Factor. The selected measurement or calculation must be a
scalar value (that is, a value without units).
•
three collinear points, the command becomes Mark Ratio. If you
select points A, B, and C in order, dilation will use the ratio of
signed distances AC/AB.
Dilations can use either a fixed ratio (such as ½) or a marked ratio
(such as AC/AB). When you dilate by ½, the ratio of dilation never
changes, but when you dilate by AC/AB, the dilation changes as the
positions of points A, B, and C change.
You can also mark a
segment ratio after
choosing Dilate. With
the Dilate dialog box
open, click on the
first and then the
second segment in
your sketch.
Marking a Ratio Using Two Segments
1. Select a segment whose length will be used as the numerator of
your ratio.
2. Select a second segment whose length will be used as the
denominator.
3. Choose Mark Segment Ratio from the Transform menu.
You will see a brief animation confirming that the ratio has been
marked.
If the first segment is shorter than the second, the ratio is less than 1
and dilation will shrink objects toward the marked center. If the first
segment is longer than the second, the ratio is greater than 1 and
dilation will stretch objects away from the marked center.
You can also mark a
scale factor after
choosing Dilate. With
the Dilate dialog box
open, click on a
measurement,
parameter, or
calculation that has
no units.
Marking a Scale Factor Using a Measurement, Parameter, or
Calculation
1. Select a measurement, parameter, or calculation to be the scale
factor. This value must be a pure number with no units (such as
inches or degrees) associated with it.
2. Choose Mark Scale Factor from the Transform menu.
The selection will flash briefly to confirm that your measurement,
parameter, or calculation has been marked.
If the magnitude of the value you marked as a scale factor is less than 1,
the dilation will shrink objects toward the marked center. If the value’s
magnitude is greater than 1, the dilation will expand objects away from
the marked center. If the scale factor is less than 0, image objects will
be dilated “through the center,” appearing on the opposite side of the
center as their pre-images.
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171
Transform Menu
Marking a Ratio Using Three Collinear Points
1. Select three collinear points in your sketch. The points must be in a
straight line.
2. Choose Mark Ratio from the Transform menu.
You will see a brief animation confirming that the ratio has been
marked.
When you mark three collinear points, A, B, and C, as a ratio,
Sketchpad dilates by the ratio of signed distances AC/AB. If B is on
the same side of A as C, the signed ratio of distances is positive; if B is
on the opposite side of A as C, the ratio is negative. You may wish to
use the Ratio command in the Measure menu to display this ratio
numerically. One handy way to remember the role of the three selected
points is to think of them as determining a ratio that, if the first
selected point was the marked center, would dilate the second selected
point to the location of the third selected point.
See also: Mark Commands (p. 168), Dilate (p. 179), Ratio (p. 196)
Mark Vector
You can also mark a
vector using two
points after choosing
Translate. With the
Translate dialog box
open, click on the
desired initial and
terminal points in
your sketch.
Marks for future translations the vector determined by the two most
recently selected points. The first of these two points is the initial point
of the vector, and the second is the terminal point of the vector.
When you choose this command, you will see a brief animation from
the initial to the terminal point confirming that the vector has been
marked.
After marking a vector, future
translations for which you specify the By
Marked Vector option will be based on
this vector. Such translations will
translate objects by a distance equal to
the distance from the initial to the
terminal point and in the same direction
as the direction from the initial to the
terminal point.
D
D'
A
C
E
B
C'
E'
For instance, if you mark the vector from
A to B as in the figure at right, then
translate ∆CDE, the result will be a second triangle translated by the
distance from A to B and in the direction from A to B. As you adjust
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Transform Menu
the vector by dragging A or B, the translated image of ∆CDE adjusts
accordingly.
See also: Mark Commands (p. 168), Translate (p. 173)
Mark Distance
Marks the one or two most recently selected distance values
(measurements, parameters, or calculations) to be used for future polar
and rectangular translations.
You can also mark a
distance after choosing
Translate. With the
Translate dialog box
open, click on a
distance
measurement,
parameter, or
calculation in your
sketch.
To mark a distance:
1. Select one or two distance measurements, parameters or
calculations (ones with distance units, such as inches).
2. Choose Mark Distance from the Transform menu.
The selected distance value(s) will flash briefly to confirm that your
distance has been marked.
If you choose Mark Distance with a single selected distance value, this new
distance will become the marked distance for polar translation.
If you choose Mark Distance with two selected distance values, the first
selected values will become the horizontal distance for rectangular
translations, and the second will become the vertical distance.
See also: Mark Commands (p. 168), Translate (p. 173)
Translate
Sketchpad has both a
Translate Arrow tool
and a Translate
command. When you
use the tool, you
translate the original
object. When you use
the command, you
create a new object—
a translated image of
the original object.
Constructs a translated image of the selected geometric object(s).
To construct a translated image:
1. Select the geometric object(s) you wish to translate.
2. Choose Translate from the Transform menu.
The Translate dialog box appears, and a translated image of your
selections appears in the sketch.
3. Choose one of the Translation Vector options (Polar, Rectangular,
or Marked), depending on how you wish to specify the translation.
The Marked option is available only if you’ve already marked a
vector; see Marked Translation Vector (p. 176) for more details.
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173
Transform Menu
4. Specify any required values for the type of translation you’ve
chosen. While you’re choosing your values, you can see in the
sketch a copy of the translated image that will be created.
5. You can click a distance measurement in the sketch to change the
marked distance; you can click an angle measurement to change the
marked angle; or you can click two points to change the marked
vector.
6. When you’re satisfied with the translation you’ve specified, click
OK.
More detailed descriptions of the three Translation Vector options
appear below.
See also: Dragging with the Translate Arrow Tool (p. 75), How to Construct a Segment
of Fixed Length (p. 177)
When you specify a
polar vector, a
positive angle
indicates counterclockwise rotation,
and a negative
distance indicates
clockwise rotation.
Polar Translation Vector
Choose this option to define the translation by specifying a distance
and an angle.
F'
H'
F
H
G'
G
m AB = 0.53 in.
m∠CDE = 26°
The distance can be a Fixed Distance (a number in your current
distance unit), or it can be a Marked Distance (a distance measurement
you’ve specified using the Mark Distance command). If you want to use a
distance measurement that exists in your sketch but has not already
been marked, you can click that measurement now to mark it.
The angle can be a Fixed Angle (a number in your current angle unit),
or it can be a Marked Angle (an angle measurement you’ve specified
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The Geometer’s Sketchpad Reference Manual
Transform Menu
using the Mark Angle command). If you want to use an angle
measurement that exists in your sketch but has not already been
marked, you can click that measurement now to mark it.
See also: Mark Distance (p. 173), Mark Angle (p. 169)
When you specify a
rectangular vector,
positive distances
indicate translation to
the right or up, and
negative distances
indicate translation to
the left or down.
Rectangular Translation Vector
Choose this option to define the translation by specifying a horizontal
distance and a vertical distance.
A'
C'
B'
A
C
B
Each of these distances can be a Fixed Distance (a number in your
current distance unit), or a Marked Distance (a distance measurement
you’ve specified using the Mark Distance command). If you want to use a
distance measurement that exists in your sketch but has not already
been marked, you can click that measurement now to mark it. When
you click a distance measurement in the sketch, the measurement you
click becomes the marked vertical distance and the previous marked
vertical distance becomes the marked horizontal distance.
See also: Mark Distance (p. 173)
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175
Transform Menu
Marked Translation Vector
Choose this option to define the translation by specifying a vector’s
initial and terminal points.
D
E
D'
E'
A
F
F'
B
This option is enabled only if you’ve already marked a vector in your
sketch. If you haven’t already used the Mark Vector command to mark a
vector, you can mark one now by clicking two points in your sketch—
first the initial point, then the terminal point.
See also: Mark Vector (p. 172)
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The Geometer’s Sketchpad Reference Manual
Transform Menu
How To . . . Construct a Segment of Fixed Length
Note that these steps
give you a fixed
length and a fixed
orientation. If you
want a segment of
fixed length whose
orientation you can
change, construct a
circle with radius
AA'. Then draw a
second radius from A
to any point B on the
circle. Segment AB
can now be dragged
to any orientation
while always having a
fixed length, because
all radii of the circle
have the same length
and the circle is
constructed to always
have a radius of fixed
length AA'.
Occasionally you may wish to create a segment of fixed length—for
instance, a segment that is exactly 1.5 cm in length. While you could
create a segment with the Segment tool, measure its length, and drag one
endpoint until its length was 1.5 cm, this segment would not be
constructed to be fixed at 1.5 cm long. That is, dragging an endpoint
again would change it from its current length to some other length.
To fix a segment’s length in Sketchpad, you need to construct the
segment in such a way that dragging cannot change its length. For an
arbitrary length, the easiest way to do this is with the Transform menu.
1. Create a point A in your sketch.
2. Select point A and choose Translate from the Transform menu. In
the dialog box, enter the fixed length as the distance by which you
wish to translate point A.
3. Click on Translate.
Sketchpad constructs a new point A' as the translated
A'
image of A by the distance that you entered (1.5 cm).
Now use the Segment tool to construct the segment
1.5 cm
between A and A'. No matter how you drag either
A
endpoint, the segment’s length will remain 1.5 cm
because it has been constructed to always have that
length.
Rotate
Sketchpad has both a
Rotate Arrow tool and
a Rotate command.
When you use the
tool, you rotate the
original object. When
you use the
command, you create
a new object—a
rotated image of the
original object.
Constructs a rotated image of the
selected object(s). To rotate objects:
1. If you haven’t already done so,
select a point to act as the center for
rotation and choose Mark Center from
the Transform menu. Alternatively,
you can double-click the desired
center point with the Selection Arrow
tool.
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Transform Menu
A brief animation indicates that the point has been marked as a
center for subsequent rotations and dilations.
If you rotate without first marking a center, Sketchpad will mark
one for you.
2. Select the object(s) you wish to rotate.
3. Choose Rotate from the Transform menu.
The Rotate dialog box appears, and a rotated image of your
selections appears in the sketch.
C'
D'
D
C
B'
B
A
4. Choose either Fixed Angle or Marked Angle, as described below.
5. You can click a point in the sketch to change the marked center, or
you can click an angle measurement to change the marked angle.
6. When you have chosen the options you want and entered any
required values, click Rotate.
The rotated image appears.
Positive angles result
in counter-clockwise
rotations, and
negative angles result
in clockwise rotations.
Fixed Angle
Choose this option to enter a fixed (numeric) angle of rotation using
your sketch’s current angle units.
Marked Angle
Choose this option to rotate your selection based on an angle you’ve
specified using the Mark Angle command. This choice is disabled if you
haven’t already marked an angle. However, if you want to use an angle
measurement that exists in your sketch but has not already been
marked, you can click that measurement now to mark it.
See also: Mark Center (p. 169), Mark Angle (p. 169), Dragging with the Rotate Arrow
Tool (p. 75)
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How To . . . Construct an Angle of Fixed Measure
Occasionally you may wish to create an angle of fixed measure—for
example, an angle that measures exactly 33º. While you could create an
angle by measuring three points and dragging them until they form an
angle of 33º, this angle would not be constructed to be fixed at 33º. That
is, dragging it again would change it from its current magnitude to
some other magnitude.
To fix an angle in Sketchpad, you need to construct the angle in such a
way that dragging cannot change its magnitude. For an arbitrary angle,
the easiest way to do this is with the Transform menu.
1. Place two points, A and B, in your sketch.
2. Select point A and choose Mark Center from the Transform menu.
Point A becomes marked as the center of future rotations and
dilations.
3. Select point B and choose Rotate. In the dialog box, enter the fixed
angle by which you wish to rotate.
4. Sketchpad constructs point B' as the rotated
B'
image of B by your requested angle. Even if
you drag A, B, or B', Sketchpad will maintain
this angle’s magnitude, because you’ve
A 33°
B
defined B' to be the rotated image of B by
this angle.
You can now construct rays or segments connecting A, B, and B' to
incorporate the fixed angle into your sketch.
Dilate
Sketchpad has both a
Dilate Arrow tool and a
Dilate command.
When you use the
tool, you dilate the
original object.
When you use the
command, you create
a new object—a
dilated image of the
original object.
Constructs a dilated image of the selected object(s). In a dilated image,
a particular ratio is used to move every point of the original closer to or
farther away from the center point. If the ratio is greater than 1, the
image points are farther away from the center than the originals and the
image is larger than the original image. If the ratio is less than 1, the
image points are nearer the center and the image is smaller.
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Transform Menu
To dilate objects:
1. If you haven’t already done so, select a point to act as the center
for dilation and choose Mark Center from the Transform menu.
Alternatively, you can double-click the desired center point with
the Selection Arrow tool.
A brief animation indicates that the point has been marked as a
center for subsequent rotations and dilations.
If you dilate without first marking a center, Sketchpad will mark
one for you.
2. Select the object(s) you wish to dilate.
3. Choose Dilate from the Transform menu.
The Dilate dialog box appears, and a dilated image of your
selection(s) appears in the sketch.
4. Choose either Fixed Ratio or Marked Ratio, as described below.
5. You can click a point in the sketch to change the marked center,
you can click a measurement with no units to set the marked scale
factor, or you can click two segments to set the marked segment
ratio.
6. When you have chosen the options you want and entered any
required values, click on Dilate.
The dilated image appears.
A ratio smaller than 1
results in an image
that’s smaller than the
original, and a ratio
greater than 1 results
in an image that’s
larger.
Fixed Ratio
Choose this option to enter a fixed ratio by entering both numerator
and denominator.
A
B'
B
E'
E
C
D'
C'
D
Marked Ratio
Choose this option to dilate your selection based on a marked ratio or
scale factor you’ve specified using the Mark Ratio command. This choice
is disabled if you haven’t already marked a ratio. However, if you want
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Transform Menu
to use a ratio of two segments that exist in your sketch, you can click
the segments now to mark them. Similarly, if you want to use a
measured scale factor that exists in your sketch, you can click it now to
mark it.
F'
F
I
I'
A
H
G
H'
G'
m BC
m DE
= 2.43
See also: Mark Center (p. 169), Mark Ratio (p. 170), Dragging with the Rotate Arrow
Tool (p. 75)
Reflect
This command constructs a mirror image of the selected object(s)
across a marked mirror. To reflect objects:
A shortcut for
marking a straight
object as a mirror is
to double-click it.
1. Select a straight object (line, segment, ray, or axis) to be the mirror
for reflection. Then choose Mark Mirror from the Transform menu.
(If you don’t already have a marked mirror and you skip this step,
Sketchpad will mark a mirror for you.)
A brief animation indicates that the straight object has been
marked as a mirror for subsequent reflections.
2. Select the object(s) you wish to reflect.
A'
A
3. Choose Reflect from the Transform menu.
The reflected image appears.
See also: Mark Mirror (p. 169)
B
C
C'
B'
Iterate
This command constructs the iterated images of a set of related
geometric objects according to an iteration rule that you define. The
command is available when you select some combination of pre-image
points and pre-image calculations.
Pre-image points must be independent points or points on path, and
must define other points (the image points) in your sketch. Pre-image
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Transform Menu
calculations must be parameters or independent calculations, and must
define both image calculations and geometric objects.
See the “Iterations and Iterated Images” topic in the Objects chapter
(p. 31) for a more detailed description of iteration and related concepts
such as iteration rules, pre-images, images, seeds, and orbits.
See also: Iterations and Iterated Images (p. 31)
Iterating by Example
In Sketchpad, you define an iteration rule by example, creating both
pre-image and image and specifying how they correspond to each
other.
For instance, the figure at right starts with
∆ABC. ∆DEF is the midpoint triangle,
constructed using as its vertices the midpoints of
the sides of ∆ABC.
To repeat this construction, you’d construct a
new midpoint triangle using the midpoints of the
sides of ∆DEF, and then another using the
midpoints of the sides of that triangle, and so
forth.
A
F
B
E
C
D
To specify this repetition as an iteration rule in
Sketchpad, you define a map that identifies the
image for each pre-image:
A
A⇒D
F
E
B⇒E
C⇒F
B
D
C
Even though the repeated construction includes
the sides of the triangles as well as the vertices, you specify the rule
using only points A, B, C, D, E, and F. Sketchpad figures out what
other objects depend on the pre-image points and includes them in the
iteration. In this example, Sketchpad will include the sides of the
triangle in the iteration.
For best results, construct the entire pre-image but construct only the
points of the image. Let Sketchpad construct the other parts of the
image.
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Transform Menu
For example, to iterate the midpoint triangle,
construct the pre-image (∆ABC) and the image
points (midpoints D, E, and F), but not the
remaining image objects (the segments
connecting D, E, and F). Once you’ve
constructed these objects, select the independent
points that define the pre-image (points A, B,
and C) and choose Iterate from the Transform
menu.
You may need to drag
the dialog box out of
the way so that you
can see your original
triangle and its
midpoints.
