Transfinite mean value interpolation Christopher Dyken and Michael S. Floater

Transfinite mean value interpolation Christopher Dyken and Michael S. Floater
Transfinite mean value interpolation
Christopher Dyken and Michael S. Floater
July 2007, revised November 2007
Abstract
Transfinite mean value interpolation has recently emerged as a simple and robust way
to interpolate a function f defined on the boundary of a planar domain. In this paper we
study basic properties of the interpolant, including sufficient conditions on the boundary of
the domain to guarantee interpolation when f is continuous. Then, by deriving the normal
derivative of the interpolant and of a mean value weight function, we construct a transfinite
Hermite interpolant and discuss various applications.
Keywords: Transfinite interpolation, Hermite interpolation, mean value coordinates,
1
Introduction
Transfinite interpolation means the construction of a function over a planar domain that matches
a given function on the boundary, and has various applications, notably in geometric modelling
and finite element methods [20]. Transfinite mean value interpolation has developed in a series of
papers [3, 10, 6, 12]. In [3] barycentric coordinates over triangles were generalized to star-shaped
polygons, based on the mean value property of harmonic functions. The motivation for these
‘mean value coordinates’ was to parameterize triangular meshes but they also give a method for
interpolating piecewise linear data defined on the boundary of a convex polygon. In [10] it was
shown that these mean value interpolants extend to any simple polygon and even sets of polygons,
possibly nested, with application to image warping. In both [6] and [12] 3D coordinates were
similarly constructed for closed triangular meshes, and in [12] the coordinates were used for
mesh deformation. Moreover, in [12] the construction was carried out over arbitrary curves and
surfaces, not just polygons and polyhedra. Further work on mean value coordinates and related
topics can be found in [1, 4, 5, 11, 13, 15, 17, 22].
The purpose of this paper is to study and further develop mean value interpolation over arbitrary curves in the plane, as proposed by Ju, Schaefer, and Warren [12]. There are two main
contributions. The first is the derivation of sufficient conditions on the shape of the boundary that
guarantee the interpolation property: conditions that ensure that the mean value interpolant really is an interpolant. This has only previously been shown for polygonal curves with piecewise
linear data, in [10]. The second is the construction of a Hermite interpolant, matching values
1
ρ (x , θ )
Ω
L (x ,θ )
p (x , θ )
θ
x
dΩ
Figure 1: A convex domain.
and normal derivatives of a given function on the boundary. The Hermite interpolant is constructed from a weight function and two Lagrange interpolants, and requires finding their normal
derivatives.
We complete the paper with applications to smooth mappings and the web-spline method for
solving PDE’s.
2
2.1
Lagrange interpolation
Interpolation on convex domains
Let Ω ⊂ R2 be open, bounded and convex. We start with the convexity assumption because
the definitions and analysis are easier. However, we make no assumption about the smoothness
of the boundary ∂Ω, nor do we demand strict convexity: three points in ∂Ω can be collinear.
Thus we allow Ω to be a convex polygon as well as a circle, ellipse, and so on. For any point
x = (x1 , x2 ) in Ω and any angle θ let L(x, θ) denote the semi-infinite line that starts at x and
whose angle from the x1 -axis is θ, let p(x, θ) denote the unique point of intersection between
L(x, θ) and ∂Ω, and let ρ(x, θ) be the Euclidean distance ρ(x, θ) = kp(x, θ) − xk; see Figure 1.
The intersection point p(x, θ) depends on the curve ∂Ω, and sometimes it will help to indicate
this by writing p(x, θ; ∂Ω). In general, p(x, θ; C) will denote the intersection (assumed unique)
between L(x, θ) and any planar curve C and ρ(x, θ; C) the corresponding distance.
Given some continuous function f : ∂Ω → R, our goal is to define a function g : Ω → R
that interpolates f . To do this, for each x ∈ Ω, we define g(x) by the following property. If
F : Ω → R is the linear radial polynomial, linear along each line segment [x, y], y ∈ ∂Ω, with
F (x) = g(x) and F (y) = f (y), then F should satisfy the mean value property
Z
1
F (x) =
F (z) dz,
(1)
2πr Γ
where Γ is any circle in Ω with centre x, and r is its radius. To find g(x), we write (1) as
Z 2π
1
g(x) =
F x + r(cos θ, sin θ) dθ,
2π 0
2
(2)
Figure 2: Mean value interpolants.
and since
ρ(x, θ) − r
r
g(x) +
f (p(x, θ)),
F x + r(cos θ, sin θ) =
ρ(x, θ)
ρ(x, θ)
(3)
equation (2) reduces to
Z
0
whose unique solution is
Z
g(x) =
0
2π
2π
f (p(x, θ)) − g(x)
dθ = 0,
ρ(x, θ)
f (p(x, θ))
dθ
ρ(x, θ)
Z
φ(x),
φ(x) =
0
2π
1
dθ.
ρ(x, θ)
(4)
Equation (4) expresses g(x) as a weighted average of the values of f around Ω. We will show
later that under reasonable conditions on ∂Ω, g interpolates f , i.e., that g extends continuously
to the boundary ∂Ω and equals f there. Thus, since F satisfies the mean value property (1) at
x, we call g the mean value interpolant to f . The interpolant g itself does not satisfy the mean
value theorem and is not in general a harmonic function. But in the spirit of [7], we can view
it as ‘pseudo-harmonic’ as it shares some of the qualitative behaviour of harmonic functions,
such as the maximum principle. Also, similar to harmonic functions, the operator I, defined by
g = I(f ), has linear precision: if f : R2 → R is any linear function, f (x1 , x2 ) = ax1 + bx2 + c,
then I(f ) = f in Ω. This follows from the fact that, if f is linear and we let g(x) = f (x), then
F = f , and so F is linear and therefore trivially satisfies (1). Figure 2 shows two examples of
mean value interpolants on a circular domain.
3
2.2
Interpolation on convex polygons
The construction of the mean value interpolant g was carried out in [3] in the special case that
Ω is a polygon and that f is linear along each edge of the polygon. In this case g is a convex
combination of the values of f at the vertices of the polygon. To see this we prove
Lemma 1 Let e = [p0 , p1 ] be a line segment and let f : e → R be any linear function. Let x be
any point in the open half-plane lying to the left of the vector p1 − p0 . Let θ0 < θ1 be the two
angles such that p(x, θi ; e) = pi , i = 0, 1, and let ρi = kpi − xk. Then
Z θ1
f (p0 ) f (p1 )
θ1 − θ0
f (p(x, θ; e))
dθ =
+
tan
.
(5)
ρ(x, θ; e)
ρ0
ρ1
2
θ0
Proof. Similar to the approach of [3], since f is linear, we have with p = p(x, θ; e),
f (p) =
A1
A0
f (p0 ) +
f (p1 ),
A
A
(6)
with A0 , A1 , A the triangle areas A0 = A([p0 , x, p]), A1 = A([p, x, p1 ]), A = A([p0 , x, p1 ]).
Letting ρ = ρ(x, θ; e), by the sine rule,
A0
sin(θ − θ0 )ρ
,
=
A
sin(θ1 − θ0 )ρ1
A1
sin(θ1 − θ)ρ
,
=
A
sin(θ1 − θ0 )ρ0
and putting these into (6), dividing by ρ, and integrating from θ0 to θ1 gives (5).
2
Since the function f ≡ 1 is linear, the lemma also shows that
Z θ1
1
1
1
θ1 − θ0
dθ =
+
tan
.
ρ0 ρ1
2
θ0 ρ(x, θ; e)
Together with (5), this implies that, if Ω is a convex polygon with vertices p0 , p1 , . . . , pn−1 , and,
indexing modulo n, if f is linear on each edge [pi , pi+1 ], then g in (4) reduces to
g(x) =
n−1
X
wi (x)f (pi ) φ(x),
φ(x) =
i=0
where
wi (x) :=
n−1
X
wi (x),
(7)
i=0
tan(αi−1 (x)/2) + tan(αi (x)/2)
,
ρi (x)
(8)
and ρi (x) = kpi − xk and αi (x) is the angle at x of the triangle with vertices x, pi , pi+1 . The
functions
X
n−1
λi (x) := wi (x)
wj (x),
j=0
4
were called mean value coordinates in [3]. By the linear precision property of I, since both
f (x) = x1 and f (x) = x2 are linear, we have
x=
n−1
X
λi (x)pi ,
i=0
which expresses x as a convex combination of the vertices pi . Thus, the coordinates λi are a
generalization of barycentric coordinates.
2.3
The boundary integral formula
It is not clear from the formula (4) how to differentiate g. Ju, Schaefer, and Warren [12] noticed
however that if a parametric representation of ∂Ω is available, the two integrals in (4) can be
converted to integrals over the parameter of the curve. Let c : [a, b] → R2 , with c(b) = c(a),
be some parametric representation of ∂Ω, oriented anti-clockwise with respect to increasing
parameter values. If c(t) = (c1 (t), c2 (t)) = p(x, θ), then θ is given by
c2 (t) − x2
,
(9)
θ = arctan
c1 (t) − x1
and differentiating this with respect to t gives
(c1 (t) − x1 )c02 (t) − (c2 (t) − x2 )c01 (t)
(c(t) − x) × c0 (t)
dθ
=
=
,
dt
(c1 (t) − x1 )2 + (c2 (t) − x2 )2
kc(t) − xk2
(10)
where × denotes the cross product in R2 , i.e, a × b := a1 b2 − a2 b1 . Using (10) to change the
integration variable in (4) yields the boundary integral representation (c.f. [12]),
Z b
Z b
g(x) =
w(x, t)f (c(t)) dt φ(x),
φ(x) =
w(x, t) dt,
(11)
a
a
where
w(x, t) =
(c(t) − x) × c0 (t)
.
kc(t) − xk3
(12)
It is now clear that we can take as many partial derivatives of g as we like by differentiating
under the integral sign in (11). Thus we see that g is in C ∞ (Ω). The boundary integral formula
is also important because it provides a way of numerically computing the value of g at a point
x by sampling the curve c and its first derivative c0 and applying some standard quadrature rule
to the two integrals in (11). A simple alternative evaluation method that only requires evaluating
c itself is to make a polygonal approximation to c and apply (7). The third alternative of using
the original angle formula (4) and sampling the angles between 0 and 2π requires computing the
intersection points p(x, θ).