Sketchpad displays the dialog box
shown at right, allowing you to define
the corresponding image for each
selected pre-image point. For each preimage point in the triangle, click on the
midpoint to which that pre-image
should map. As you click each midpoint,
Sketchpad displays the results of
iterating the pre-image triangle towards
your mapped images. Once you’ve
mapped each of the three pre-image
points to a unique midpoint, click Iterate
to confirm the iteration rule and close
the dialog box.
You can use Iterate to
create fractals by
specifying more than
one mapping of your
pre-images to first
images. See Multiple
Iteration Maps (p.
185) for more
information.
Sketchpad produces a set of iterated images for
each object affected by your mapping. In this
example there are six iterated images, one for
each vertex and side of your original triangle.
You can select and manipulate each iterated
image separately. For example, you could hide or
delete the three iterated images of your original
triangle’s vertices, or you could color each of the
three iterated images of your original triangle’s
edges with a different color. You can also use
Iteration Properties to change the number of
iterations that Sketchpad displays.
A
F
B
E
C
D
A
F
B
E
D
C
See also: Iterations and Iterated Images (p. 31), Iteration Properties (p. 134)
Using the Iterate Dialog Box
To construct the iterated images of one or more objects:
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Transform Menu
1. Select the initial pre-image objects whose positions or values define
the iteration. You may select independent points, points
constructed on paths, or independent parameters as your initial
objects. (In other words, initial objects must be objects whose
positions or values are not fully determined by other objects.
Objects whose position or value depends on these initial objects
will be iterated automatically as you iterate the initial objects.)
2. Choose Iterate from the Transform menu.
The Iterate dialog box appears.
Drag the Iterate
dialog box out of the
way if it’s hiding the
destination images
you want to click.
3. For each selected pre-image, click
the corresponding first image of
that object to which the pre-image
maps during the iteration. For a preimage point, click a dependent
image point to which it should
move during the iteration. For a
pre-image parameter value, click a
dependent calculated value. Images
must be objects that are defined in
terms of your pre-images, so that
their position or value changes as
the pre-image changes.
As you specify each destination image, a partial
iteration appears in the sketch corresponding to
the matches you’ve specified so far.
4. You can use the Display pop-up menu to
A
F
E
change the appearance of the iteration. See
Display Options (p. 184) for a description of the
available options.
B
D
C
5. You can use the Structure pop-up menu to
change the structure of the iteration. See Structure Options (p. 185)
for a description of the available choices.
6. Once you’ve specified the destination for each of the selected preimages, click Iterate to finalize the iteration.
The constructed iteration appears. Use Iteration Properties to
change the depth or other characteristics of the iteration.
See also: Iterations and Iterated Images (p. 31), Iteration Properties (p. 134)
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Transform Menu
Display Options
While you’re using the Iterate dialog box, you can use the Display
pop-up menu to control the appearance of the iteration. You can:
While there are
exceptions, in general
you may find it
convenient to display
all iterations when
you’re iterating a
single map, and to
display only the final
iteration when you’re
iterating more than
one map
simultaneously. See
Multiple Iteration
Maps (p.185) for
more information.
•
increase or decrease the number of times your rule is iterated.
•
display all iterations, or only the final iteration. The set of all
iterated images of an object is sometimes called the orbit of that
object.
Structure Options
While you’re using the Iterate dialog box, you can use the Structure
pop-up menu to control the structure of the iteration. You can:
•
add a new iteration map or remove the current map (see Multiple
Iteration Maps, p. 185).
•
create images of all objects that depend on the selections or create
images of only the nonpoint objects. (Often—especially when
working with multiple maps—you won’t want to see the iterated
images of points, but only of segments, polygons, and so forth.
Sketchpad automatically switches this option for you—to only
create nonpoint object images—when you start working with
multiple maps, though you can switch it back if you prefer to create
point images as well as nonpoint images.)
•
set iterated points on objects to stay at the same relative location as
the original or set them to use new random locations each time
they’re iterated. (See Random Iterated Points, p. 189.)
Multiple Iteration Maps
By specifying a destination point for each of the iterated independent
points, you create a single iteration map. This map describes how to
transform the pre-image to create a transformed copy of the original
objects. For most iterations, each iteration step produces a single copy
of the original objects. For such iterations, the iteration rule consists of
a single map.
But for other iterations, one step of the iteration produces two or more
copies of the original objects. Each such copy of the original objects
requires its own map, so such iterations require multiple maps. For
example, a parallelogram tessellation requires you to iterate the original
parallelogram both horizontally and vertically.
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Transform Menu
Fractals and
tessellations are the
most common
geometric
constructions for
which the iteration
rule requires multiple
iteration maps.
To construct an iteration with multiple maps, specify the first map as in
steps 1–3 above. Then choose Add New Map from the Structure pop-up
menu and specify a second map by clicking in turn on a new
destination for each of the original points. When you’re finished
specifying all the maps that make up your iteration rule, click Iterate.
C'
D
C
A
B
B'
How To . . . Construct a Sierpinski Gasket
Because iteration can be applied to any type of Sketchpad construction,
the options that support it may at first seem bewilderingly complex.
The best way to develop an understanding of iteration is to work
through examples. In this example, you’ll use iteration to define a
fractal known as the Sierpinski gasket. This fractal is the limit of the
process of replacing a triangle by three smaller interior triangles; then
replacing each of these three interior triangles by three even smaller
triangles; and so on. Since at each stage you are replacing a pre-image
triangle by three different image triangles, you’ll need three mappings to
define the fractal.
1. In a new sketch, use the Segment tool to construct a ∆ABC in your
sketch.
2. Construct the midpoints of your triangle’s edges. Use the Text tool
to label the vertices A, B, and C, and the midpoints D, E, and F, as
in the illustration below.
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Transform Menu
You now have a pre-image triangle and—
implicitly—many smaller triangles, such as ∆AFE,
∆EBD, and so forth. Note that the three smaller
triangles, ∆AFE, ∆FBD, and ∆EDC, form the
interior “corners” of your original triangle.
3. Select the three points A, B, and C, and choose
Iterate
Note that in this step
you map B to itself, as
this vertex is the same
in both the original
triangle and in the
lower-left corner
triangle.
Be careful not to
increase the number
of iterations too
quickly. Since each
iteration adds three
times as many new
triangles as the
previous iteration, the
construction quickly
becomes very
complex! Sketchpad
will start to slow
down if your sketch
contains iterations
more complex than
your computer can
handle gracefully.
A
F
B
E
D
C
from the Transform menu.
4. In the Iterate dialog box, map A ⇒ F, B ⇒ B, and C ⇒ D. This
maps the original triangle to the lower-left corner ∆ FBD. You
should see a series of triangles iterating into the lower-left corner of
your original triangle.
5. Use the Structure pop-up menu to add a new mapping to your
iteration rule. In the new mapping, map A ⇒ E, B ⇒ D, and
C ⇒ C. This now iterates your pre-image triangle to the lowerright, while simultaneously—by the previous map—iterating each
image to the lower-left.
6. Use the Structure pop-up menu again to add a third and final
mapping to your iteration rule. In this third mapping, map A ⇒ A,
B ⇒ F, and C ⇒ E. This iterates your previous mappings toward
the upper corner of the triangle.
7. Click the Iterate button to dismiss the dialog.
With your completed iteration selected, you can
A
increase or decrease the number of displayed
iterations by pressing the + or – key on the
keyboard.
F
E
If you could iterate an infinite number of times,
the result of this process would be a Sierpinski
gasket. If you imagine the area of your initial
triangle as having been replaced by the area of
B
C
D
three smaller triangles at each step, think for a
moment about what happens to the total area of
all of the smaller triangles as you increase the number of iterations.
Since the three smaller triangles didn’t cover the initial triangle, the area
must be getting smaller. Thus with each iteration the area becomes
smaller. What’s the limit of the area? How do you know? What happens
to the perimeter? Fractals frequently give rise to surprising properties.
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187
Transform Menu
You can visualize the areas by repeating the above steps in a new
sketch, using a ∆ABC in which you have used Polygon Interior to
construct the triangle interior. When you’re done specifying your three
mappings and have dismissed the Iterate dialog box, be sure to select
and hide the original interior of ∆ABC so you can see the iterated
smaller triangle interiors “inside” your original triangle interior.
Tables of Iterated Values
If one or more visible measured values change as a result of an
iteration, Sketchpad creates a table of iterated values when you create
the iteration. This table contains one column for each visible value
affected by the iteration, as well as an initial column—labeled n—that
indicates the level of iteration. Each row in this table describes the
values of the measurements at the indicated level of iteration.
For example, in a sketch containing
a parameter Seed, with an initial value
of 100 and a calculation Seed/2, if
you iterate the pre-image parameter
Seed to the first image calculation
Seed/2, Sketchpad produces the
table shown at right, containing the
iterated images of Seed/2 as Seed ⇒
Seed/2.
seed = 100.00
seed
2
= 50.00
n
seed
2
0
50.00
1
25.00
2
12.50
3
6.25
The number of rows in a table of iterated values changes automatically
as you increase or decrease the level of iteration. If you don’t want to
use the table of iterated values, uncheck Tabulate Iterated Values in the
Iterate dialog box’s Structure menu while defining the iteration, or
select and delete the table after creating your iteration.
See also: Tables (p. 34)
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Transform Menu
After you’ve
constructed an
iteration, you can use
its Iteration
Properties to
experiment with both
choices for how to
iterate points
constructed on path
objects.
Random Iterated Points
Sometimes you may find yourself
specifying a point on an object as the
first image of an iterated pre-image.
For instance, rather than map a
triangle’s vertices to its midpoints as
in the previous example, you might map a triangle’s vertices to arbitrary
points constructed on its three segment edges. In this case, the
Structure pop-up menu gives you the choice of how you want
Sketchpad to iterate these arbitrary points on objects. When you choose
To Same Relative Location, Sketchpad displays each iterated image point at
the same relative location as that first image you chose in the Iterate
dialog box. If you drag that first image point to a new position on the
path on which it’s constructed, all of the iterated images adjust to the
same relative new position. On the other hand, if you choose To New
Random Locations, each iterated image of your initial point appears at a
new, random location on its (iterated) path, independent of the location
of the first image. This choice is useful if you’re exploring geometric
probability or other applications of randomness.
See also: Point On Object (p. 156), Iteration Properties (p. 134)
If an iteration’s depth
is determined by the
value of a parameter
or calculation, you
can’t use Properties to
change the depth.
Parametric Depth
When you define an iteration, you can use a measurement, parameter,
or calculation in your sketch to determine the depth of the iteration.
After you select the points or parameters that define the pre-image, but
before you choose the Iterate command, also select the value to be used
in defining the iteration depth. Hold down the Shift key while you pull
down the Transform menu, and the Iterate command becomes Iterate To
Depth. Choose this command and use the Iterate dialog box to define
the iteration normally. Once the iteration is defined, the integer part of
the value of the selected measurement, parameter, or calculation is used
to determine the iteration depth. (If the value is negative, the depth is
set to zero. If the value is too large to display all the iterated images,
Sketchpad uses the maximum depth at which the images can be
displayed.)
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Transform Menu
You cannot construct
a terminal point on an
iteration that uses
multiple iteration
maps.
When you select an
iterated point image,
the Iterate command
changes to Terminal
Point.
Terminal Point
On occasion you may want to make
use of the very last point of an iterated
point object. You may want to attach a
construction to this point, or you may
want to measure some quantity that
depends on the point.
P
To construct the terminal point of an
ground
iterated point image, select the iterated
Distance P to ground = 1.75 cm
image of the point. From the
Transform menu, choose Terminal Point.
The terminal point of the iterated point image is constructed. If the
depth of the iteration changes, the terminal point moves accordingly.
In the example above, an iteration was used to construct the flight path
of a thrown ball. (The pre-image points, not shown here, defined the
initial velocity of the ball and the strength of gravity.) To determine the
height of the ball after some number of iterations, the terminal point
was constructed and used to measure the distance between the terminal
point and the ground.
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Measure Menu
The Measure menu commands allow you to
measure numeric properties of selected
objects. The commands in the top portion
of the menu measure objects’ geometric
properties; the commands in the bottom
portion measure analytic properties. The
Measure menu also contains a powerful
calculator that allows you to derive new
properties by calculating relationships between
existing measurements. For example, you can
use Calculate to sum the measured interior
angles of a triangle or compute the ratio of a
circle’s measured circumference to its radius.
If you want to
measure a specific
property and its
command is not
available, make sure
you have only that
command’s
prerequisites selected.
(You may have too
many or too few
objects selected.)
To measure an object’s properties, select
that object and choose from the available
commands in the Measure menu. Sketchpad
produces a measurement—a named numeric value in the proper
units—as the result. When you drag or change an object that you’ve
measured, the measured value changes accordingly.
A
m∠ABC = 55°
m∠BCA = 34°
B
C
m∠CAB = 91°
m∠ABC+m∠BCA+m∠CAB = 180°
Three measurements and a calculation
Each command in the Measure menu is available only when objects
appropriate to that measurement are selected. In other words, like
those in the Construct menu, commands in the Measure menu require
certain selection prerequisites.
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Measure Menu
Here’s a summary of the selection prerequisites for each command:
To use this
command:
Select:
Length
one or more segments
Distance
two points, or one point and one straight object
Perimeter
one or more polygon, arc sector, or arc segment interiors
Circumference
one or more circles or circle interiors
Angle
three points (select the vertex second)
Area
one or more interiors or circles
Arc Angle
one or more arcs, or a circle and two or three points on
the circle
Arc Length
one or more arcs, or a circle and two or three points on
the circle
Radius
one or more circles, circle interiors, arcs, or arc interiors
Ratio
two segments or three collinear points
Calculate
(always enabled)
Coordinates
one or more points
Abscissa (x)
one or more points
Ordinate (y)
one or more points
Coordinate Distance
two points
Slope
one or more straight objects
Equation
one or more lines or circles
See also: Measurements, Calculations, and Parameters (p. 18), Selection Arrow Tools
(p. 70)
Length
The length of a
segment is equal to
the distance between
its endpoints.
Selection prerequisites: One or more segments.
Measures the length of each selected segment, using the distance units
chosen on the Units panel of the Preferences dialog box.
See also: Units Preferences (p. 136)
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Measure Menu
Distance
Selection prerequisites: Two points, or one point and one straight object.
To measure the
distance between
two points on a
coordinate system in
grid units, use the
Coordinate Distance
measurement instead.
Measures the distance between two
points, or the distance from a point to a
straight object, using the distance units
chosen on the Units panel of the
Preferences dialog box.
A
B
C
0.48 in.
Distance A to BC = 0.48 in.
The distance from a point to a line is the
shortest distance from the point to the
line and is measured along the perpendicular. The distance from a point
to a ray or segment is defined to be the same as the distance from the
point to the straight line that contains the ray or segment.
See also: Units Preferences (p. 136), Coordinate Distance (p. 199)
Perimeter
Selection prerequisites: One or more polygon, arc sector, or arc segment interiors.
Measures the perimeter of each selected polygon or arc interior, using
the distance units chosen on the Units panel of the Preferences dialog
box. The perimeter of an arc sector is the sum of the arc length and the
lengths of the two radii bounding the arc sector. The perimeter of an
arc segment is the sum of the arc length and the length of the chord
bounding the arc segment.
See also: Units Preferences (p. 136), Coordinate Distance (p. 199)
Circumference
Selection prerequisites: One or more circles or circle interiors.
Measures the circumference of each selected circle or circle interior,
using the distance units chosen on the Units panel of the Preferences
dialog box.
See also: Units Preferences (p. 136)
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Measure Menu
Angle
Selection prerequisites: Three points.
When measuring an
angle, always select
the vertex second.
Measures the angle defined by the three selected points, using the angle
units chosen on the Units panel of the Preferences dialog box. The first
selected point defines the initial side of the angle, the second defines
the vertex, and the third defines the terminal side of the angle.
If angle units in Preferences are set to directed degrees or radians, the
value of the measurement can be either positive or negative. A counterclockwise angle results in a positive measurement, and a clockwise
angle results in a negative measurement. Possible values range from
–180° to +180°, or from –π radians to π radians.
If the angle units are set to degrees, all angle measurements are positive,
between 0° and 180°.
See also: Units Preferences (p. 136), Arc Angle (p. 195)
Area
Selection prerequisites: One or more interiors or circles.
Measures the area of each selected polygon interior, circle, circle
interior, arc segment interior, and arc sector interior, using the distance
units chosen on the Units panel of the Preferences dialog box.
See also: Units Preferences (p. 136)
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Measure Menu
Arc Angle
Arc Angle (sometimes
also called arc measure)
refers to the central
angle—that is, the
angle formed by the
radii connecting the
circle’s center to the
endpoints of the arc,
as shown at right.
Selection prerequisites: One or more arcs, or a circle and two or three points on
the circle.