5
The numerator in w can also be written as the scalar product of (c(t) − x) and rot(c0 (t)) =
the rotation of c0 (t) through an angle of −π/2. Then, since the outward normal
to the curve c is rot(c0 (t))/kc0 (t)k, another way of representing g is in terms of flux integrals:
Z
Z
f (y)F(y) · N(y) dy φ(x),
φ(x) =
F(y) · N(y) dy,
g(x) =
(c02 (t), −c01 (t)),
∂Ω
∂Ω
where F is the vector field
F(y) =
y−x
,
ky − xk3
and N(y) is the outward unit normal at y and dy denotes the element of arc length of ∂Ω. The
Gauss theorem could then be applied to these expressions to give further formulas for g and φ.
Recently, Lee [16] has studied more general formulas of this type.
2.4
Non-convex domains
We now turn our attention to the case that Ω is an arbitrary connected open domain in R2 , not
necessarily convex. In the case that Ω is a polygon, it was shown in [10] that the mean value
interpolant g defined by (7–8) has a natural extension to non-convex polygons if we simply
allow αi (x) in (8) to be a signed angle: negative when x lies to the right of the vector pi+1 − pi .
The main point is that φ continues to be strictly positive in Ω so that g is well defined.
To deal with arbitrary (non-polygonal) domains, suppose initially that Ω is simply-connected,
i.e., has no holes, in which case its boundary can be represented as a single parametric curve
c : [a, b] → R2 , with c(b) = c(a), oriented anti-clockwise. Then, similar to the construction
in [12], we define g in this more general setting by the boundary integral (11). Note that for
arbitrary x ∈ Ω the quantity w(x, t) may change sign several times as t varies.
We can also express g in this general setting using angle integrals. Recall that an intersection
point of two smooth planar curves is said to be transversal if the curves have distinct tangents at
that point. We call θ a transversal angle with respect to x if all the intersections between L(x, θ)
and ∂Ω are transversal. For example, in Figure 3a all angles at x are transversal except for θ1 and
θ2 . We make the assumption that ∂Ω is such that there is a finite number of non-transversal angles
at each x ∈ Ω. If θ is a transversal angle, let n(x, θ) be the number of intersections of L(x, θ)
with ∂Ω which will be an odd number, assumed finite, and let pj (x, θ), j = 1, 2, . . . , n(x, θ), be
the points of intersection, ordered so that their distances ρj (x, θ) = kpj (x, θ)−xk are increasing,
ρ1 (x, θ) < ρ2 (x, θ) < · · · < ρn(x,θ) (x, θ).
(13)
For example, for θ ∈ (θ1 , θ2 ) in Figure 3a, there are three such intersections, shown in Figure 3b.
Now for fixed x ∈ Ω, let
S+ = {t ∈ [a, b] : w(x, t) > 0}
and
S− = {t ∈ [a, b] : w(x, t) < 0},
and observe that both integrals in (11) reduce to integrals over S+ and S− . Moreover, the sets S+
and S− are unions of intervals, and thus the integrals in (11) are sums of integrals, one integral
6
p 2(x ,θ)
p 1(x ,θ)
θ1 θ 2
x
p 3(x ,θ)
x
θ
Figure 3: (a) Example with two non-transversal angles and (b) an angle with three intersections.
for each interval, and w(x, ·) has constant sign in each interval. By changing the variable of
integration for each interval from t to θ, using (10), it follows that g can be expressed as
Z
g(x) =
0
2π n(x,θ)
X
j=1
(−1)j−1
f (pj (x, θ)) dθ
ρj (x, θ)
Z
φ(x),
φ(x) =
0
2π n(x,θ)
X
j=1
(−1)j−1
dθ.
ρj (x, θ)
(14)
Here, if θ is not a transversal angle, we set n(x, θ) = 0. We now use (14) to deduce the positivity
of φ and therefore the validity of g in the non-convex case.
Theorem 1 For all x ∈ Ω, φ(x) > 0.
Proof. The argument is similar to the polygonal case treated in [10]. Since the sequence of
distances in (13) is increasing, if n(x, θ) ≥ 3,
1
1
−
> 0,
ρ2j−1 (x, θ) ρ2j (x, θ)
j = 1, 2, . . . , (n(x, θ) − 1)/2,
and so (14) implies
Z
φ(x) ≥
0
2π
1
dθ > 0.
ρn(x,θ) (x, θ)
2
2.5
Bounds on φ
Having shown that g, given by either (11) or (14), is well-defined for non-convex domains, our
next goal is to show that g interpolates the boundary data f under reasonable conditions on the
shape of the boundary. A crucial step in this is to study the behaviour of φ near the boundary.
In this section we show that φ behaves like the reciprocal of the distance function d(x, ∂Ω), the
minimum distance between a point x ∈ Ω and the set ∂Ω. First we derive an upper bound.
7
Theorem 2 For any x ∈ Ω,
φ(x) ≤
2π
.
d(x, ∂Ω)
(15)
Proof. If n(x, θ) ≥ 3 in equation (13), then
−1
1
+
< 0,
ρ2j (x, θ) ρ2j+1 (x, θ)
and so
Z
φ(x) ≤
0
2π
j = 1, 2, . . . , (n(x, θ) − 1)/2,
1
dθ ≤
ρ1 (x, θ)
Z
2π
0
1
dθ.
d(x, ∂Ω)
2
To derive a lower bound for φ, we make some assumptions about ∂Ω in terms of its medial
axis [2]. Observe that ∂Ω divides R2 into two open and disjoint sets, the set Ω itself, and its
complement ΩC . The interior / exterior medial axis MI ⊂ R2 / ME ⊂ R2 of ∂Ω is the set of all
points in Ω / ΩC whose minimal distance to ∂Ω is attained at least twice. For any set M ⊂ R2 ,
we let
d(M, ∂Ω) = inf d(y, ∂Ω),
y∈M
and to derive a lower bound, we will make the assumption that d(ME , ∂Ω) > 0. Note that this
condition holds for convex curves because in the convex case, ME = ∅ and d(ME , ∂Ω) = ∞.
We will also make use of the diameter of Ω,
diam(Ω) =
sup
ky1 − y2 k.
y1 ,y2 ∈∂Ω
Theorem 3 If d = d(ME , ∂Ω) > 0, there is a constant C > 0 such that for x ∈ Ω,
φ(x) ≥
C
.
d(x, ∂Ω)
(16)
With β the ratio β = D/d, where D = diam(Ω), we can take
C=
2
(1 + β)(1 + β +
p
β 2 + 2β)
.
Note that C ≤ 2 and if Ω is convex then β = 0 and C = 2. On the other hand, if d is small
relative to D, then C will be small.
Proof. Let y be some boundary point such that d(x, ∂Ω) = ky − xk, and let δ = ky − xk and
let θy ∈ [0, 2π) be the angle such that L(x, θy ) intersects y. Then the open disc B1 = B(x, δ)
is contained in Ω. By the assumption that d > 0, let xC be the point in ΩC on the line L(x, θy )
whose distance from y is d; see Figure 4. Then the open disc B2 = B(xC , d) is contained in ΩC .
Let α1 , α2 , with α1 < θy < α2 , be the two angles such that the lines L(x, α1 ) and L(x, α2 ) are
8
a2
d
α δ
α
x
y
d
d
xC
a1
Figure 4: Lines in proof of Theorem 3.
tangential to ∂B2 , and let ai , i = 1, 2, be the point where L(x, αi ) touches ∂B2 . Let q1 be the
polygon consisting of the two line segments [a1 , y] and [y, a2 ], and q2 the polygon consisting of
[a1 , xC ] and [xC , a2 ].
Let θ be any transversal angle in (α1 , α2 ). Then there is some odd number k, say with
k ≤ n(x, θ), such that the intersection points p1 (x, θ), . . . , pk (x, θ) lie between B1 and B2
while the remaining ones pk+1 (x, θ), . . . , pn(x,θ) (x, θ) lie beyond B2 . Then, similar to the proof
of Theorem 1, if k = n(x, θ), the sum in φ in (14) satisfies the inequality
n(x,θ)
X (−1)j−1
1
≥
,
ρ
(x,
θ)
ρ
(x,
θ)
j
k
j=1
while, if k < n(x, θ), it satisfies
n(x,θ)
X (−1)j−1
1
1
≥
−
.
ρ
(x,
θ)
ρ
(x,
θ)
ρ
(x,
θ)
j
k
k+1
j=1
Consequently, in either case
n(x,θ)
X (−1)j−1
1
1
≥
−
,
ρ
(x,
θ)
ρ(x,
θ;
q
)
ρ(x,
θ;
q
)
j
1
2
j=1
and therefore, from (14),
Z
α2
φ(x) ≥
α1
1
1
−
ρ(x, θ; q1 ) ρ(x, θ; q2 )
dθ.
We now use the explicit formula from Lemma 1, and setting α = (α2 − α1 )/2, we find
1
1
1
1
φ(x) ≥ 2
+
tan(α/2) − 2
+
tan(α/2)
ka1 − xk ky − xk
ka1 − xk kxC − xk
9
=2
1
1
−
δ δ+d
tan(α/2) =
2d
tan(α/2).
δ(δ + d)
Moreover, since
1 − cos α
tan(α/2) =
,
sin α
we have
tan(α/2) =
and therefore
p
d
sin α =
,
δ+d
1
φ(x) ≥
δ
d
δ+d+
2d
δ+d
p
(δ + d)2 − d2
δ+d+
(δ + d)2 − d2
,
δ+d
cos α =
d
√
,
δ 2 + 2δd
.
(17)
Since δ ≤ D, this implies
1
φ(x) ≥
δ
2d
D+d
d
√
D + d + D2 + 2Dd
.
and, putting D = βd and cancelling the d’s, proves the theorem.
2.6
2
Proof of interpolation
We can now prove that g really interpolates f under the medial axis condition of Theorem 3. We
also make the mild assumption that
N := sup sup n(x, θ) < ∞,
(18)
x∈Ω θ∈T (x)
where T (x) is the subset of [0, 2π) of those angles that are transversal with respect to x. Note
that this holds for convex Ω, in which case N = 1.