If one or more arcs are selected, this command
measures the angle of each selected arc, using the
angle units chosen on the Units panel of the
Preferences dialog box. If the angle units in
Preferences are set to degrees or directed degrees,
the value ranges from 0° to 360°. If the angle
units are set to radians, possible values range
from 0 radians to 2π radians.
m AB on c1 = 90°
A
B
P
m∠APB = 90°
c1
If a circle and two points are selected, this
command measures the angle of the minor arc
on the circle defined by the two selected endpoints. If the angle units in
Preferences are set to degrees, possible values range from 0° to 180°. If
the angle units in Preferences are set to directed degrees, values range
from –180° to 180°; clockwise arcs result in negative values, and
counter-clockwise arcs result in positive values. If the angle units are set
to radians, possible values range from –π radians to π radians.
If a circle and three points are selected, this
command measures the angle of the minor or
m ACB on c1 = 270°
major arc that starts at the first selected point,
B
A
passes through the second, and ends at the
third. If the angle units in Preferences are set
P
to degrees, possible values range from 0° to
360°. If the angle units in Preferences are set
to directed degrees, possible values range from
c1
C
–360° to 360°; clockwise arcs result in negative
values, and counter-clockwise arcs result in
positive values. If the angle units are set to radians, possible values
range from –2π radians to 2π radians.
See also: Units Preferences (p. 136), Angle (p. 194)
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195
Measure Menu
Arc Length
Selection prerequisites: One or more arcs, or a circle and two or three points on
the circle.
If one or more arcs are selected, this command measures the length of
each selected arc, using the distance units chosen on the Units panel of
the Preferences dialog box. If a circle and two points are selected, this
command measures the length of the minor arc on the circle defined by
the two selected endpoints. If a circle and three points are selected, this
command measures the length of the minor or major arc on the circle
that starts at the first selected point, passes through the second, and
ends at the third.
See also: Units Preferences (p. 136)
Radius
Selection prerequisites: One or more circles, circle interiors, arcs, or arc interiors.
The radius of an arc is
the same as the radius
of the circle on which
the arc falls.
Measures the radius of each selected circle, circle interior, arc, arc sector
interior, or arc segment interior, using the distance units chosen on the
Units panel of the Preferences dialog box.
See also: Units Preferences (p. 136)
Ratio
Collinear points are
points that lie on the
same straight line.
196
Selection prerequisites: Two segments or three collinear points.
If two segments are selected, this command measures the ratio of the
lengths of the segments. The length of the first selected segment is the
numerator of the ratio, and the length of the second is the
denominator.
The Geometer’s Sketchpad Reference Manual
Measure Menu
If three collinear points A, B, and C
are selected, this command
measures the ratio of the distance
from A to C divided by the distance
from A to B. If points B and C are
on the same side of A, the ratio is
positive; if B and C are on opposite
sides of A, the ratio is negative.
AC
= 0.75
AB
A
C B
AC
= 1.50
AB
A
B
C
Another way to think of the ratio
AC
= -0.25
defined by three collinear points is
AB
to think of a number line with its
C A
B
origin at point A and its unit point
at B. The position of point C on this
number line determines the value of the measurement.
How To . . . Construct a Slider
You can also create a
numeric value
directly, using the New
Parameter command
from the Graph
menu. The advantage
of the slider described
here is that sliding the
control point back
and forth provides a
powerful, visual way
to change the value.
Often it is useful to construct a slider for controlling a numeric value.
For example, if you wanted to create a graph of the line y = mx + b in
which you can slide a point back and forth to control the values of m
and b, you might want to set up sliders for m and b. To make a slider
for m:
1. Construct a line through two points A and B.
2. Use the Point tool to construct a point C on the line.
3. Select A, B, and C in that order, and choose Ratio from the Measure
menu.
AC
= 0.61
AB
A
C
B
4. Select and hide the line and point B, leaving only two points and
the measured ratio.
5. Construct a segment from A to C.
AC
= 0.61
AB
A
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C
197
Measure Menu
6. Double-click the measured ratio with the Text tool, and set its label
to m.
m = 0.61
A
C
Your basic slider is complete. As you drag C, the value of m changes
accordingly. If you want, you can improve on the slider by hiding A, by
hiding the label of C or changing C’s label to something more
meaningful, or by using Calculate to multiply m by a constant, creating a
new value that spans a larger (or smaller) range than the original value
of m as you drag C.
Calculate
Selection prerequisites: None.
This command displays the
m∠ABC+m∠BCA+m∠CAB = 180.00°
Calculator and allows you
to create a calculation in
the sketch. The calculation can use constants and mathematical
operations, and it can use measurements, calculations, and parameters
that already exist in the sketch. The calculation can also use Sketchpad’s
standard functions as well as any user-defined functions that already
exist in the sketch.
See also: Calculator (p. 49), New Function (p. 207), Calculations (p. 18)
Coordinates
Selection prerequisites: One or more points.
Measures the coordinates of each selected point with respect to the
marked coordinate system. If there is no marked coordinate system,
Sketchpad marks an existing coordinate system or creates a new one.
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Measure Menu
If the coordinate system is square or rectangular, the coordinates are
measured in (x, y) form. If the coordinate system is polar, the
coordinates are measured in (r, θ ) form.
See also: Abscissa (p. 199), Ordinate (p. 199), Grid Form (p. 203), Coordinate Systems
and Axes (p. 21)
Abscissa (x)
Selection prerequisites: One or more points.
Measures the abscissa (x value) of each selected point with respect to
the marked coordinate system. If there is no marked coordinate system,
Sketchpad marks an existing coordinate system or creates a new one.
See also: Coordinates (p. 198), Ordinate (p. 199), Grid Form (p. 203), Coordinate
Systems and Axes (p. 21)
Ordinate (y)
Selection prerequisites: One or more points.
Measures the ordinate (y value) of each selected point with respect to
the marked coordinate system. If there is no marked coordinate system,
Sketchpad marks an existing coordinate system or creates a new one.
See also: Coordinates (p. 198), Abscissa (p. 199), Grid Form (p. 203), Coordinate
Systems and Axes (p. 21)
Coordinate Distance
Selection prerequisites: Two points.
Measures the distance between the two points based on the marked
coordinate system. If there is no marked coordinate system, Sketchpad
marks an existing coordinate system or creates a new one.
This command differs from the Distance command because the
coordinate distance is based, not on physical distance (that is, not on
inches or centimeters), but on the unit size of the marked coordinate
system. A coordinate distance measurement has no units.
See also: Distance (p. 193), Coordinates (p. 198), Coordinate Systems and Axes (p. 21)
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199
Measure Menu
Slope
Selection prerequisites: One or more straight objects.
Measures the slope of each selected line with respect to the marked
coordinate system. If there is no marked coordinate system, Sketchpad
marks an existing coordinate system or creates a new one.
See also: Coordinate Systems and Axes (p. 21)
Equation
Selection prerequisites: One or more lines or circles.
Measures the equation of each selected object with respect to the
marked coordinate system. If there is no marked coordinate system,
Sketchpad marks an existing coordinate system or creates a new one.
The equation of a line is expressed in one of the following three forms:
In other words, a
Euclidean circle
described by an
equation in square
coordinates has a
circle’s equation. The
same shape, described
in nonsquare
coordinates, has the
equation of an ellipse.
200
•
y = c for a horizontal line
•
x = c for a vertical line
•
y = m ⋅ x + b for any other line
For a circle, the equation is expressed in one of the following two
forms:
•
(x − h )2 + ( y − k ) 2 = r 2 if the coordinate system is square
•
( x − h) 2 ( y − k ) 2
+
= 1 if the coordinate system is nonsquare
a2
b2
See also: Coordinate Systems and Axes (p. 21), Grid Form (p. 203)
The Geometer’s Sketchpad Reference Manual
Graph Menu
The Graph menu allows you to create
and manipulate coordinate systems, to
create parameters and functions, to
find the derivative of a function, to
plot points and functions on the
coordinate axes, and to tabulate
measured values. This menu offers a
variety of commands that serve as a
foundation for investigations and
activities in analytic geometry and
algebra.
In addition to the commands on the
Graph menu, the commands at the
bottom of the Measure menu allow
you to measure various quantities on
the coordinate plane, including coordinates of points, coordinate
distances between points, slopes of straight objects, and equations of
lines and circles.
See also: Coordinates (p. 198), Abscissa (p. 199), Ordinate (p. 199), Coordinate Distance
(p. 199), Slope (p. 200), Equation (p. 200)
Define Coordinate System
If you choose a
command, such as
Measure | Coordinates,
that requires a
coordinate system but
you haven’t created
one in your sketch,
Sketchpad defines
and marks a
coordinate system for
you.
This command creates a new coordinate system and marks it as the
active coordinate system. The type and scale of the created coordinate
system depends on your selections, as described in this table.
Selections
Command
Result
One point
Define Origin
Square coordinate system centered on
the selected point with default unit
scale
One circle
Define Unit Circle
Square coordinate system centered on
the selected circle with unit scale
determined by the circle’s radius
One defining
distance*
Define Unit
Distance
Square coordinate system centered on
a default origin with unit scaling
determined by the defining distance
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Graph Menu
One point and
one defining
distance*
Define Unit
Distance
Square coordinate system centered on
the selected point with unit scaling
determined by the defining distance
Two defining
distances*
Define Unit
Distances
Rectangular coordinate system
centered on a default origin, with
horizontal unit scaling determined by
the first selected distance and vertical
unit scaling determined by the second
selected distance
One point and
two defining
distances*
Define Unit
Distances
Rectangular coordinate system
centered on the selected point with
horizontal unit scaling determined by
the first selected distance and vertical
unit scaling determined by the second
selected distance
Nothing or
anything other
than the above
Define Coordinate
System (disabled
Square coordinate system with default
origin and unit scaling
if there’s already
a marked
coordinate
system)
* A defining distance can be either a segment or a distance measurement or calculation.
As described in the table, some commands define a coordinate system
with a default origin or default unit scaling. Sketchpad constructs a
default origin as an independent point in the center of your window.
The default unit scale is equal to the current distance unit chosen in the
Preferences dialog box, but can be adjusted by dragging the coordinate
system’s unit point or axis tick labels.
Since most activities require only a single coordinate system, the last
command choice (which defines an entirely default coordinate system)
is disabled if your sketch already has a coordinate system. If you really
want a second coordinate system, you must select the appropriate
objects as listed in the first column of the table.
See also: Coordinate Systems and Axes (p. 21), Mark Coordinate System (p. 203)
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Graph Menu
Mark Coordinate System
Selection prerequisites: a coordinate system’s axis, origin point, unit point, unit
circle, or grid
This command marks the coordinate system associated with the
selected object as the coordinate system on which to measure or plot
new objects.
When you have multiple coordinate systems, one of those coordinate
systems is the marked coordinate system. All of the Graph menu
commands apply to the marked coordinate system, as do those
Measure menu commands that require a coordinate system.
The marked coordinate system is normally the most recently created
coordinate system; you use this command to mark a different
coordinate system.
Normally you’ll have only a single coordinate system, and you won’t
have to worry about which coordinate system is marked.
See also: Define Coordinate System (p. 201), Coordinate Systems and Axes (p. 21)
Grid Form
Use this command to change the grid
appearance and scaling of the marked
coordinate system.
Sketchpad’s coordinate systems can
have three possible forms:
If the horizontal and
vertical axes of a polar
coordinate system
have different scaling,
the constant-distance
grid lines appear as
ellipses rather than
circles.
•
•
Polar Grid:
A polar coordinate system has a set
of grid lines that are circles (constant distance
from the origin, or r value) and a set of grid
lines that pass through the origin (constant
angle from the origin, or θ value). Any
coordinate system can be made polar.
Square Grid:
A square coordinate system has the
same scaling on the horizontal and vertical axes
and has horizontal (constant y value) and
vertical (constant x value) grid lines. Any
coordinate system except one defined in terms
of two different distances can be made square.
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1
0.5
1
1
0.5
1
203
Graph Menu
•
Rectangular Grid:
A rectangular coordinate system
has independent scaling for the horizontal and
vertical axes and has horizontal (constant y
value) and vertical (constant x value) grid lines.
Any coordinate system except one defined in
terms of a unit circle can be made rectangular.
6
4
2
1
If you choose one of these commands with a sketch that doesn’t yet
have a coordinate system, Sketchpad creates a default coordinate
system with the chosen form.
See also: Show Grid (p. 204)
Show Grid
This command shows or hides the grid lines of the marked coordinate
system.
When the grid of the marked coordinate system is showing, the
command changes to Hide Grid.
If you choose Show Grid with a sketch that doesn’t yet have a coordinate
system, Sketchpad creates a default coordinate system.
If you press the Shift key when you activate the Graph menu, the
command changes to Show Coordinate System or Hide Coordinate System, and
has the effect of showing or hiding the grid, the axes, and the origin of
the marked coordinate system.
See also: Grid Form (p. 203), Snap Points (p. 204)
Snap Points
Use Snap Points when
you want to work
with “nice” wholenumber coordinates.
When active, this command causes independent points to snap to
nearby locations when you drag them. Choose this command once to
activate it. When snapping is active, a check-mark appears next to the
command. Choose the command a second time to deactivate snapping.
The locations to which points snap when this command is active
depend on the grid form of the marked coordinate system. For square
and rectangular grids, points snap to locations with whole number
coordinates (that is, to the integer lattice). For polar grids, points snap to
locations whose distance from the origin is a whole number and whose
angle with respect to the horizontal axis is a multiple of 15°. In either
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Graph Menu
case, points you drag when Snap Points is active only snap to locations
that are relatively close by. In other words, if you have rescaled your
coordinate system so that the distance between integer coordinates is
great, points you drag will only snap to integer locations that are close
to them.
See also: Grid Form (p. 203)
Plot Points/Plot Table Data/Plot As (x, y)/
Plot As (r, theta)
This command plots one or more points on the marked coordinate
system at the specified coordinate location. If there is no marked
coordinate system, Sketchpad creates a default coordinate system.
Depending on the objects you select, the command appears as Plot As
(x,y), Plot As (r, theta), Plot Table Data, or Plot Points.
Use plotted values to
graph measurements
or calculations that
change over time.
Plot As (x,y)/Plot As (r, theta)
If you have two selected measurements or calculations, the command
appears as Plot As (x, y) or (if there’s a marked polar coordinate system)
as Plot As (r, theta). When you choose the command, Sketchpad plots the
point determined by the two selected values, with the first value used as
the x or r coordinate and the second as the y or θ coordinate. As the
measurements change, the position of the plotted point changes to
match.
Plot Table Data
If you have a single selected table,
the command appears as Plot Table
Data. This command displays a
dialog box that allows you to plot
a point on the marked coordinate
system for each row of the
selected table. The dialog box
allows you to choose whether to
plot the table data using
rectangular or polar coordinates,
and to choose which column of
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205
Graph Menu
the table to plot as the x (or r) coordinate and which to plot as the y (or
θ) coordinate.
See also: Tables (p. 34), Tabulate (p. 209)
Plot Points
If you don’t have a selected table or
two selected values, the command is
Plot Points. This command displays a
dialog box that allows you to plot
one or more points on the marked
coordinate system by entering the
coordinates of each point to plot.
In the dialog box:
You can type
numbers with decimal
points, or you can
type expressions such
as 2+3 and π/4.
•
Choose Rectangular to plot points using (x, y) values or choose
Polar to plot them using (r, θ ) values.
•
Click Plot to plot the current values; the dialog box remains open
to allow you to plot additional points.
•
After plotting your last point, click Done to close the dialog box.
New Parameter
Use this command to create a new parameter in your sketch. A
parameter is a number that can easily be changed. It’s convenient to use
parameters in places where you need to have a number but want to be
able to change that number easily.
You can use parameters in calculations, in functions, and as values by
which to transform objects. For example, you might create two
parameters, named m and b, and use them in plotting the function
y = mx + b . Or you might create a parameter that varies from 0º to 360º
and use it as a marked angle to rotate a polygon.
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Graph Menu
When you choose New Parameter, a dialog
box appears.
Name: You can type a new name for
the parameter. If you want the name to
have a subscript, enter the subscript
within square brackets at the end of the
name. For example, the default name
t[1] appears as t1 in the sketch.
To change the value
of a parameter, you
can double-click it
with the Arrow tool,
use the Animate or Edit
Parameter commands,
or create an
Animation action
button.
Value: You can set the initial value of the parameter by typing a value in
this field.
Units: You can choose whether the parameter’s value uses no units, uses
the current angle unit, or uses the current distance unit.
See also: Measurements, Calculations, and Parameters (p. 18), Units Preferences (p. 136),
Animate (p. 153), Edit Parameter (p.120)
New Function
This command displays the Calculator and
allows you to create a new function in the
f(x) = a⋅x2+b⋅x+c
sketch. The function can use constants and
mathematical operations, and it can use
measurements, calculations, and parameters that already exist in the
sketch. The function can also use Sketchpad’s standard functions as
well as other user-defined functions that already exist in the sketch.