Theorem 4 If f is continuous on ∂Ω and d(ME , ∂Ω) > 0, then g interpolates f .
Proof. Let c(s) be any boundary point and observe that for x ∈ Ω,
1
g(x) − f (c(s)) =
φ(x)
Z
b
w(x, t) f (c(t)) − f (c(s)) dt.
(19)
a
Rb
R R
We will choose some small γ > 0 and split the integral into two parts, a = I + J , where
I = [s − γ, s + γ] and J = [a, b] \ I. Then, with M := supy∈∂Ω |f (y)|,
Z
Z
1
1
|g(x) − f (c(s))| ≤ max |f (c(t)) − f (c(s))|
|w(x, t)| dt + 2M
|w(x, t)| dt.
t∈I
φ(x) I
φ(x) J
10
Considering the first term on the right hand side, note that
Z
Z b
1
1
|w(x, t)| dt ≤
|w(x, t)| dt =: R,
φ(x) I
φ(x) a
which we will bound above. The argument used to derive (14) also shows that
Z
b
Z
|w(x, t)| dt =
a
2π n(x,θ)
X
0
j=1
1
dθ,
ρj (x, θ)
and so
Z
b
Z
|w(x, t)| dt = φ(x) + 2
a
2π (n(x,θ)−1)/2
X
0
j=1
1
2(N − 1)π
dθ ≤ φ(x) +
,
ρ2j (x, θ)
d(x, ∂Ω)
with N as in (18). Dividing by φ(x) and applying the lower bound (16) to φ(x), then leads to
R≤1+
2(N − 1)π
2(N − 1)π
≤1+
,
φ(x)d(x, ∂Ω)
C
which is independent of x. Note that when Ω is convex, N = 1 and R = 1.
Let > 0. We must show that there is some δ > 0 such that if x ∈ Ω and kx − c(s)k ≤ δ
then |g(x) − f (c(s))| < . Since f ◦ c is continuous at t = s, we can choose γ > 0 such that
|f (c(t)) − f (c(s))| < (/2)/(1 + 2(N − 1)π/C) for t ∈ I. Then
Z
1
|w(x, t)| dt.
(20)
|g(x) − f (c(s))| < + 2M
2
φ(x) J
Finally, since
Z
|w(x, t)| dt =
lim
x→c(s)
Z
J
|w(c(s), t)| dt < ∞,
J
and
lim φ(x) = ∞,
x→c(s)
it follows that there is some δ > 0 such that if x ∈ Ω and kx − c(s)k ≤ δ then
Z
1
|w(x, t)| dt <
,
φ(x) J
4M
2
in which case |g(x) − f (c(s))| < .
3
Differentiation
In some applications we might need to compute derivatives of g. Let α = (α1 , α2 ) be a
multi-index, and let Dα = ∂ α1 +α2 /(∂ α1 x1 ∂ α2 x2 ). We start by expressing g in (11) as g(x) =
σ(x)/φ(x), where
Z b
σ(x) =
w(x, t)f (c(t)) dt,
a
11
and we reduce the task of computing derivatives of g to that of computing derivatives of σ and
φ, which are given by
Z b
Z b
Dα w(x, t)f (c(t)) dt, and Dα φ(x) =
Dα w(x, t) dt,
Dα σ(x) =
a
a
α
α1
with Dα operating with respect to the x variable. Letting β = β1 αβ22 , and defining β ≤ α
to mean that βi ≤ αi for both i = 1, 2, and β < α to mean that β ≤ α and α 6= β, we take the
Dα derivative of the equation φ(x)g(x) = σ(x), and the Leibniz rule gives
X α
Dβ φ(x)Dα−β g(x) = Dα σ(x),
β
0≤β≤α
and by rearranging this in the form
1
Dα g(x) =
φ(x)
!
X α
Dα σ(x) −
Dβ φ(x)Dα−β g(x) ,
β
0<β≤α
(21)
we obtain a rule for computing all partial derivatives of g recursively from those of σ and φ.
Letting
d = d(x, t) = c(t) − x,
r = r(x, t) = kd(x, t)k,
(22)
so that r3 w = d × c0 , an approach similar to the derivation of (21) gives
!
X
1
Dα w = 3 Dα d × c0 −
Dβ r3 Dα−β w ,
r
0<β≤α
(23)
a rule to compute the partial derivatives of w recursively. Since it is easy to differentiate r2 , we
can use the Leibniz rule to differentiate r3 :
X α
3
2
Dα r = Dα r r =
Dβ r2 Dα−β r.
β
0≤β≤α
By applying the Leibniz rule to r2 , we obtain derivatives of r:
Dα r =
1
2r
Dα r
2
!
X α
−
Dβ rDα−β r .
β
0<β<α
(24)
In the case that ∂Ω is a polygon, we can differentiate the explicit formula of g in (7), which
boils down to differentiating wi in (8). Similar to (21) we have
1
1 X α
1
Dα
=−
Dβ ρi Dα−β
,
ρi
ρi 0<β≤α β
ρi
12
and the formula for Dα ρi is given by (24) with r replaced by ρi . Derivatives of tan(αi /2) can be
found by rewriting it in terms of scalar and cross products of di (x) = pi − x,
tan
α i
2
=
ρi ρi+1 − di · di+1
.
di × di+1
Then, by viewing this as a quotient, we have
Dα tan
4
α i
2
1
=
di × di+1
Dα (ρi ρi+1 − di · di+1 ) −
X
Dβ (di × di+1 ) Dα−β tan
0<β≤α
α i
2
!
.
Hermite interpolation
We now construct a Hermite interpolant based on mean value interpolation. As we will see, the
interpolant is a natural generalization of cubic Hermite interpolation in one variable, and it helps
to recall the latter. Given the values and first derivatives of some function f : R → R at the
points x0 < x1 , cubic Hermite interpolation consists of finding the unique cubic polynomial p
such that
p(xi ) = f (xi )
and
p0 (xi ) = f 0 (xi ),
i = 0, 1.
(25)
One way of expressing p is in the form
p(x) = g0 (x) + ψ(x)g1 (x),
(26)
where g0 is the linear Lagrange interpolant
g0 (x) =
x − x0
x1 − x
f (x0 ) +
f (x1 ),
x1 − x0
x1 − x0
ψ is the quadratic weight function
ψ(x) =
(x − x0 )(x1 − x)
,
x1 − x0
and g1 is another linear Lagrange interpolant,
g1 (x) =
x1 − x
x − x0
m0 +
m1 ,
x1 − x0
x1 − x0
whose data m0 and m1 are yet to be determined. To see this, observe that since ψ(xi ) = 0,
i = 0, 1, p in (26) already meets the Lagrange conditions in (25), and since ψ 0 (xi ) 6= 0 for
i = 0, 1, the derivative conditions in (25) are met by setting
mi =
f 0 (xi ) − g00 (xi )
,
ψ 0 (xi )
13
i = 0, 1.
Now observe that for x ∈ (x0 , x1 ) we can express g0 and ψ as
X
1
1
X
f (xi )
1
g0 (x) =
|xi − x| i=0 |xi − x|
i=0
and
X
1
ψ(x) = 1
i=0
1
,
|xi − x|
(27)
and similarly for g1 . Therefore, by viewing |xi − x| as the distance from x to the boundary point
xi of the domain (x0 , x1 ) we see that the mean value interpolant g in (4) is a generalization of the
linear univariate interpolant g0 to two variables. Similarly, φ in (4) generalizes the denominator
of ψ above. This suggests a Hermite approach for the curve case. Given the values and inward
normal derivative of a function f defined on ∂Ω, we seek a function p : Ω → R satisfying
p(y) = f (y)
and
∂p
∂f
(y) =
(y),
∂n
∂n
y ∈ ∂Ω,
(28)
in the form
p(x) = g0 (x) + ψ(x)g1 (x),
(29)
where g0 is the Lagrange mean value interpolant to f in (11), ψ is the weight function
ψ(x) =
1
,
φ(x)
(30)
with φ from (11), and g1 is a second Lagrange mean value interpolant whose data is yet to be
decided. Similar to the univariate case, we need to show that ψ(y) = 0 and ∂ψ
(y) 6= 0 for
∂n
y ∈ ∂Ω. Then we obtain (28) by setting
∂g0
∂ψ
∂f
(y) −
(y)
(y),
y ∈ ∂Ω.
(31)
g1 (y) =
∂n
∂n
∂n
0
Thus we also need to determine ∂ψ
(y) and ∂g
(y). We treat each of these requirements in turn.
∂n
∂n
First, observe that Theorems 2 and 3 give the upper and lower bounds
1
1
d(x, ∂Ω) ≤ ψ(x) ≤ d(x, ∂Ω),
2π
C
x ∈ Ω,
(32)
and so ψ(x) → 0 as x → ∂Ω, and so ψ extends continuously to ∂Ω with value zero there.
Figure 5 shows the upper and lower bounds on ψ with C = 2 in the case that Ω is the unit disk.
The figure shows a plot of ψ and the two bounds along the x-axis. Next we show that the normal
derivative of ψ is non-zero.
Theorem 5 If d(ME , ∂Ω) > 0 and d(MI , ∂Ω) > 0 and y ∈ ∂Ω, then
∂ψ
1
(y) = .
∂n
2
14
0.5
0.4
0.3
0.2
0.1
0
−1
0
1
Figure 5: Upper and lower bounds for the unit disk.
a1
a2
R
R
h
x
h
δ
y
a3
Figure 6: Lines in proof of Theorem 5.
15
Proof. Let R = d(MI , ∂Ω). Then the open disc B of radius R that is tangential to ∂Ω at y on
the inside of ∂Ω is empty. For small enough δ > 0, the point x = y + δn is in B. Let a1 , a2 , a3
be the three points on ∂B such that a2 6= y lies on the line through x and y, and a1 and a3 lie
on the line perpendicular to it, see Figure 6. Let q be the four-sided polygon passing through
y, a1 , a2 , a3 . Then
Z 2π
Z 2π
1
1
dθ ≤
dθ.