See also: Calculator (p. 49), Functions (p. 27), Plot Function (p. 207)
Plot Function/Plot New Function
Use this command to plot one or
more selected functions or to
create and plot a new function if
no objects are selected.
If you’re plotting a new function,
the Calculator appears, just as it
does for the New Function
command, to allow you to define
the function to be plotted.
6
4
a = 1.00
b = 1.00
c = 1.00
2
f(x) = a⋅x2+b⋅x+c
5
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207
Graph Menu
Once you’ve plotted a function, you can change the domain or the
number of samples by using the Plot Properties dialog box. You can
also change the domain by dragging the arrows at the ends of the
function plot.
To change the form of the function plot—to plot your function f as a
polar function r = f (θ ) or as an inverse plot x = f (y )—change the
equation of the function itself. (To do this, select the function and
choose Edit Function from the Edit menu. When the Calculator appears,
choose the form you want from the Equation pop-up menu.)
See also: Calculator (p. 49), Properties (p. 120), Editing Functions (p. 29), Functions and
Function Plots (p. 27), Plot Properties (p. 124), New Function (p. 207)
Derivative
This command creates a new function that is the
derivative of the selected function with respect
to that function’s independent variable.
•
Derivative functions update automatically when the function they
differentiate is edited.
•
In most ways derivative functions behave just like other functions
you create. You can calculate with derivative functions, evaluate
derivative functions for specific arguments, and plot derivative
functions. You can even create the derivative of a derivative. The
only thing you can’t do with derivative functions is to edit them
directly; you must edit the original function instead.
•
Differentiation of complicated
functions can be time-consuming.
In these cases, a dialog box like
the one at right appears. Click
Cancel if you wish to interrupt
computation of an exact derivative. If you cancel, Sketchpad will
provide you with an approximate derivative, in the form
f ′( x) =
208
f(x) = x⋅sin(x)
f'(x) = x⋅cos(x)+sin(x)
f ( x + 0.005) − f ( x − 0.005)
0.01
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Graph Menu
See the chapter on
Sketchpad’s Internal
Mathematics (p. 246)
for a more detailed
discussion of
derivatives.
This approximate derivative can still be plotted, and can even be
differentiated itself. Although there is some loss in accuracy, in most
cases the approximation resembles the exact derivative.
See also: Functions and Function Plots (p. 27)
Tabulate
This command creates a new table of the selected measurements’
values. Sketchpad’s tables contain entries describing a set of
measurements’ values at different points in time. In addition to
measured values, you can also tabulate parameters, calculations,
measured equations, functions, and composite captions. This command
is available when the selection contains only objects that can be
tabulated.
By default, new tables contain one row that dynamically tracks the
current values of all tabulated measurements. Values in this row change
as measured objects are dragged or animated. You can turn off
dynamically changing values by using the table’s Table Properties panel.
See also: Add Table Data (p. 209), Remove Table Data (p. 210), Table Properties
(p. 127), Composite Captions (p. 36)
Add Table Data
Shortcut: Add a
single row to a table
by double-clicking it
with the Arrow tool.
This command displays a dialog box allowing you to add one or more
rows of data to a table. Each row of data captures the values of the
tabulated measurements at the moment the row is added. Add Table Data
is available when a single table created by the Tabulate command is
selected.
In the dialog box:
1.
Choose Add One
Entry Now to add
a single row to
your table,
capturing the
current values of
tabulated
measurements.
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209
Graph Menu
Sketchpad lets you
collect up to 25 rows
of new data at a time.
If you want to collect
more than that,
collect them in groups
of 25 rows at a time.
Choose Add Entries As Values Change to add one or more rows
automatically when the tabulated measurements next change. In
the edit boxes, enter how many rows you wish to add to the table,
and the rate at which to add them.
Note: You cannot use Add Table Data to add data to a table of iterated
measurements created using the Iterate command. If you want to add
data to a table of iterated measurements, you must increase the level of
iteration using Iteration Properties (or by selecting the iterated
measurement table and pressing the + key).
2.
See also: Tabulate (p. 209), Remove Table Data (p. 210), Iterate (p. 181)
Remove Table Data
Shortcut: Remove
the most-recently
added row from a
table by pressing the
Shift key while
double-clicking the
table with the Arrow
tool.
This command displays a dialog box allowing you to remove data from
a table. It is available when a single table created by the Tabulate
command is selected. In the dialog box, choose whether to remove the
most recently added row or all previously added rows from the table.
When you remove all previously added rows, Sketchpad adds a new
first row containing the tabulated measurements’ current values.
Note: You cannot use Remove Table Data to remove data from a table of
iterated measurements created using the Iterate command. If you want
to remove data from a table of iterated measurements, you must
decrease the level of iteration using Iteration Properties (or by selecting
the iterated measurement table and pressing the – key).
See also: Tabulate (p. 209), Iterate (p. 181)
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Window Menu
This menu is available only with Microsoft
Windows and Mac OS X editions of
Sketchpad. It includes commands for
arranging sketch windows on the screen and
for navigating among those windows.
Zoom, Minimize (Mac OS X)
These commands zoom or minimize the
active sketch window.
Bring All to Front (Mac OS X)
This command brings all Sketchpad
windows in front of all non-Sketchpad windows.
Cascade, Tile (Microsoft Windows)
These commands arrange all open document windows in either an
overlapping or a tiled fashion, starting from the upper left corner of the
main Sketchpad window.
Window List
Choose a document from this list to bring that document’s window in
front of all other document windows. This list contains up to ten
documents, with a checkmark indicating the document whose window
is in front of all other document windows.
More Windows (Microsoft Windows)
This command appears at the bottom of the window list when you
have more than ten open document windows. Use this command to
bring to the front a window that doesn’t appear among the first ten
in the list.
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Help Menu
The Help menu contains commands that you can
use to get help on using Sketchpad. The Sketchpad
help system is an electronic version of this Reference
Manual, including information on all of
Sketchpad’s commands and tools, contains a
myriad of useful tips and techniques, and provides
a table of contents and index to make it easy for
you to find the information you’re looking for.
Contents takes you to the first page of the help
Microsoft Windows
users can press F1
system, from which you can get an overview of Sketchpad’s help
while pointing to a
system, access the various help topics, and look up specific terms and
menu command to
phrases.
get help on that
command, even if the What’s New gives you a summary of the differences between Sketchpad
command is disabled.
Version 3 and Sketchpad Version 4, and also provides information
Windows users can
about any new features or late-breaking news which is not in the
also press F1 when a
dialog box is open to
printed documentation.
get help on that dialog
Elements describes the windows, pages, and tools of Sketchpad
box.
documents; the various kinds of objects you can create; and the Motion
Controller, Text Palette, and Calculator.
Toolbox
Sketchpad’s help
system is only
available on
computers that have a
web browser installed.
provides information on using the tools in Sketchpad’s toolbox.
allows you to quickly find information about using any of
Sketchpad’s menu commands.
Menus
describes the various keyboard shortcuts available in
Sketchpad, and the special features you can take advantage of by using
the keyboard.
Keyboard
describes several advanced features of Sketchpad that
are of particular interest to experienced Sketchpad users.
Advanced Topics
appears here in the Windows version only; it appears on
the Apple menu in the Macintosh version. This command displays a
dialog box identifying the registered user name, the specific version
number of the program, and assorted program information and credits.
About Sketchpad
In addition to the Help menu, most of Sketchpad’s dialog boxes have
Help buttons that you can click to get specific information about using
that dialog box.
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Context Menu
The Context menu provides a convenient
shortcut to many of the menu commands that
apply to a specific document or object. The
Context menu does not appear in the main
menu itself; it appears only when you request it.
To display the Context menu:
When the Context
menu appears, if
there’s room on the
screen, the menu
appears in such a
position that either
Preferences or
Properties is directly
under the mouse.
This makes it
especially easy to
choose these
commands.
Microsoft Windows: Click the right mouse
button in your sketch.
Mac: Hold down the Ctrl key while you click in
your sketch.
The commands available in the Context menu
depend on where in the sketch you click. If you
click in empty space the Context menu contains
commands that apply to your sketch as a whole.
If you click on an object in your sketch, the
Context menu contains commands that apply to
that object. The Context menu always shows only available commands.
For information on any of the commands in the Context menu, refer to
that specific topic in the Menu Reference.
See also: Menu Reference (p. 101)
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Keyboard Reference
The keyboard provides a convenient way to access many of
Sketchpad’s functions, providing shortcuts to common menu
commands as well as performing other operations. This section
describes all of Sketchpad’s keyboard actions.
Keyboard Menu Command Shortcuts
Many menu commands have shortcut keys, allowing you to access the
menu command directly from the keyboard whenever that command is
available. Most of these shortcuts require a modifier key (such as Alt or
Ctrl on Windows or a on Mac) to be held down while you type the
command’s shortcut key. Shortcut keys (and their modifiers) are listed
in the menu, directly across from the name of each command that can
be accessed by a keyboard shortcut.
For example, in the Windows File menu the Save command lists Ctrl+S
as its keyboard shortcut. In the Macintosh File menu it lists a+S as its
keyboard shortcut. Thus, you can save the active document by typing
the S key while holding down the Ctrl modifier key (in Windows) or the
a modifier key (on Macintosh).
See also: Menu Reference (p. 101)
The Esc Key
The Esc key provides a powerful general-purpose means to “escape
from” your current activity. Depending on the activity you’re presently
engaged in, Sketchpad’s response to the Esc key varies, but in general,
Esc reverts Sketchpad to a less “special” state. Each press of the Esc
key performs one of the following actions:
•
If a caption is being edited, Esc stops editing the caption.
•
If the Arrow tool is not active, Esc activates the Arrow tool.
•
If any object is selected, Esc deselects all objects.
•
If any object is animating, Esc stops all animations.
•
If any traces are visible, Esc erases all traces.
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Keyboard Reference
Press Esc repeatedly to revert your document to a “normal” state, with
no objects animating, no traces visible, no objects selected, and the
Arrow tool active.
See also: Editing Captions (p. 88), Selection Arrow Tools (p. 70), Selecting and
Deselecting Objects (p. 70), Stop Animation (p. 154), Erase Traces (p. 152)
Other Special Keys
Other keys perform special operations that affect your document’s view
and selected objects, or your choice of active tool in the Toolbox.
Key
Action
Delete or
Backspace
Deletes selected object(s) (same as Clear).
↑, ↓, ←, or →
Drags the selected object(s) one pixel in the indicated
direction. (Press and hold keys to drag longer distances.)
+ or –
When one or more loci or function plots are selected,
increases (+) or decreases (–) the number of samples in
those objects by a fixed percentage.
+ or –
When one or more iterated images are selected, increases
(+) or decreases (–) the number of iterations by one.
+ or –
When one or more parameters are selected, increases (+)
or decreases (–) those parameters’ values. (Press and hold
keys to continue adjusting values; use Parameter
Properties to choose the keyboard adjustment factor for
each parameter.)
Shift+↑ or
Shift+↓
Shift+← or
Shift+→
Alt (Windows) or
Option (Macintosh)
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Changes the active tool to the next higher (Shift+↑) or
lower (Shift+↓) tool in the Toolbox.
When the active tool is a Selection Arrow tool or a
Straightedge tool, changes the active tool to the next or
previous Selection Arrow or Straightedge in the Toolbox.
Temporarily invokes drag-scrolling. Press and drag in
your document to scroll it in an arbitrary direction.
Release the Alt (Windows) or Option (Macintosh) key to
resume the active tool.
The Geometer’s Sketchpad Reference Manual
Keyboard Reference
Optional menu commands:
Changes Save As to Save As HTML.
Changes Undo to Undo All.
Shift+menu
Changes Redo to Redo All.
Changes Preferences to Advanced Preferences.
Changes Iterate to Iterate To Depth.
Changes Show/Hide Grid to Show/Hide Coordinate System.
Shift+drag
Constrains straightedge tools.
Maintains a picture’s aspect ratio while resizing.
When changing object or text appearance, preserves
default settings for
• Line Width
Shift+format
• Color
• Font
• Style
• Size
Shift+Enter
Aligns the selected text objects or action buttons below
the first selected text object. If the objects are already
aligned, increases the vertical spacing between objects.
Shift+double-click
table
Removes the most recently added row from the table.
Alt+Arrow (Windows)
Sets text alignment when editing a caption. Use the left
arrow for left alignment, the right arrow for right
alignment, and either the up or down arrow for centered
alignment.
or
Option+Arrow
(Macintosh)
p (Windows)
Option+p (Macintosh)
Types π as part of a numeric value in a dialog box.
See also: Clear (p. 111), Dragging Objects (p. 73), Loci (p. 24), Functions and Function
Plots (p. 27), Plot Properties (p. 124), Parameters (p. 19), Parameter Properties (p. 126),
Iterations and Iterated Images (p. 31), Iteration Properties (p. 134), Toolbox Overview
(p. 68), Save As HTML (p. 104), Undo (p. 109), Redo (p. 110), Advanced Preferences
(p. 139), Parametric Depth (p. 189), Show Grid (p. 204), Pictures (p. 39), Constructing
Straight Objects (p. 14), Aligning Text Objects (p. 226), Remove Table Data (p. 210)
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Advanced Topics
Once you become familiar with the basic ideas behind Sketchpad—the
organization of its tools and menus, and the process of creating and
exploring mathematical ideas through Dynamic Geometry
constructions—you may wish to explore some of the program’s
advanced options. This section contains expert tips for using Sketchpad
efficiently; instructions for using Sketchpad to create both interactive
Dynamic Geometry web pages for posting on the Internet and highquality mathematical illustrations for use in other programs or in
printed documents; and an overview of the program’s internal
mathematics.
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Tips for Experts
As you gain experience in Sketchpad, you’ll naturally seek ways to
maximize your efficiency and productivity within the software
environment. This chapter contains a miscellaneous assortment of
recommendations and advanced techniques for using Sketchpad
productively.
About Sketchpad’s Menu Structure
Sketchpad’s menus are arranged thematically. While the File, Edit, and
Display menus contain commands relating to your Sketchpad
documents and workflow, the other menus and tools are more
mathematical in nature. Each of these menus presents a distinct
mathematical viewpoint and commands appropriate to that viewpoint.
Familiarizing yourself with their organizational structure can help you
plan your approach to a given construction problem or mathematical
challenge.
In that the Compass
tool does not retain a
fixed radius, it
technically provides a
collapsible, rather
than a noncollapsible,
compass.
220
•
The Toolbox’s Compass and Straightedge tools provide the
fundamental tools of compass-and-straightedge Euclidean
geometry.
•
The Construct menu contains additional commands for working in
compass-and-straightedge Euclidean geometry. (Many of the
objects you create with this menu could be created with the
Compass and Straightedge tools alone, though some would take many
steps to create with those tools.) Where the Circle By Center+Point
command is equivalent to the Compass tool, Circle By Center+Radius
allows you to construct circles of a given radius, acting as a
noncollapsible, rather than a collapsible, compass.
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Tips for Experts
Since compass-andstraightedge geometry
does not include tools
for specifying lengths
metrically, you cannot
construct a segment
of a given length—
say, 5.0 cm—using
only the Construct
menu. However, since
the Transform menu’s
operations are metric
in nature, you can use
its commands to
produce such a result.
•
The Transform menu contains commands drawn from the
perspective of a metric transformational geometry. Use them to
construct or explore symmetries and other transformational
relationships. You may specify transformational parameters—such
as angles of rotation or scaling factors of dilation—either
geometrically, by referring to existing objects, or metrically, by
entering numeric angles and lengths (or by referring to numerical
values and calculations already defined in your sketch).
•
The Measure menu’s commands continue the metric theme and
offer a variety of ways to determine numeric relationships in your
construction. The commands that appear in this menu above the
Calculate command can be thought of as ruler-and-protractor
operations: They measure distances, areas, and angles using the
metric units you choose in Preferences. The commands that appear
after the Calculate command are analytic in nature and measure
quantities in relationship to some (existing or newly-defined)
coordinate system.
•
Finally, the Graph menu’s commands continue the analytic
perspective and pursue it into algebra and calculus, offering
operations relating to coordinate systems and to variables and
functions considered abstractly.
While each of these menus reflects a unique mathematical perspective,
in the course of any Sketchpad activity you may move back and forth
between perspectives to focus on different aspects of your activity. In
particular, move from a geometric or spatial visualization to a numeric
perspective using commands from the Measure menu. (Think of these
commands as “turning shapes into numbers.”) Move from numbers
back into geometric or spatial visualizations (“turn numbers into
shapes”) by using the Plot As (x, y) and Plot Function commands, or use
Transform menu commands with marked numeric values as
transformational parameters.
Finally, in addition to the commands that produce or construct specific
mathematical relationships in your sketch, each menu contains one
command that produces a generalization of an arbitrary set of such
relationships over some change.
•
The Construct menu’s Locus command lets you visualize the position
of a constructed object over a change in one point’s position.