φ(x) ≤
ρ1 (x, θ)
ρ(x, θ; q)
0
0
Then, by Lemma 1 applied to each edge of q, and since tan(π/4) = 1, we have
1
1
1
1
φ(x) ≤ 2
+
+2
+
.
ky − xk ka1 − xk
ka1 − xk ka2 − xk
So, since ky − xk = δ and ka2 − xk = 2R − δ, and letting h = ka1 − xk = ka3 − xk, we find
2δ
δ
δφ(x) ≤ 2 1 +
+
.
h
2R − δ
p
√
Moreover, since h2 = R2 − (R − δ)2 , we have h = (2R − δ)δ ≈ 2Rδ for small δ, and
therefore
lim sup δφ(x) ≤ 2.
(33)
δ→0
On the other hand, for small δ, y is the closest point to x in ∂Ω, and then (17) gives
2d
d
√
δφ(x) ≥
,
δ+d
δ + d + δ 2 + 2δd
where d = d(ME , ∂Ω), and thus
lim inf δφ(x) ≥ 2.
δ→0
(34)
The inequalities (33) and (34) show that δφ(x) → 2 as δ → 0, and thus
∂ψ
ψ(x) − ψ(y)
1
1
(y) = lim
= lim
= .
δ→0
δ→0 δφ(x)
∂n
δ
2
2
We have now shown that the Hermite construction (29) works, and that the normal derivative
of ψ is 1/2. To apply (31) we still have to compute the normal derivative of g0 .
Theorem 6 Let g be as in (11). If d(ME , ∂Ω) > 0 and d(MI , ∂Ω) > 0, and y ∈ ∂Ω then
Z
∂g
1 b
(y) =
w(y, t) f (c(t)) − f (y) dt.
∂n
2 a
Proof. For small δ > 0, let x = y + δn. Then dividing both sides of equation (19) by δ, and
letting δ → 0, gives the result, using Theorem 5.
2
16
Figure 7: The weight function ψ on various domains.
We plotted the weight function ψ on four different domains, shown in Figure 7. In the first
three, we used numerical quadrature on the integral formula for φ in (11). We use an adaptive approach, where for each x, we split the integral into a fixed number of pieces, and apply Romberg
integration to each piece, i.e., the extrapolated trapezoidal rule. If at some stage we detect that x
is on the boundary, within a given numerical tolerance, we terminate the integration and return 0
for the value of ψ. For the fourth domain, which is a regular pentagon, we simply use the exact
polygonal formula in (7). We apply similar approaches to evaluate the interpolant g in (11).
The weight function ψ is itself a Hermite interpolant with value 0 and normal derivative 1/2
on the boundary. Figure 8 shows other Hermite interpolants.
5
A minimum principle
A useful property of harmonic functions is that they have no local maxima or minima on arbitrary
domains. Lagrange mean value interpolants, however, do not share this property on arbitrary
domains, but we conjecture that they do on convex domains. We are not able to show this, but
we can establish a ‘minimum principle’ for the weight function ψ on arbitrary domains. Since ψ
is positive in Ω and zero on ∂Ω, it must have at least one maximum in Ω, and the S example in
Figure 7 illustrates that it can have saddle points. But we show that it never has local minima.
Lemma 2 For arbitrary Ω, with φ given by (14),
Z
∆φ(x) = 3
0
2π n(x,θ)
X
j=1
17
(−1)j−1
dθ.
ρ3j (x, θ)
Figure 8: Hermite mean value interpolants.
Proof. With the notation of (22) we have w = (d × c0 )/r3 in (11) and differentiation gives
∇w =
(−c02 , c01 ) 3(d × c0 )d
+
r3
r5
and
∆w = 3
d × c0
.
r5
Integrating the latter expression with respect to t and using (10) and the notation of (14), gives
the claimed formula.
2
Lemma 2 shows that ∆φ > 0 in Ω due to (13). From this we deduce
Theorem 7 In an arbitrary domain Ω, the weight function ψ has no local minima.
Proof. Suppose x∗ ∈ Ω is a local minimum of ψ. Then ∇ψ(x∗ ) = 0 and ∆ψ(x∗ ) ≥ 0. But since
ψ = 1/φ, we have
∇ψ = −
∇φ
φ2
and
∆ψ = −
∆φ
|∇φ|2
+
2
.
φ2
φ3
Therefore, ∇φ(x∗ ) = 0 and ∆ψ(x∗ ) = −∆φ(x∗ )/φ2 (x∗ ) < 0, which is a contradiction.
6
2
Domains with holes
So far in the paper, we have assumed that Ω is simply connected. In the case that Ω is multiply
connected, all the previously derived properties and formulas continue to hold with only minor
changes. In fact, the angle formula for g in (14) is unchanged in the presence of holes as long as
the points pj (x, θ) are understood to be the ordered intersections of L(x, θ) with all components
of ∂Ω. Thus, all angle formulas and associated properties are also valid for multiply connected
domains. However, the boundary integral formula (11) needs to be extended as follows. Suppose
that Ω has m holes, m ≥ 0, so that ∂Ω has m + 1 components: the outer one and the m inner
ones. We represent all these pieces parametrically as ck : [ak , bk ] → R2 , k = 0, 1, . . . , m,
18
c2
c1
Ω
c0
Figure 9: Multiply connected domain.
with ck (ak ) = ck (bk ). The outer curve c0 of ∂Ω is oriented anti-clockwise and the inner pieces
c1 , . . . , cm are oriented clockwise, see Figure 9. Then (11) should be replaced by
m Z bk
m Z bk
X
X
wk (x, t) dt.