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Tips for Experts
•
The Transform menu’s Iterate command lets you visualize the orbit
of one or more objects over some number of repetitions of a
construction.
•
The Measure menu’s Calculate command lets you express a general
relationship arithmetically between two or more measured
quantities.
•
The Graph menu’s Plot Function command lets you visualize a
general function evaluated over a changing domain.
•
The Toolbox’s Custom tool lets you generalize a set of relationships
constructed between objects into a new tool that you can use to
replicate that construction on a new set of objects.
Mastering these advanced commands allows you to move beyond the
specific mathematical relationships, objects, tools, and commands that
form Sketchpad’s starting points and opens up a set of mathematical
curves, shapes, and construction tools limited only by your imagination.
See also: Menu Reference (p. 101), Locus (p. 166), Iterate (p. 181), Calculate (p. 198),
Plot Function (p. 207), Plot As (x, y) (p. 205), Custom Tools (p. 90)
Selection Techniques
The Select Parents and Select Children keyboard shortcuts can be very
useful when used together. For instance, if you have one side of a
triangle selected, you can quickly select all three sides as follows:
1. Select the two adjacent vertices using the keyboard shortcut for
Ctrl+U (Windows) or a+U (Mac).
2. Select all three sides using the keyboard shortcut for Select Children:
Ctrl+D (Windows) or a+D (Mac).
Thus with one side selected you can quickly select all three sides by
holding down the Ctrl or a key and pressing the keys U and then D.
Similarly, with one side selected you can quickly select all three vertices
by holding down the Ctrl or a key and pressing the keys U, D, and U.
Select Parents:
The selection rectangle also provides a way of selecting several objects
at once, and makes possible some useful shortcuts. Shown below are
two common cases in which using a selection rectangle is significantly
easier than clicking on the objects individually.
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Tips for Experts
Select segment and point before
constructing perpendicular.
Select 3 sides and no vertices before
constructing midpoints.
See also: Select Parents (p. 115), Select Children (p. 115), Selecting Objects Using a
Selection Rectangle (p. 71), Object Relationships: Parents and Children (p. 10)
Using Shortcuts
Using Sketchpad’s shortcuts can significantly reduce the amount of
time you spend navigating the menus and manipulating tools in the
Toolbox. Menu command keyboard shortcuts are listed in the menus
next to the commands themselves, but be sure to review the Keyboard
Reference section for a list of keyboard shortcuts beyond those that
correspond to menu commands.
Two shortcuts are worth special mention to the expert.
To display an
element’s Context
menu when clicking
on that element, click
with the right mousebutton in Windows,
or click while holding
down the Ctrl key on
a Macintosh.
•
The Esc key is a general-purpose shortcut for “escaping from” any
special program activity and returning the program to a more basic
state. Use the Esc key to deselect all objects and to revert to the
Arrow tool without using the mouse. If you are animating or
displaying the traces of animated objects, Esc can stop your
animations and erase their traces as well. Repeatedly pressing Esc
causes Sketchpad eventually to revert to its most basic state, with
no objects animating or selected, no traces visible, and the Arrow
tool active in the Toolbox.
•
The Context menu is a general-purpose shortcut for displaying
only the menu commands that are available and appropriate to a
clicked-on program element. Use the Context menu to display
commands appropriate to a single sketch object by clicking on that
object or to a single document by clicking in blank space within
that document’s window. When you click on a sketch object,
Sketchpad displays the Context menu for that object with Properties
as the default command choice. When you click in a document’s
blank space, Sketchpad displays the Context menu for that
document with Preferences as the default command choice. These
default choices make the Context menu the most convenient way
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Tips for Experts
to access these two powerful dialog boxes, each of which allows
you to modify the appearance and behavior of your chosen object
or document.
See also: Keyboard Reference (p. 215), Context Menu (p. 213), Properties (p. 120),
Preferences (p. 135)
Expressions in Dialog Boxes
To enter π in a dialog
box, type p (in
Windows) or
Option+p (on a
Macintosh).
Many of Sketchpad’s dialog boxes permit or require you to enter
various numeric quantities. Wherever you’re required to enter a
number, you can substitute an arithmetical expression—such as (1/3)
or 2π—instead. Type expressions combining numbers, parentheses,
addition (+), subtraction (-), multiplication (*), division (/), and
exponentiation (^). Sketchpad evaluates your expression and uses the
resulting value for the dialog box quantity.
Command-Line Flags for Sketchpad
(Windows Version)
With the Windows version of Sketchpad you can set several commandline flags that determine Sketchpad’s start-up behavior. You can use
command-line flags to maximize the Sketchpad application window, to
maximize the frontmost document window, to specify a default
document for Sketchpad to open, to determine the location of the
Sketchpad Preferences file, to specify the folder within which
Sketchpad will first open or save a document, or to specify a folder to
use as the Tool Folder.
To set command-line flags, you must create a shortcut to the program
itself. Windows automatically creates such a shortcut when you drag the
Sketchpad icon to the Start button to install Sketchpad in the Start
menu. You can also create such a shortcut on the desktop by rightclicking the Sketchpad icon and choosing Send To | Desktop. Consult your
computer’s Windows manual or online help to determine other ways to
create a shortcut. Once you have created a shortcut, follow these steps
to set its command-line flags.
1. If the shortcut is on the desktop, right-click the shortcut and
choose Properties.
If the shortcut is in the Start menu, activate the Start menu, rightclick on Sketchpad, then choose Properties.
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Tips for Experts
2. On the Shortcut panel of the Properties dialog box, click in the
Target edit box and type the desired command-line flags at the end
of the existing target. Here are examples of the command-line flags
you can use.
-ma Maximize the application window. The main Sketchpad
window will fill your screen.
-md Maximize the first document window. When you start
Sketchpad, the document window will fill the application
window.
"Read Me.gsp" Open the sketch named Read Me.gsp.
Normally, Sketchpad
uses the preferences
in the file Sketchpad
Preferences.dat from
the Windows folder.
-pref "h:\Sketchpad\Modified Preferences.dat" Start
Sketchpad using the preferences in the specified file.
-f "c:\Program Files\Sketchpad\Samples" Sketchpad will use
the Samples folder when it first opens or saves a sketch.
-t "Triangle Tools" Start Sketchpad using the folder named
Triangle Tools as the Tool Folder.
3. Click OK.
Your command-line flags will be in effect every time you run Sketchpad
from the modified shortcut.
For example, if Sketchpad is installed in the folder c:\Sketchpad, the
following target entry in the Shortcut Properties panel will start
Sketchpad with the document maximized in the application window,
using the folder named My Tools as the Tool Folder, and with the file
named Read Me.gsp open:
"c:\Sketchpad\GSP 4.0.exe" -md "Read Me.gsp" –t "My Tools"
The following target entry in the Shortcut Properties panel will start
Sketchpad with the file GSP Algebra Prefs.dat as the preferences file:
"c:\Sketchpad\GSP 4.0.exe" –pref "GSP Algebra Prefs.dat"
The following entry will start Sketchpad so that the first time the user
chooses File | Open or File | Save, the dialog box will show the folder
containing the Geometry sample sketches:
"c:\Sketchpad\GSP 4.0.exe" –f "Samples\Sketches\Geometry"
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Advanced Text Topics
Sketchpad’s text-alignment shortcut allows you to line up selected text
objects so that their left sides are aligned and they are spaced evenly
beneath the first selected object.
Sketchpad’s Merge Text command gives you the ability to combine
various captions, measurements, and labels into a single caption. Two
advanced applications of Merge Text offer much more precise control
over the formatting and layout of merged pieces of text, and allow you
to control the location and appearance of merged text in your sketch.
Sketchpad also allows you to modify the behavior of the Label Multiple
Objects dialog box by creating your own custom sequences.
Aligning Text Objects
When a sketch contains a number of text objects (captions,
measurements, parameters, calculations, functions, and action buttons),
you may want to line them up neatly to improve the appearance of the
sketch. In the following examples, the aligned arrangement on the right
looks neater than the unaligned arrangement on the left.
Measurements in
m∠ABC = 69.96°
ABC
m CA = 3.54 cm
m∠CAB = 54.79°
m∠BCA = 55.24°
m BC = 3.08 cm
Measurements in
ABC
m AB = 3.10 cm
m BC = 3.08 cm
m CA = 3.54 cm
m∠CAB = 54.79°
m AB = 3.10 cm
m∠ABC = 69.96°
m∠BCA = 55.24°
To align text objects, select them in the order in which you want them
to appear, from top to bottom. Then hold down the Shift key and press
the Enter key. The first selected object remains in its original position,
and the remaining selected objects line up immediately below the first.
If you accidentally
create too much
space, continue
holding the Shift key
and pressing the
Enter key until the
objects return to their
minimum spacing.
226
To increase the spacing between the newly-aligned objects, make sure
they are still selected. Then hold the Shift key and press Enter again.
Each time you press Enter, a small amount of space is added between
the objects. Continue holding Shift and pressing Enter until you have
the vertical spacing you want.
The Geometer’s Sketchpad Reference Manual
Advanced Text Topics
Merging Text with a Custom Template
When you choose Merge Text, by default Sketchpad merges the text of
your various selected objects into a single, horizontal result, in which
the text of each object runs into the text immediately following it, and
in which all of the resulting text shares the same text style. If you prefer
to use a more complex layout, or if you would like to control the text
style of separate components of your merged caption, you can use a
custom template to define the overall appearance of the final result of
using Merge Text.
A custom template is a caption that begins with the character “=” and
includes parameters—represented as “{1},” “{2},” and so forth—
indicating the various selected text components you wish to merge
together. For example, the custom template “={1}{2}{3}” describes
the default appearance when you use Merge Text to merge three pieces of
text together horizontally.
A custom template such as the one shown on left of the following
illustration, on the other hand, describes the appearance of a fraction,
in which the first selected text component appears italicized in the
numerator and the second selected text component appears in the
denominator. When applied to two selected pieces of text—for
example, captions A and B—the custom template on the left produces
an appearance like that shown at right.
=
{1}
{2}
+
A
+
B ⇒
A
B
To apply a custom template to merged text:
1. Create a custom template using the Text tool. A custom template is
any caption beginning with “=” and including “{1}.” Use “{1},”
“{2},” “{3},” and so on to refer to indicate the first, second, and
third pieces of text to which you’ll apply the template. Use the Text
Palette and symbolic notation tools to style your template or apply
other formatting. When you type a parameter into a custom
template, be sure to use the keyboard to type “{1},” not the
symbolic notation tools. Also, be sure all of the characters you
enter for a single parameter share a common text style. Otherwise,
Sketchpad won’t know how to format the result.
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Advanced Text Topics
2. Select the custom template first, and then select the various objects
Sketchpad only
applies a custom
template if the
template is the first
selected object when
you use Merge Text.
(captions, measurements, labeled objects, calculations, and
functions) to which you wish to apply it.
3. Choose Merge Text from the Edit menu.
Sketchpad displays the resulting merged text with the style and with the
layout determined by the template. Once the result appears, the
template no longer controls the result. (Changing the template after
you’ve applied it will not change the result.) You can re-apply the same
template to other pieces of selected text or delete it from your sketch
when you’re done with it. The resulting merged text can be split back
into its component pieces by using Split Merged Text, but if you split text
merged with a custom template, the style and layout determined by the
original template will be lost.
Merging Text to a Point
Merge Text can also be used to position a copy of a selected text object—
a caption, measurement, calculation, or function—at a location
determined by a geometric point in your sketch.
1. Select one text object and one point.
2. While holding down the Shift key, choose Merge Text To Point from
the Edit menu.
A merged copy of your selected text appears, centered on the point you
selected. As you cause the point to move (by dragging it, animating it,
or dragging or animating its parents), the merged text moves with it;
and as you drag the merged text, the point moves with it. (You may
wish to hide the point to make the text more legible.)
A caption merged to a point can be used to provide a richer description
of the point (or the object on which the point is constructed) than can
be entered in a label, because you can use the style buttons and
symbolic notation tools from the Text Palette to style the caption. A
measurement merged to a point can provide a useful visualization of
measured properties. For instance, you can merge the measure of an
angle to that angle’s vertex, or you can merge the length of a segment
to a (hidden) point on that segment. Note that the copy of your original
text merged to the point is not the measurement itself, however. It’s
only a display copy of your original measurement. If you want to use
the original measurement elsewhere in your sketch, you should not hide
it after merging its text to a point.
See also: Merge Text (p. 118), Split Merged Text (p. 118)
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Advanced Text Topics
Labeling Objects with a Custom Sequence
When you use the Label command from the Display menu to label
several objects, Sketchpad normally generates a sequence of labels by
changing only the ending of whatever you type as the first label. Thus if
you type A1 as the first label, Sketchpad generates a sequence in which
all the labels start with A, but with different numbers following the A:
A1, A2, A3, and so forth. Similarly, if you type 123a as the first label,
only the endings change: 123a, 123b, 123c, and so forth.
To generate a sequence in which the changing portion is not at the end,
select the objects to label and choose Label Objects from the Display
menu. In the Label Multiple Objects dialog box, define a custom
sequence by entering a special first label. This special custom sequence
label—similar to the custom template used when merging text—must
begin with an equal sign (“=”), and contain “{...}” immediately
following the part of the label that should change over the sequence.
Thus you can type “=P{...}1” to generate the sequence starting
with P1, Q1, and R1. Similarly, type “=A{...}[x]” to generate a
sequence that begins Ax, Bx, and Cx.
See also: Label Command (p. 150), Label Multiple Objects dialog box (p. 150), Merging
Text with a Custom Template (p. 227)
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Advanced Tool Topics
This chapter describes advanced options available to you when creating
custom tools.
Automatically Matching a Given Object
When you create a custom tool, there may be a particular given object
that you would like to always match to the same object in your sketch.
Normally you must match that given object each time you use the tool,
even though you’re matching it to the same object each time.
To save the trouble of clicking the object each time and to make the
tool easier to use, you can specify that the given object should be
automatically matched to the same sketch object each time the tool is
used. For example, suppose you create a tool that constructs one
segment on a Poincaré disk model of the hyperbolic plane. Such a tool
might have three given objects: a circle defining the Poincaré disk and
two points defining the segment’s endpoints. Since you’d like to use the
same tool repeatedly to construct multiple segments on the same
Poincaré disk, it’s inconvenient to have to match the given circle to the
same Poincaré disk each time you use the tool. You can change your
tool to automatically match this given circle to the appropriate circle in
your sketch, resulting in a tool that requires you to match only two
points each time you use it. Since the Poincaré disk circle is matched
automatically, this tool will always create segments on the same
Poincaré disk.
To match a given object automatically:
1. Open the Label Properties panel for the sketch object you want to
2.
3.
4.
5.
230
match automatically. Assign a distinctive label to the object.
Choose Show Script View from the Custom Tools menu to show the
Script View for the tool containing the given object to be matched
automatically.
Double-click the given object in the object list to open the
Properties dialog box.
On the Label Properties panel, assign the same label to the script
object that you assigned to the sketch object.
Check the Automatically Match Sketch Object checkbox and close
the Properties dialog box.
The Geometer’s Sketchpad Reference Manual
Advanced Tool Topics
If all the given objects
for a tool match
automatically, the tool
completes its
construction as soon
as you choose the
tool. The active tool is
then changed back to
the Selection Arrow
tool.
When you use the tool, it will automatically match the tool’s given
object to the sketch object with the same label. Provided the
appropriate checkbox is checked in Label Properties, a tool containing
a given point labeled “Center” will automatically match the point
labeled “Center” in your sketch. If no object in the sketch has a
matching label, the tool will require you to match the given object
manually.
When a given object in a tool is set to match automatically, it appears in
the Assuming section of the script view, rather than the Given section.
See also: Label Properties (p. 122), Script View (p. 59), Custom Tools (p. 90), Custom
Tools Menu (p. 91)
Generating Specific Labels
Normally when you use a tool, the results are assigned labels just the
way they would be if they were constructed normally, without using a
tool. But sometimes you may want to control the labels that are used
for the results of a tool. There are two types of labels you can specify
when you make the tool: constant labels and variable labels.
Setting a Constant Label
A result with a constant label is assigned the same label every time the
tool is used. For example, a particular line produced by a tool might
always be labeled “Mirror.”
To set a constant label for an object before you make the tool, use the
Label Properties panel of the Properties dialog box for the sketch
object that should have the constant label. Enter the desired label and
check the Use Label in Custom Tools checkbox.
To set a constant label for a tool object after you make the tool, use the
Custom Tools menu to show the tool’s script view. Double-click the
step corresponding to the resulting object that you want to have a
constant label. On that object’s Label Properties panel, enter the
desired label and check the Use Label in Sketches checkbox.
Setting a Variable Label
When a tool has a result with a variable label, the label of the resulting
object in the sketch is set based on the labels of other objects—objects
that correspond to givens or other results of the tool. For instance, you
can create a centroid tool that labels the centroid of a triangle based on
the labels of the vertices of the triangle.
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Advanced Tool Topics
A variable label specifies how the labels of the givens and other results
of the tool should be used in constructing the desired label. To specify
that a particular label should be used, start the label with an equal sign
(=) and enclose in curly brackets the numeric index of the object whose
label should be used. The number of an object is its position in the
tool’s script view, starting with 1 for the first given. If a tool has three
givens, the givens are numbered from 1 to 3 and the number of the
first step is 4.
For example, the following variable label combines the labels of the
first and second given objects of the tool:
={1}{2}
Square brackets at the
end of a label indicate
that the portion of the
label within square
brackets should
display as a subscript.
As a second example, if you create a
tool to construct the centroid of a
triangle, you can set the resulting
centroid’s label so that, if the tool is
used on points A, B, and C, the
centroid is labeled GABC. To
accomplish this, set the centroid label
in the tool to
=G[{1}{2}{3}]
B
A
GABC
C
To set a variable label for a tool object,
show the tool’s script view using the Custom Tools menu. Double-click
the step that should have the variable label to show its Properties. On
the Label Properties panel, enter the desired label and check the Use
Label in Sketches checkbox.
When you create a variable label for a resulting object, be sure to
specify in the curly brackets only objects on which the labeled object
depends. Otherwise it’s possible that the labeled object will be
produced before one of the objects that determines its label, and the
variable label will not be created correctly.
See also: Custom Tools (p. 90), Custom Tools Menu (p. 91), Label Properties (p. 122)
Alternate Tool Folders
In some environments, multiple users may share a single computer or
networked volume, and that volume may be configured to grant
different levels of access to different users. Depending on how the
computer or network is configured, some users may not have the ability
to modify folders inside the Sketchpad application folder. For example,
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Advanced Tool Topics
your network or computer may be configured to prevent student users
from modifying the contents of the Tool Folder inside the Sketchpad
application folder. Any tools that you (or the computer’s administrator)
choose to store in that Tool Folder will be accessible to all Sketchpad
users on the computer, but other users will not be able to add or
change those tools.
Under Mac OS X, if users without access to the central Sketchpad
application directory wish to have “personal” Tool Folders of their
own, they can create a folder named Tool Folder inside a folder named
Sketchpad in their personal Documents folder. Any tools they store in
this folder (that is, in Documents | Sketchpad | Tool Folder) will
appear in their Custom Tool menu along with tools from the main
(application directory) Tool Folder, but will not affect the contents of
the Custom Tool menu of other users of that Macintosh.
Under Microsoft Windows, you can use a command-line flag in
conjunction with a shortcut to Sketchpad to change the location of the
Tool Folder to a writable folder. You can also use this command-line
flag to create multiple shortcuts to Sketchpad, each of which starts
Sketchpad using a different set of tools. See Command-Line Flags for
Sketchpad (p. 224) for more information.
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233
Advanced Graphics Export
You can use Sketchpad to create images to paste into other programs—
word processors, illustration programs, page-layout programs, and
programs that create web pages. You can use Sketchpad to produce
illustrations for handouts, tests, and quizzes for your class, to illustrate
articles or books about geometry or algebra, or to decorate your
personal web site or a web site for your class or school.
This chapter describes several useful tips and techniques that can help
you produce attractive, high-quality images for such purposes.
Screen Captures
You can create a graphics file of the entire Sketchpad window,
including even menus and the cursor, by using the screen capture
capabilities built into Microsoft Windows and Macintosh. Press the
Print Screen key (Windows) or a+Shift+3 (Macintosh) to take a
picture of the screen. In Windows, the result is a bitmap on the
clipboard; you can paste this bitmap into any program which recognizes
bitmap files. On Macintosh, the result is a PICT file on the outer level
of the hard drive. The first such file is named Picture 1; subsequent files
are named Picture 2, Picture 3, and so forth.
Screen captures are excellent for showing what the entire screen looks
like and for including a menu or the cursor in your graphic. But a
screen capture does a crude job on Sketchpad graphics. Diagonal lines
and circles look jagged and blocky, text is of poor quality, and resizing
the image may give unexpected results.
Copying and Pasting Graphics
The clipboard
graphics format in
Windows is both
EMF (Enhanced
Metafile) and WMF
(Windows Metafile);
on Macintosh it’s
PICT. Most advanced
graphics programs
can use these formats.
234
You can produce high-quality images of geometric constructions using
the Copy and Paste commands. For instance, you can construct a triangle
in Sketchpad, select the triangle, and choose Copy from the Edit menu.
Switch to your word processor and use Paste to put an image of the
triangle into your word-processor document. When you use the Copy
command, the image placed on the clipboard is not a bitmap, but
encodes the actual graphics commands used to draw the object in the
sketch. The resulting image displays smooth diagonal lines, circles, and
text, even when scaled or when printed on a high-resolution printer.
The Geometer’s Sketchpad Reference Manual
Advanced Graphics Export
You can even use illustration programs to edit or embellish individual
components of images copied to the clipboard.
See also: Copy (p. 110)
Using Export Preferences
The Export Preferences panel has several settings you can use to obtain
high-quality graphics. Hold down the Shift key and choose Advanced
Preferences from the Edit menu to change the settings on this panel.
The graphics in this
manual were copied
using an export
setting of 400%. After
pasting they were
reduced to 25% in the
layout program.
Improving Positioning of Objects
Although each object is drawn smoothly, the accuracy with which
copied objects are positioned remains limited by the screen resolution.
When such images are scaled or printed at high resolution, visible
misalignment of the objects can result. To minimize misalignment, you
can set the Clipboard Image Scale to copy objects at 200%, 400%, or
800% of their normal on-screen size. When you paste the image into
your word processor or graphics program, reduce it by a corresponding
percentage. The result is an image with the same dimensions as the
original, but with greatly increased accuracy, suitable for high-resolution
printing and publication.
Arrowheads
If you want arrowheads to appear on your printed or copied image, you
can check the Include Arrowheads on Lines, Rays… checkbox.
Locus Quality
When exported or printed on a high-resolution printer, the positions of
the samples for point loci and functions can become visible, since the
intervals between samples are drawn as segments. To produce
smoother functions and loci, set the Locus/Plot Export Quality to 5x
or 10x to produce either five or ten times the normal number of
samples when printing or copying.
See also: Advanced Preferences (p. 139), Export Preferences (p. 139)
Cropping
Lines and rays extend far beyond the limitsof the screen. Depending on
the specifics of a sketch, other objects, particularly loci and function
plots, may have similar extents. But when you copy such objects, you
don’t want an image of infinite size. Accordingly, if an image being
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235
Advanced Graphics Export
copied extends beyond the edges of the window, Sketchpad crops the
image to the window dimensions.
You can use this behavior to control the exact extent of the image
copied to the clipboard. Before copying, resize and scroll your sketch
window so that it shows the desired portion of the sketch. When you
choose Copy, the clipboard image will be correctly cropped.
See also: Copy (p. 110)
Metafile Export (Windows Only)
You can use Save As to save your sketch as a graphics file that can be
used by many other Windows programs. The exported graphics include
all objects that appear within the window. Use the edges of the window
to crop the image that will be exported.
Both WMF and EMF
are common graphics
formats for Windows.
Newer applications
are more likely to take
advantage of the
features of enhanced
metafiles.
After you choose Save As, use the Save As Type menu to choose
Enhanced Metafile (*.emf) or Windows Metafile (*.wmf). Then click the Save
button.
If you want to export some but not all of the visible objects, use the
command.
Copy
See also: Save As (p. 103), Copy (p. 110)
PostScript and EPS Files
Under Mac OS X,
you can also use the
standard printing
dialog box to create a
PDF Preview file.
This file can be
opened by Adobe
Acrobat or other
PDF readers.
236
On both Macintosh and Windows, you can use PostScript printer
drivers to produce PostScript files—files that encode the graphics to be
printed in the form of a PostScript program. Such files are most
commonly used for printing, either at a later time or from a different
computer. But many such printer drivers also have the capability of
producing EPS (Encapsulated PostScript) files, which can be placed in
a wide variety of graphics programs on a wide variety of platforms.
To use your printer driver to produce an EPS file, first hide all objects
in the sketch except those you want to include in the EPS file. Then
choose Print, and use the Print dialog box to set your printer driver to
produce an EPS file. The settings to produce such a file vary according
to your platform and PostScript printer driver, and may not be available
with all drivers. (With the Macintosh Adobe PostScript printer driver
8.7, set the Destination to File and use the PostScript settings to change
the format to EPS. With certain Windows HP printer drivers, check the
The Geometer’s Sketchpad Reference Manual
Advanced Graphics Export
Print To File checkbox and use the Advanced button on Layout
Properties to set the Document PostScript options to EPS.)
See also: Print (p. 108)
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237
JavaSketchpad and Web-Based
Dynamic Geometry
JavaSketchpad is a
Sketchpad extension
that allows you to
place simple sketches
inside web pages you
publish on the
Internet. These
sketches appear as
illustrations in your
pages, and anyone
who visits your web
page can interact with
them—by dragging
points and pressing
action buttons—
directly from their
web browsers, even if
they don’t have a
copy of Sketchpad. If
you create web pages,
consider using
JavaSketchpad as a way to enhance any mathematics—such as personal
discoveries, journal articles, class syllabi, or assignments—that you post
to the web.
JavaSketchpad is
intentionally small in
comparison to The
Geometer’s
Sketchpad so that it
downloads quickly
when someone visits
your web page.
Not every sketch or activity you create in The Geometer’s Sketchpad
can be turned into an interactive JavaSketchpad illustration. Many of
Sketchpad’s advanced features require the full power of the program
itself, rather than the slimmed-down version that visitors to your web
page interact with. Also, while visitors can drag points and press action
buttons in sketches you post as JavaSketchpad illustrations, they cannot
draw or construct new objects. Despite these limitations,
JavaSketchpad can enhance your web-publishing options significantly.
Web Publishing Overview
While the full intricacies of web publishing fall beyond the scope of this
book, if you’ve created a web page before, using JavaSketchpad should
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JavaSketchpad
be relatively straightforward. (If you’ve not yet started web publishing,
consider purchasing a book on the subject or searching the web itself
for more information. There are thousands of sites devoted to
becoming a web author.) Before getting started with JavaSketchpad,
you should be familiar with these terms.
Because your browser
can open local files as
well as URLs, a folder
on your own hard
disk can serve as a
web directory for
JavaSketchpad testing
purposes, even if it’s
not accessible to the
general public via the
Internet.
•
HTML file: This is a document that describes the fundamental
appearance of a web page and is written in HTML, the internal
language of the web.
•
Web directory: This is a storage area on your computer, or a
computer you have access to, that contains HTML files that
describe web pages, as well as image files and other components
that appear within a web page or set of web pages that visitors can
see in their browsers.
•
Applet: This is a special computer program that resides in a web
directory and that provides extended functionality to the web pages
(HTML files) stored on that site. Applets are different from
browser “plug-ins.” Visitors have to install plug-ins to their
browsers themselves, before they can access sites requiring plugins. In other words, plug-ins reside in the visitor’s computer.
Applets, on the other hand, reside in the same web directory as the
HTML files that require them, so visitors don’t need to worry
about installing applets or reconfiguring their browsers to support
them. Instead, applets are like HTML files: a visitor’s browser
accesses them from your web directory as needed to display your
applet-enhanced web pages; the visitor doesn’t have to worry about
what happens “behind the scenes.”
Putting this all together, JavaSketchpad is an applet that you can place
in your web directory so that your HTML files present dynamic,
interactive Sketchpad illustrations to your visitors.
Essential JavaSketchpad Folder Structure
Two components work behind the scenes to provide a dynamic
Sketchpad illustration in a web page. The HTML file contains
information that describes the geometric construction to be visualized
in a language that JavaSketchpad understands. The applet—a separate
set of files—provides the functionality that interprets this description,
displays the figure in your visitors’ browsers, and lets them interact with
it. You can have many HTML files containing different illustrations
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239
JavaSketchpad
that all refer to the same applet, just as on your local computer you can
have many sketch documents that can all be opened by the same copy
of Sketchpad.
If you can’t find the
JSP folder, you can
reinstall it from your
Sketchpad CD-ROM.
Before you create your first web page containing a construction, locate
the JavaSketchpad applet itself. The applet consists of a folder titled JSP
and the complete collection of files within that folder.
By default, when you installed Sketchpad, this folder was installed in
the same directory as the Sketchpad application on your hard disk. You
can copy the JSP folder from there to wherever you choose—into
another folder, or onto your web server—but you should never change
the contents of the folder itself. The applet only works if all of its files
are in the JSP folder, with the same file and subfolder names as they
had when the applet was first installed.
It’s essential that you know where the JSP folder is because a web
browser must be able to access the JSP folder when it displays any web
page containing a Sketchpad illustration. By default, web browers assume
that the JSP folder is located in the same place as your HTML file. Therefore,
they’ll only work if you store the JSP folder (or a copy of it) in the same
folder as the web page itself.
Experts. If you don’t
want to store your
HTML files in the
same folder as the JSP
applet, specify a
relative URL from the
HTML file’s base
directory to the JSP
applet directory
anywhere on your
server by modifying
the <CODEBASE>
parameter in your
HTML file. See your
HTML reference
manual for more
details.
For example, in the following illustration, the Triangle web page has
been stored in a folder (in this case, named Web Folder), also contains a
copy of the applet folder (JSP). This is the correct relationship between
web pages you create and the JSP applet that a web browser requires,
whether the containing directory (in this case, Web Folder) is on your
local hard disk or your web server. If you don’t store the JSP folder in
the same folder as your HTML file, the web browser will not be able to
locate the applet and, therefore, won’t display your Sketchpad
illustration when you open the HTML file that describes it.
Proper folder structure (Windows and Macintosh)
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JavaSketchpad
Creating a JavaSketchpad Web Page
Once you’ve located your JSP folder, you’re ready to use The
Geometer’s Sketchpad to create your first JavaSketchpad web page:
1. Start Sketchpad.
If this is your first
time creating a
JavaSketchpad
illustration, start with
a simple construction
like a triangle.
The size of the
illustration in your
web page is the same
as the size of your
sketch window at the
moment you saved
the HTML file. Be
sure to decrease the
size of the window
before saving unless
you want very large
web page illustrations!
2. Create or open a sketch containing the construction you wish to
show in a web page.
3. Resize the sketch window to be the size of your intended
illustration on your web page. Adjust your construction so that it
appears as you’d like your visitors to first see it.
4. Choose Save from the File menu and save your document normally.
This way you’ll have a saved copy of the sketch from which you’re
about to create the web page.
5. Choose Save As from the File menu. (If you hold the Shift key while
pulling down the File menu, Save As becomes Save As HTML.)
6. In the Save As dialog box, change the file format or type to
HTML/JavaSketchpad Document.
Before saving, be sure to navigate to the folder that contains your
JSP directory. When you’re in the right place, enter an appropriate
filename and click Save.
Sketchpad creates a new web page—an HTML file—describing the
Sketchpad illustration.
Note: Sketchpad cannot open HTML documents, so saving your
document as HTML does not save your document in a way that
Sketchpad can reopen. This is the reason you did a normal save in
step 4 above.
7. Assuming all goes well, Sketchpad will ask if you want to preview
the web page in your browser. Click Yes.
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JavaSketchpad
8. Your browser opens and displays the web page. The first time your
browser opens a page containing JavaSketchpad illustrations, it may
take a few seconds to load.
9. Once your illustration appears, drag its independent points to
explore your construction. You’ve successfully created your first
Dynamic Geometry web page!
What Can Go Wrong
Because of the many factors involved, it may take a few tries to get
things right. Here are some common JavaSketchpad mishaps and steps
you can take to avoid them.
Unsupported Objects
Problem: When you save, Sketchpad warns you that not all objects were
successfully saved to JavaSketchpad format.
Cause: Because JavaSketchpad is smaller than “desktop” Sketchpad, it
supports fewer ways of defining objects than you can use in the
desktop application. If your sketch contains objects that JavaSketchpad
doesn’t support, Sketchpad warns you about them—and selects the
unsupported objects and their children in your sketch so that you can
tell which ones were not supported. (Even when your sketch has
unsupported objects, Sketchpad will save the objects that
JavaSketchpad does support, so you can continue testing your web
page.)
If your sketch or
activity requires objects
not supported by
JavaSketchpad, you
won’t be able to share
that sketch as an
illustration. However,
you can still post your
original Sketchpad
sketch—your .gsp
file—as a
downloadable file, so
that visitors who use
The Geometer’s
Sketchpad can
download it and open
it in Sketchpad, rather
than in their browser.
242
To work around unsupported objects, explore different ways of
constructing the same illustration. For example, at present,
JavaSketchpad does not support iterations or iterated images. If your
sketch contains a construction that you’ve iterated using the Iterate
command, you may be able to replace it with one that you’ve iterated
“manually” by actually constructing the first several iterations.