(35)
wk (x, t)f (ck (t)) φ(x),
φ(x) =
g(x) =
k=0
ak
k=0
ak
Previous formulas involving the single parametric curve c need to be extended accordingly, but
this is straightforward and left to the reader.
7
Applications
We discuss two applications of mean value Hermite (and Lagrange) interpolation.
7.1
Smooth mappings
Smooth mappings from one planar region to another are required in reduced basis element methods for PDE’s that model complex fluid flow systems [18]. The reduced basis element method
is a domain decomposition method where the idea is to decompose the computational domain
into smaller blocks that are topologically similar to a few reference shapes. We propose using
mean value interpolation as an efficient way of generating suitable smooth mappings. Figure 10
shows on the top left a reference shape for a bifurcation point in a flow system studied in [18]
that could model for example blood flow in human veins. Top right shows the reference shape
mapped to the computational domain, using the method of [18]. The mapping is continuous
but not C 1 along certain lines in the interior of the domain. However, the result of using Lagrange mean value interpolation is a C ∞ mapping, bottom left. Finally, it may be desirable to
control the normal derivative of the mapping along the boundary. This can be achieved using
Hermite mean value interpolation. Bottom right shows the Hermite mean value mapping where
the normal derivative of the mapping at each boundary point equals the unit normal vector at the
corresponding point of the computational domain boundary.
There appears to be no guarantee that these mappings will in general be one-to-one. However,
we have tested Lagrange mean value mappings from convex domains to convex domains and
have always found them to be injective. We conjecture that this holds for all convex domains.
19
Figure 10: A bifurcation prototype is mapped to a deformed bifurcation using different transfinite
interpolants.
7.2
A weight function for web-splines
Recently, Hollig, Reif, and Wipper [9, 8] proposed a method for solving elliptic PDE’s over
arbitrarily shaped domains based on tensor-product B-splines defined over a square grid. In
order to obtain numerical stability, the B-splines are ‘extended’, and in order to match the zero
boundary condition, they are multiplied by a common weight function: a function that is positive
in Ω and zero on ∂Ω. Various approaches to choosing a weight function for this kind of finite
element method have been discussed in [14, 19, 9, 21]. The weight function ψ we used in Hermite
interpolation satisfies these basic properties, and in view of the upper and lower bounds (32) and
the constant normal derivative in Theorem 5, it behaves like half the signed distance function
near the boundary. So ψ is a good candidate for the weight function in the web-spline method.
We used bicubic web-splines to solve Poisson’s equation ∆u = f on various domains Ω with
zero Dirichlet boundary condition and various right-hand sides f . The top two plots of Figure 11
show approximate solutions u over an elliptic domain with a circular hole, defined by the zeros
of r1 and r2 where
r1 (x1 , x2 ) = 1 − x21 /16 − x22 /9,
r2 (x1 , x2 ) = (x1 + 3/4)2 + (x2 − 1/2)2 − 1,
and with right hand side f = sin(r1 r2 /2), a test case used in [9]. The top left plot shows the
result of using the weight function ψ = r1 r2 , while the top right plot shows the result of using
the mean value weight function ψ. The error for the two methods is similar, with both having a
numerical L2 -error of the order O(h4 ) with h the grid size, as predicted by the analysis of [9].
At the bottom of Figure 11 are plots of the approximate numerical solution to ∆u = −1 on
other domains using the mean value weight function. On the left is the solution over a regular
20
Figure 11: Numerical solution using bicubic web-splines.
Figure 12: Solving inhomogeneous problems.
pentagon, and on the right is the solution over the domain defined by the ‘S’ in the Times font,
with piecewise-cubic boundary. The numerical L2 error in these two cases was O(h2 ), which is
expected when the domain boundary has corners.
One can extend the web-spline method to deal with inhomogeneous problems using Lagrange
mean value interpolation. If we want to solve ∆u = f in Ω with u = u0 on ∂Ω, we can let g be
the mean value interpolant (11) to u0 , and express the solution as u = g + v where v solves the
homogeneous problem ∆v = fˆ in Ω with v = 0 on ∂Ω, and fˆ = f − ∆g. This approach requires
computing the Laplacian of the mean value interpolant g in (4) and this can be done using the
formulas of Section 3. We used bicubic web-splines to solve the inhomogeneous problem with
f = −1/2 and u0 (y) = 1 − (y12 + y22 )/8. In Figure 12, the left plot shows the true solution
u(x) = 1 − (x21 + x22 )/8 and the right plot shows the numerical solution.
Acknowledgement. We thank Ulrich Reif, Kai Hormann, and Solveig Bruvoll for helpful ideas
21
and comments in this work.
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23
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