A complete list of objects supported by JavaSketchpad is available on
the JavaSketchpad web site. Also, new versions of the applet—that
support more and more Sketchpad objects—occasionally appear on the
web site. See More Information (p. 245) for details.
The Geometer’s Sketchpad Reference Manual
JavaSketchpad
No Preview Offered
Problem: Rather than ask whether you want to preview your file in a
browser, when you save, Sketchpad warns that you saved to a folder
that does not contain a copy of the JSP folder.
Cause: The HTML file must be located in the same directory as the JSP
directory in order to be viewed successfully in a browser. Review the
material on Essential JavaSketchpad Folder Structure (p. 239) and make
sure you stored your HTML file beside your JSP folder, not inside it.
Correct the situation either by moving your saved file to a folder
containing the JSP applet folder or by copying the applet folder into the
folder containing your saved file. Or repeat the steps described in
Creating a JavaSketchpad Web Page (p. 241), choosing a different
location in step 5.
Non-Java Browsers
Problem: When you preview your web page in a browser, the page
contains a message reading “Sorry, this page requires a Java-capable
browser.”
Cause: If you have an old web browser, it may not support the Java
language and, therefore, won’t work with JavaSketchpad. Contact your
browser manufacturer for a newer version. Alternately, it may be that
your browser supports Java, but that it’s currently set to disable Java
applets. Go to your browser’s Preferences or Options to turn on
support for Java applets.
Java Exception or Error Occurs
Problem: When you preview your web page in a browser, a dialog box
appears saying that a Java error or “exception” occurred.
Cause: If the message goes on to say that a “class” was not found—for
example “GSP.class: class not found”—then your JSP folder is either in
the wrong location or has become corrupted. (For example, vital files
within it may have been accidentally deleted or moved.) Make a fresh
copy of the JSP folder in the appropriate location, as described in
Essential JavaSketchpad Folder Structure (p. 239).
If the error message says something else, a different problem has
occurred. While JavaSketchpad has been extensively tested on current
versions of Internet Explorer and Netscape Navigator, older browsers
are erratic in their support of Java, and other browsers may have similar
problems. (Java is a relatively new language and undergoes frequent
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JavaSketchpad
modifications; different browsers support it to different degrees.)
Check with your browser manufacturer to see if a more recent version
of your browser is available. Errors or exceptions may also indicate a
problem with JavaSketchpad itself. Visit the JavaSketchpad web site to
see if a more recent version of the applet is available.
Appearance Discrepancies
Problem: Your sketch saves correctly and previews in your browser, but
certain details in the JavaSketchpad illustration—such as choice of
fonts, size of exact position of text—do not match your original sketch.
Cause: These minor discrepancies are inevitable in Java applets, where
less sophisticated graphic and text services are available to an applet
than to a nonapplet program (like your browser or desktop Sketchpad).
What you sacrifice in appearance flexibility you gain in generality:
Applets like JavaSketchpad work well on a much wider variety of
computers than are capable of running the full desktop version of
Sketchpad.
Modifying and Publishing Your Pages
Once you’ve successfully previewed a JavaSketchpad web page, you’ll
find that although it may be exciting to have an interactive illustration,
the rest of the page is rather dull. Sketchpad adds some default text to
your illustration, but otherwise leaves the page blank.
You can even put
multiple
JavaSketchpad
illustrations on the
same page by copying
an <APPLET> …
</APPLET> block
from one HTML file
to another.
244
Using your favorite HTML editor, you can replace the default text and
add any new HTML content to the page that you want: a description,
images, links, and so forth. When editing the HTML, be careful to
preserve the large <APPLET>…</APPLET> block you’ll find in the
middle of the page. This block describes the JavaSketchpad illustration,
and the illustration may no longer function if you alter any of the
contents between the <APPLET> and the </APPLET> tags.
When you’re ready to share your page on the Internet, copy it and the
JSP folder to your web server. Remember to keep the JSP folder in the
same folder as your HTML file, even on your server. (You can store
multiple HTML files in that folder and only one copy of the applet, but
they must be in the same folder for JavaSketchpad to work.)
The Geometer’s Sketchpad Reference Manual
JavaSketchpad
For information
about other licenses,
visit the
JavaSketchpad web
site.
Your license for The Geometer’s Sketchpad includes the use of
JavaSketchpad on the Internet for noncommercial purposes. The
purpose of this license is to allow you to post sketches that you or your
students have created using The Geometer’s Sketchpad. This license is
granted provided your site can be freely visited by anyone on the
Internet (that is, it’s not password-protected or available only to
subscribers) and that no direct or indirct profit is being made by having
people visit your site.
More Information
As an Internet technology, JavaSketchpad evolves rapidly, and the best
source of information about it is on the web itself. The JavaSketchpad
web site is the place to go for additional information; it includes sample
applications, technical support, and a full description of the
JavaSketchpad construction language that appears in your HTML files.
You’ll even find information about some features of JavaSketchpad
that aren’t available in The Geometer’s Sketchpad.
Visit the site at
http://www.keypress.com/sketchpad/java_gsp/
The Geometer’s Sketchpad Reference Manual
245
Sketchpad’s Internal Mathematics
Sketchpad’s internal mathematics determine how the program
computes and represents numbers, geometric figures, functions, and
other mathematical quantities. This in turn determines how these
objects appear graphically, numerically, or symbolically.
Don’t confuse the
displayed precision of a
value with its internal
precision or accuracy.
When you display a
value (such as a
measurement), it
appears only to the
number of decimal
places you choose in
Preferences or
Properties. Internally,
that number is
represented to much
greater precision, as
described here.
At the numeric level, Sketchpad represents point coordinates and other
quantities using 64-bit floating-point arithmetic. This standard
representation for scientific computation allows your computer to
represent a value with 14 to 16 significant digits of decimal precision
over a wide range of magnitudes (roughly, as large as ±10300 and as
small as ±10–300). While this is very precise, it is not exact in a
mathematical sense. (For example, π cannot be represented exactly with
only 15 significant digits.) Sketchpad uses tuned algorithms to attempt
to represent numbers as close to their exact value as possible, and to
minimize the inevitable error introduced by calculating with only a
finite number of significant digits. Nonetheless, you may witness
numerical error effects in the least significant digits of numbers in
sketches involving a lot of internal calculation. Regrettably, no
computer or computer program can represent every number exactly:
there are an infinite number of numbers, of course, and—at least
today!—computers have only a finite amount of memory. Thus, while
Sketchpad’s numeric calculations are generally reliable and can serve as
the basis of a convincing argument or conjecture, a Sketchpad result
should never be mistaken for constituting a mathematical proof.
At the graphical level, Sketchpad transforms its internal numeric
representations into the shapes and positions that appear in your sketch
window. For objects such as circles, points, and segments, the resulting
images are as accurate as can be displayed on your computer screen.
(When you print to a printer with higher resolution than your screen,
you’ll see the images are even more accurate than their on-screen
representations.)
However, for plotted functions and loci, Sketchpad displays only a
visual approximation of the curve’s ideal mathematical shape. Primarily
to maintain responsiveness when you’re dragging objects, Sketchpad
employs the same technique to plot these objects as a person might use
if plotting them by hand: it evaluates the ideal curve at a number of
different positions (called samples), then plots the curve by
interpolating between these known samples. The samples themselves
are very accurate, but the interpolations may or may not be, depending
246
The Geometer’s Sketchpad Reference Manual
Sketchpad’s Internal Mathematics
on the ideal shape of the mathematical object. The Plot Properties
panel gives you control over how many samples Sketchpad uses to plot
a function or a locus, as well as whether it displays that plot only as the
(discrete) collection of accurate samples or by including the
(continuous) interpolations between samples. If you consistently prefer
a higher number of samples than Sketchpad uses by default, you can
increase the default on the Sampling Preferences panel. Also, while
Sketchpad uses approximate interpolations for display purposes, be
aware that it never relies on them for mathematical purposes. Thus, if
you construct a point on a function plot or on a locus (or use the
Calculator to evaluate a function at given values), that point’s
coordinates (or function’s values) are always based on the exact curve
or function and not on its visual approximation.
Finally, at the symbolic level, Sketchpad performs simple computer
algebra to differentiate functions when you use the Derivative command
from the Graph menu. While these computed derivatives are generally
reliable for graphing and evaluation purposes, they may not be exact. In
particular, when differentiating intricate functions, Sketchpad may fail
to simplify the result fully, introducing point discontinuities in the
derivative; and Sketchpad does not compute domain restrictions on the
derivative function. Use Derivative to compute the slope of a function at
an arbitrary point for graphing purposes or for mathematical
constructions, but be sure to verify the result before using it as the basis
of a mathematical proof.
See also: Accuracy vs. Precision (p. 137), Plot Properties (p. 124), Sampling Preferences
(p. 140), Derivative (p. 208), Unit Preferences (p. 136)
The Geometer’s Sketchpad Reference Manual
247
Index
π, 217, 224
entering in dialog boxes, 224
in function plotting, 125
abs (absolute value function), 52
Abscissa (x) measurement
command, 192, 199
Absolute value symbol, 57
Accuracy and precision, 124, 137,
209, 246–47
Action Buttons, 37–39, 77
Action Buttons submenu, 111
aligning, 217, 226
Animation Buttons, 38, 112,
128
creating, 111
Hide/Show Buttons, 38, 112,
127
Link Buttons, 38, 114, 133
Movement Buttons, 38, 113,
130
Presentation Buttons, 39, 113,
131
pressing, 77
Scroll Buttons, 39, 114, 134
sequenced, 131
simultaneous, 131
Add New Map command, 186
Add Table Data, 209
Advanced Preferences, 26, 47, 64,
135, 137, 139, 144, 235
invoking, 217
Advanced text topics, 226–28
Advanced tool topics, 230–33
Aligning text objects, 217, 226
Alt key scrolling, 69, 216
Angle Bisector command, 14, 85,
161
Angle measurement command,
192, 194
The Geometer’s Sketchpad Reference Manual
Angles
construct an angle of fixed
measure, 179
measuring, 192, 194
using in transformations, 169
Animate command, 19, 153
Animate Properties, 128
Animation, 9, 11, 17, 41
direction, 47
independent points, 45
Motion Controller, 41–48, 42
showing and hiding, 154
once-only motion, 48
parameters, 19, 46, 126
continuous or discrete, 130
points on paths, 46
principles, 44–48
role of object relationships in,
45
selecting moving objects, 43
speed, 42, 44, 47, 142, 153
stopping, 43, 154, 215
Animation Action Button
command, 112
Animation Buttons, 38, 112, 128
direction, 129
once only, 129
speed, 129
Anti-aliased graphics, 6, 143
Arc Angle measurement
command, 192, 195
Arc Length measurement
command, 192, 196
Arc On Circle command, 15, 163
Arc Sector command, 17, 165
Arc Segment command, 17, 165
Arc Through 3 Points command,
15, 163
Arccos (inverse cosine function),
52
249
Index
Arcs, 15
constructing, 15
measuring, 16
using, 15
Arcsin (inverse sine function), 52
Arctan (inverse tangent function),
52
Area measurement command, 192,
194
Arrow Tool. See Selection Arrow
Tools
Arrowheads, 235
including in exported images,
140
Assumed objects, 61, 231
Automatically Match Sketch
Object checkbox, 230
Axes, 21
rescaling, 78
Braces, 57
Brackets, 57
Bring All To Front command
(Mac OS X), 211
Buttons. See Action Buttons
Calculate command, 18, 30, 49,
192, 198, 222
Calculations, 18, 49, 51
aligning, 217, 226
changing into parameters, 19,
50
editing, 120
using, 20
using to set color, 147
using to set iterative depth, 189
using to transform objects, 170,
171, 173
Calculator, 49–54
Equation menu, 52
Functions menu, 52
how to use signum, 53
inserting values, parameters and
functions, 54
parts of, 50
250
pop-up menus, 51
Units menu, 52
Values menu, 51
Captions, 36
adding symbolic notations to,
36
aligning, 217, 226
alignment, 217
composite, 36, 37
creating, 88
editing, 88
math symbols in, 57
resizing, 36, 87, 89
shortcut for editing, 76
Cascade command (Windows),
211
Cassiopeia™ Sketchpad Format,
103
Children, 10, 115, 121
selecting, 115, 222
splitting from parents, 116
viewing, 122
Circle By Center+Point
command, 15, 82, 161
Circle By Center+Radius
command, 15, 82, 162
Circle Interior command, 17, 164
Circle tool. See Compass tool
Circles, 15
attaching, 81
constructing, 15, 81
measuring, 16, 193, 196
selecting all, 115
using, 15
Circumference measurement
command, 192, 193
Clear command, 111, 216
Clipboard image scale, 140, 235
Close box, 5
Close command, 104
Collapse box, 6
Collapsible compass, 220
The Geometer’s Sketchpad Reference Manual
Index
Color
default object colors, 138
editing available colors, 143
label vs. object, 147
object, 8, 146
other, 148
parametric, 147
preserving default color, 146
text, 9, 56
Color menu, 146
editing, 143
editing, Windows, 144
Color Picker, 64–66
Macintosh, 65
overview, 64
Windows, 66
Color Preferences, 138
Command-line flags, 224, 233
Compass tool, 15, 68, 81–82
Composite captions, 36, 209
Constant labels in custom tools,
231
Construct menu, 156–66, 220
using the commands, 156
Context menu, 213, 223
Script View, 61
Coordinate Distance measurement
command, 193, 199
Coordinate Systems, 21, 199, 200,
201, 202, 203, 205
changing scale of, 22
creating, 21
defining, 14
modifying, 22
multiple, 23
rescaling, 78
showing and hiding, 204
using, 22
Coordinates measurement
command, 22, 192, 198
Copy command, 110
cropping graphics, 235
exporting graphics, 234
exporting table data, 110
cos (cosine function), 52
The Geometer’s Sketchpad Reference Manual
Create New Tool command, 91,
94, 95, 97
Cropping exported graphics, 235
Custom label sequences, 229
Custom tools, 90–99, 222
advanced topics, 230–33
assigning labels to results, 123
automatic matching, 123, 230
comments, 59
constant labels, 231
copying, 107
creating, 94, 97
creating a Tool folder, 98
defining, 91
labels, 230
making, 94
managing, 96, 105
matching given objects, 93
organizing, 7, 91
overview, 90
removing, 107
renaming, 105
samples, 92
Script View, 7, 59, 96, 106
sharing, 6
specifying labels, 231
Tool Folder, 7
using, 92
variable labels, 231
Custom Tools icon, 68
Custom Tools menu, 91
Cut command, 110
Dashed Line Width command,
146
Decrease All Speeds command,
153
Decrease Size command (text),
148
Decrease Speed command, 42, 47,
153
Define Coordinate System
command, 21, 201, 202
Define Origin command, 201
Define Unit Circle command, 16,
201
251
Index
Define Unit Distance command,
14, 20, 201, 202
Degrees, 136, 194, 195
Dependent point, 11
Derivative command, 31, 208
Deselecting objects, 70, 71
Esc key, 215
Dilate Arrow tool, 74, 75, 179
zooming, 76
Dilate command, 167, 179
Directed degrees, 136, 194, 195
Direction
controlling in animations, 129
Display menu, 145–55
Display Options menu, 185
Distance measurement command,
192, 193
compared to Coordinate
Distance command, 199
Document Options command, 4,
6, 7, 104
Documents, 4–7
page tabs, 6
pages, 4, 6
tools, 6
Dotted Line Width command, 146
Dragging, 73
dilation, 74
rotation, 74
translation, 74
Drag-scrolling
Alt key, 69
Option key, 69
Drive path, 25
Driven object, 25, 166
Driver point, 25, 166
Edit Calculation command, 49,
120
shortcut for, 76
Edit Definition command, 19, 120
Edit Function command, 49, 120,
208
shortcut for, 76
252
Edit menu, 109–44
Edit Parameter command, 49, 120,
124
Edit Plotted Point command, 120
EPS files, 236
Equation form, 52
Equation measurement command,
192, 200
Erase Traces command, 138, 151,
152
Esc key, 215, 223
Euclidean geometry, 220
Exponents, 57
Export Preferences, 139, 235
Exporting graphics. See Graphics
export
Expressions
dialog boxes, 224
Fading of traces, 138
Family tree, 10
File menu, 102–8
Font, 148
Fractals, 186
Function Plots, 27, 31
number of samples in exported
image, 140
properties, 125
resizing, 79
Sampling Preferences, 140
Functions, 27, 28, 49, 52
aligning, 217, 226
composite, 30
creating, 27
defining, 30
derivatives, 31
editing, 29, 120
families of, 28
plotting, 22, 29
properties of, 124
transforming, 30
Geoboard, 23
Given objects, 60, 93
matching, 230
Graph menu, 201–10
The Geometer’s Sketchpad Reference Manual
Index
Graphics
anti-aliased (high quality), 6,
143
Graphics export, 234–37
arrowheads, 140, 235
cropping, 235
EPS files, 236
Export Preferences, 139
Metafiles, 104, 236
pasting, 234
PostScript, 236
scaling, 140
Grid Form command, 203
Grid forms, 22, 204
Grids
changing appearance of, 22
dot paper, 146
showing and hiding, 22
Help menu, 212
Hide Buttons, 112
Hide command, 148
Hide Coordinate System
command, 204, 217
Hide Grid command, 22, 204
Hide Labels command, 8, 149
Hide Motion Controller
command, 154
Hide Script View command, 59,
91, 96
Hide Text Palette command, 55,
154
Hide Toolbox command, 69, 155
Hide/Show Action Button
command, 112
Hide/Show Buttons, 38
controlling selection status of
shown objects, 128
fading control, 128
setting controls for, 127
High-quality graphics. See Antialiased graphics
Horizontal lines, 85
How To
Construct a Geoboard, 23
The Geometer’s Sketchpad Reference Manual
Construct a Segment of Fixed
Length, 177
Construct a Sierpinski Gasket,
186
Construct a Slider, 197
Construct an Angle of Fixed
Measure, 179
Make a Perpendicular Bisector
Tool, 97
Show Just One Hidden Object,
149
Use Signum to Construct a
Piecewise Function, 53
Use Split and Merge to Explore
Constructions, 119
Zoom In and Out on a Sketch,
76
HTML, 241
adding to JavaSketchpad
sketch, 244
saving sketches, 104
Image, 181
of iteration, 32
result of transformation, 168
Increase All Speeds command,
153
Increase Size command (text), 148
Increase Speed command, 42, 153
Independent point, 11
Interior command, 164
Interiors, 16
Arc Sector, 165
Arc Segment, 165
as paths, 17
Circle, 164
construcing, 17, 164
measuring, 17
Polygon, 164
using, 17
Intermediate Objects
in Custom tools, 94
Internet
exporting to, 238–45
linking to a web page from a
sketch, 133
253
Index
Intersection command, 12, 14, 158
and Arrow tool, 77
Iterate command, 33, 168, 181–90,
217, 222
Display Options, 185
Iterate dialog box, 183
multiple maps, 185–88
Structure Options, 185
Iterate To Depth command, 20,
189, 217
Iterations, 31, 134, 181–90
by example, 182–83
creating, 32
depth, 20, 134, 135, 141, 185,
189, 216
final image, 135
fractals, 186
full orbit, 135
image, 32
iteration map, 182
measurements, 188
parametric depth, 189
points on paths, 135, 185, 189
pre-image, 32
properties, 33
random vs. relative location,
189
randomizing, 135
Sierpinski gasket, 186
tables, 188
terminal point, 190
values, 188
working with, 33
Java Error, 243
JavaSketchpad, 238–45
adding HTML content, 244
appearance problems, 244
objects not supported by, 242
saving sketches, 104
trouble-shooting, 242–44
URL, 245
254
Keyboard shortcuts, 215–17
Label command, 9, 150, 229
Label Multiple Objects, 150, 229
Label Properties, 122
Labels, 8
changing, 8, 88, 122, 150
color, 56
custom sequence, 229
defaults, 87
for multiple objects, 150
generating in Custom Tools,
231
hiding, 87, 122, 149
moving, 88
setting in Custom Tools, 231
shortcut for editing, 76
showing, 87, 122, 139, 149, 150
showing automatically, 139
styling, 122
subscripting, 122
Length measurement command,
192
Line command, 14, 85, 158
Line tool, 83
Line Width submenu, 9, 145
Lines, 14
constructing, 83, 159, 160, 161
including arrowheads in
exported images, 140
selecting all, 115
width, 9
Link Action Button command,
114
Link Buttons, 38, 114, 133
linking to a URL, 133
linking to another page, 133
linking to another Sketchpad
document, 133
linking to the Internet, 133
Link Properties, 133
ln (natural logarithm function), 52
The Geometer’s Sketchpad Reference Manual
Index
Loci, 24
constructing, 26, 166
continuous and discrete, 26
driven objects, 24, 25
driver points, 24, 25
examples, 166
export quality, 235
modifying, 26
number of samples in exported
image, 140
properties of, 125
resizing, 27, 79
Sampling Preferences, 140
understanding, 25
Locus command, 166, 221
log (common logarithm function),
52
Mac OS X, 101, 211, 233, 236
Macintosh, 6, 55, 57, 65, 103, 104,
136, 142, 212, 216, 234, 236
Mark Angle command, 12, 20,
169, 170, 175
Mark Angle Measurement
command, 169, 170
Mark Center command, 12, 169
shortcut, 76, 169
Mark commands, 167, 168
Mark Coordinate System
command, 22, 23, 203
Mark Distance command, 20, 173,
174, 175
Mark Mirror command, 14, 169,
181
shortcut, 76, 169
Mark Polar Vector command, 172
Mark Ratio command, 12, 170,
172
Mark Scale Factor command, 20,
170, 171, 172
Mark Segment Ratio command,
170, 171
Mark Vector command, 172
Marked translation vector, 176
Matching, 230
The Geometer’s Sketchpad Reference Manual
Math symbol font, 142
Maximize box, 5
Measure menu, 191–200, 221
Measurements, 18, 51
accuracy, 137
aligning, 217, 226
precision, 137
using, 20
using to set color, 147
using to set iterative depth, 189
using to transform objects, 170,
171, 173
Menu structure, 220
Merge command, 12, 14, 15, 17,
116, 117
example of, 119
Merge Text command, 37, 118,
226
to a point, 228
with custom template, 227
Metafiles, 104, 234, 236
Microsoft Windows, 5, 42, 50, 55,
57, 66, 69, 101, 103, 104, 125,
136, 142, 144, 211, 212, 213,
216, 224, 233, 234, 236
Midpoint command, 12, 14, 157
Minimize box, 6
Minimize command (Mac OS X),
211
Motion Controller, 41–48
animation principles, 44–48
changing speed, 44, 47
direction, 47
hiding, 154
parts of, 42
reversing direction, 44
showing, 41, 154
Mouse wheel, 69
Move Properties, 130
Movement Action Button
command, 113
Movement Buttons, 38, 113
properties of, 130
255
Index
Multi-page documents
adding pages, 106
linking between pages, 133
managing, 105
removing pages, 107
New Function command, 30, 49,
207
New Parameter command, 19, 28,
51, 54, 206, 207
compared to slider, 197
New Sketch command, 102
Non-collapsible compass, 162, 220
Object list, 60
Object Properties, 9, 121
Objects, 8–40
animating, 41, 153
arcs, 15
axes, 21
calculations, 18
children, 115, 121
circles, 15
coordinate systems, 21
default color, 138
deselecting, 70
dilating, 74
dragging, 73
family tree, 10
function plots, 27
functions, 27
hiding and showing, 9, 38, 122,
148
interiors, 16, 17
label properties, 122
labels, 8, 87–88, 139, 150
loci, 24
measurements, 18
measuring, 16, 193, 194
parameters, 18, 19
parents, 115, 121
path objects, 13
points, 11
polygons, 16
preventing selection, 122
relationships, 10
rotating, 74
256
selecting, 70
selecting overlapping, 72
showing. See Objects, hiding
and showing
straight objects, 14
translating, 74
using interiors, 17
values, 18
Open command, 102
Operators in captions, 57
Option key
scrolling, 69, 216
typing π, 125, 224
Orbit, 185
Ordinate (y) measurement
command, 192, 199
Other Color command, 148
Overbars in captions, 57
Page Setup command, 107
Page tabs, 5
Pages
adding, 106
rearranging, 105
removing, 107
renaming, 105
tabs, 106
Parallel Line command, 14, 85,
159
Parameter Properties, 126
Parameters, 18, 19, 51, 216
aligning, 217, 226
animating, 19, 126
changing value of, 19
creating from calculations, 19,
50
editing, 20, 76, 120
inserting into the Calculator, 54
setting domains of, 126
setting the value of, 124
using, 20
using to set color, 147
using to set iterative depth, 189
using to transform objects, 170,
171, 173
Parametric color, 147
The Geometer’s Sketchpad Reference Manual
Index
Parametric depth, 189
Parentheses, 57
Parents, 10, 115, 121
selecting, 115, 222
viewing, 122
Paste command, 111
into other programs, 234
Paste Picture command, 39, 111
Path Objects, 13
Pause Animation command, 153
Pause button in Motion
Controller, 42
PDF, 236
Perimeter measurement
command, 192, 193
Perpendicular Line command, 14,
85, 160
Pictures, 39
maintaining aspect ratio, 39,
217
pasting into a sketch, 39, 111
resizing, 39, 78
Piecewise function, 53
Plot As (r, theta) command, 22,
205
Plot As (x, y) command, 20, 22,
205
Plot Function command, 22, 29,
31, 207, 222
Plot New Function command, 22,
31, 53, 207
Plot Points command, 12, 22, 205
Plot Properties, 26, 124
Plot Table Data command, 205
Plotted points
editing, 120
Poincaré disk, 230
Point locus, 26
Point On Object command, 12,
13, 14, 15, 17, 156
in animations, 44
in locus constructions, 166
Point tool, 12, 68, 80
Points, 11
animating, 11
The Geometer’s Sketchpad Reference Manual
constraining, 204
constructing, 12, 156
creating, 80
dependent, 11
independent, 11
merging, 11, 117
midpoint, 12, 157
moving, 11
of intersection, 12, 77, 80, 158
plotting, 12
Point On Object, 12
point on path, 11
selecting all, 115
showing labels automatically,
139
splitting, 11, 116, 117
using in measurements, 13
using in transformations, 12,
169
Polar Grid command, 22, 203
Polar translation vector, 174
Polygon Interior command, 17,
164
Polygons, 16
post-image, 168
PostScript export, 108, 236
Precision, 137
changing, 20
controlling, 124
Preferences command, 135
Advanced. See Advanced
Preferences
Color Preferences, 138
Export Preferences, 139
resetting, 143
Sampling Preferences, 140
System Preferences, 141
Text Preferences, 139
Units Preferences, 136
pre-image, 167, 168, 181
of iteration, 32
Presentation Action Button
command, 113
257
Index
Presentation Buttons, 39, 113, 131
pausing between actions, 132
setting Before Starting actions,
132
simultaneous or sequential
actions, 131
specifying stopping point, 132
Presentation Properties, 131
Print command, 108
Print Preview command, 108
Probability, 189
Properties
Tables, 127
Properties command, 9, 26, 33,
112, 120
Animate Properties, 127
Hide/Show Properties, 127
Iteration Properties, 134
Label Properties, 122
Link Properties, 133
Move Properties, 130
Object Properties, 121
Parameter Properties, 126
Plot Properties, 124
Presentation Properties, 131
Script View and, 61
Scroll Properties, 134
Value Properties, 124, 136
Quit command, 108
Radians, 136, 194, 195
Radius measurement command,
192, 196
Randomness, 189
Ratio measurement command,
192, 196
Ray command, 14, 85, 158
Ray tool, 83
Rays, 14
constructing, 83
selecting all, 115
Rectangular Grid command, 22,
204
Rectangular translation vector, 175
Redo All command, 110, 217
258
Redo command, 109, 110
Reflect command, 167, 181
Relabeling objects, 150
Remove Table Data, 210
Resize area, 5
Results of custom tools, 93
Resume Animation command, 153
Reverse button in Motion
Controller, 42
Rotate Arrow tool, 74, 75, 177
Rotate command, 167, 177, 178,
179, 180
round, 52
Sample Tools, 92
Samples, 26, 141, 216
Sampling Preferences, 140
Save As command, 103
Save As HTML, 104, 217, 241
Save command, 103
Screen captures, 234
Screen resolution, 142
Script View, 7, 59–63, 106, 231
applying, 62
assumed objects, 61
automatically matching a given
object, 230
given objects, 60
Object List, 60
Object Properties, 61
printing, 63
showing and hiding, 59
steps, 61
Scroll Action Button command,
114
Scroll bars, 5
Scroll Button Properties, 134
Scroll Buttons, 39, 114, 134
Scrolling
Alt key, 69
Option key, 69
Segment command, 14, 85, 158
Segment tool, 83, 177
using to construct segment of
fixed length, 177
The Geometer’s Sketchpad Reference Manual
Index
Segments, 14
constructing, 83
selecting all, 115
Select All command, 71, 115
Select Children command, 10, 115,
222
Select Parents command, 10, 115,
222
Selection, 70, 71
disabling selection of
individual objects, 122
of coincident objects, 72
of overlapping objects, 72
using a selection rectangle, 71
Selection Arrow tools, 68, 70–79
activation by Esc key, 215
changing parameter value, 19
choosing, 69
Dilate Arrow Tool, 75
double-clicking, 76
editing calculations, 19, 49
editing function definitions, 49
other actions, 76
rescaling axes, 22
resizing function plots, 208
resizing pictures, 39
resizing point loci, 27
Rotate Arrow tool, 75
Translate Arrow tool, 75
using, 71
Selection prerequisites, 156
Selection rectangle, 222
sgn (signum function), 52, 53
Shift key, 216, 217
choosing tools via keyboard,
69, 74
constraining straightedge
objects, 85
maintaining aspect ratio, 39, 78
preserving default color, 138,
146
preserving default line width,
145
selecting Arrow tools, 83
The Geometer’s Sketchpad Reference Manual
Shortcuts, 223
Selection Arrow tools, 76
Show All Hidden command, 148,
149
Show Buttons, 112
Show Coordinate System
command, 204, 217
Show Grid command, 22, 204
Show Labels command, 8, 149,
150
Show Motion Controller
command, 41, 154
Show one hidden object only, 149
Show Script View command, 7,
59, 91, 96
Show Text Palette command, 55,
139, 154
Show Toolbox command, 69, 155
Sierpinski gasket, 186
signum function, 53
sin (sine function), 52
Sliders, 197
Slope measurement command,
192, 200
Snap Points command, 22, 204
Speed
changing, 153
controlling default, 142
in Motion Controller, 42
Split command, 12, 116
example of, 119
Split Merged Text command, 118
Split Point command, 117
Split Text command, 37
sqrt (square root function), 52
Square Grid command, 22, 203
Square root symbol, 57
Stop All Motions command, 154
Stop Animation command, 42,
154
Stop button in Motion Controller,
42
259
Index
Straight objects, 14
attaching, 84
constraining, 85
constructing, 83, 85, 158
measuring, 14
using, 14
using in transformations, 169
Straightedge tools, 14, 68, 83–85
choosing a Straightedge tool, 69
constraining, 217
Subscripts, 57
Symbolic notation
inserting into captions, 56
Symbols, 58
System Preferences, 141
Tab divider, 5
Table Properties, 127
Tables, 34, 188
copying, 110
of Iterated Values, 188
Tabulate, 34, 209
tan (tangent function), 52
Target of Motion Controller, 42
Terminal Point command, 190
Tessellations, 186
Text
aligning, 217, 226
alignment of captions, 217
changing appearance, 55
color, 56
merging, 118
selecting all, 115
splitting merged text, 118
Text Palette, 9, 36, 55–58
diagram of parts, 55
editing captions, 57
hiding, 55, 154
mathematical symbols, 57, 58
moving, 55
showing, 55, 154
showing automatically, 139
260
Text Preferences, 139
Text submenu, 148
Text tool, 36, 68, 86–89
appearance of, 86
Thick Line Width command, 146
Thin Line Width command, 146
Tips for experts, 220–25
Title bar, 5
Tool Folder, 7, 91, 98
alternate, 232
Tool Options command, 91, 96
Toolbox, 67–99, 216, 220
choosing a tool, 68
docking, 69
hiding, 69
hiding and showing, 155
moving, 69
overview, 68
resizing, 69
showing, 69
Trace command, 151, 152
Traces, 9
erasing, 9, 152, 215
fading, 138
turning off, 152
Transform commands
basic instructions for use, 168
Transform menu, 167–90, 221
Transformational geometry, 221
Translate Arrow tool, 74, 75, 173
Translate command, 167, 173
example of using, 177
trunc (truncate function), 52
Undo All command, 109, 217
Undo command, 109, 110
Units, 52
changing, 20
Units Preferences, 20, 136
URL
linking to, 114, 133
Value Properties, 124
The Geometer’s Sketchpad Reference Manual
Index
Values, 20, 51
See also Measurements,
Calculations, Parameters.
aligning, 217, 226
display name, 124
precision, 136
units, 136
using in calculations, 20, 54
using to set color, 147
using to set iterative depth, 189
using to transform objects, 170,
171, 173
Variable labels in custom tools,
231
The Geometer’s Sketchpad Reference Manual
Vertical lines, 85
Visibility of objects, 9
Window border, 5
Window list, 211
Window menu, 211
Windows, Microsoft. See
Microsoft Windows
World Wide Web (WWW)
export, 238–45
linking to, 114, 133
Zoom, 76
Zoom box, 6
Zoom command (Mac OS X), 211
261