Eurographics Symposium on Geometry Processing 2008 Pierre Alliez and Szymon Rusinkiewicz (Guest Editors) Volume 27 (2008), Number 5 Pointwise radial minimization: Hermite interpolation on arbitrary domains M. S. Floater and C. Schulz CMA / IFI, University of Oslo, Norway Abstract In this paper we propose a new kind of Hermite interpolation on arbitrary domains, matching derivative data of arbitrary order on the boundary. The basic idea stems from an interpretation of mean value interpolation as the pointwise minimization of a radial energy function involving first derivatives of linear polynomials. We generalize this and minimize over derivatives of polynomials of arbitrary odd degree. We analyze the cubic case, which assumes first derivative boundary data and show that the minimization has a unique, infinitely smooth solution with cubic precision. We have not been able to prove that the solution satisfies the Hermite interpolation conditions but numerical examples strongly indicate that it does for a wide variety of planar domains and that it behaves nicely. Categories and Subject Descriptors (according to ACM CCS): G.1.1 [Numerical Analysis]: Interpolation – Interpolation formulas; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling – Curve, surface, solid, and object representations; J.6 [Computer-Aided Engineering]: Computer-Aided Design (CAD) Additional Key Words and Phrases: Transfinite interpolation, Hermite interpolation, mean value interpolation 1. Introduction Mean value interpolation has emerged as a simple and robust way to smoothly interpolate a function defined on the boundary of an arbitrary domain in Rn , and provides an alternative to solving a PDE such as the Laplace equation. This kind of interpolation started from a generalization of barycentric coordinates in the plane [Flo03], and was later extended to general polygons [HF06] and to triangular meshes [FKR05, JSW05]. These constructions were further generalized to continuous boundaries in [JSW05, SJW07]. More work on mean value coordinates and related topics can be found in [Bel06,FHK06,FH07,JSWD05,LBS06,LKCOL07, WSHD07]. Mean value interpolation only matches the values of a function on the boundary, not its derivatives. However, there are many applications in geometric modeling and scientific computing in which it is also desirable to match certain derivatives at the boundary, and again it is worth looking for methods that avoid solving a PDE such as the biharmonic equation. An early approach was that of Gordon and Wixom [GW74], which is based on averaging the values of interpolatory univariate polynomials across the domain. This c 2008 The Author(s) c 2008 The Eurographics Association and Blackwell Publishing Ltd. Journal compilation Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. method is simple conceptually but only applies to convex domains and requires computing intersections between lines and the domain boundary. A more recent approach to higher order interpolation by Langer and Seidel [LS08] is an extension of barycentric coordinates to incorporate derivative data given at a set of vertices. Generalized barycentric coordinates such as Wachspress, Sibson, and mean value coordinates are composed with a modifying function and then used to interpolate first order Taylor expansions at the vertices. This method is easy to compute and was applied successfully to give greater control when deforming polygonal meshes. However, even in the univariate case this method lacks cubic precision. Another recent idea, developed in [DF07, BF], was to build on properties of the mean value weight function, i.e., the reciprocal of the denominator in the rational expression for the interpolant, in order to construct C1 Hermite interpolants on arbitrary domains. This method has the advantage that it reduces to cubic polynomial interpolation in the univariate case, and in R2 and R3 it can be evaluated approximately through explicit formulas for polygons and triangular M. S. Floater & C. Schulz / Pointwise radial minimization: Hermite interpolation on arbitrary domains p (x ,v ) v Ω x dΩ Figure 1: Definition of p. meshes. However, the method does not have cubic precision in Rn for n > 1. In this paper we propose a new kind of Hermite interpolation on arbitrary domains in Rn . The method has cubic precision for C1 boundary data for all n, and in R2 the method appears to perform very well in practice. We start by viewing mean value interpolation as the pointwise minimization of a radial energy function involving first derivatives of linear polynomials. We then generalize this to match boundary derivatives up to order k by minimizing a radial energy function involving (k +1)-st derivatives of polynomials of degree 2k + 1. The case k = 0 is simply mean value interpolation. We analyze the cubic case, k = 1, and show that the minimization has a unique solution which has cubic precision. We cannot prove that the solution satisfies the Hermite interpolation conditions but numerical examples show that the solution interpolates the derivative boundary data in R2 for a wide variety of domain shapes. Figure 2: Mean value interpolants. 2. A new look at mean value interpolation We start by showing that mean value interpolation can be expressed as the solution to a radial minimization problem. 2.1. Convex domains Consider the case that Ω ⊂ Rn is a bounded, open, convex domain and that f : ∂Ω → R is a continuous function. For any point x = (x1 , x2 , . . . , xn ) in Ω and any unit vector v in the unit sphere S ⊂ Rn , let p(x, v) be the unique point of intersection between the semi-infinite line {x + rv : r ≥ 0} and the boundary ∂Ω; see Figure 1. Let ρ(x, v) be the Euclidean distance ρ(x, v) = kp(x, v) − xk. As developed in the papers [Flo03,JSW05,DF07,BF], the mean value interpolant g : Ω → R is given by the formula Z f (p(x, v)) g(x) = dv φ(x), S ρ(x, v) Z 1 φ(x) = dv, x ∈ Ω. ρ(x, v) S It was shown in [DF07,BF] that under mild conditions on the shape of the boundary ∂Ω, the function g interpolates f when f is continuous. Figure 2 shows two examples of the mean value interpolant g on a circular domain. In previous papers, g was derived from the mean value property of harmonic functions. In this paper we take a different viewpoint in order to generalize to Hermite interpolation. We claim that at a fixed point x ∈ Ω, the value g(x) is the unique minimizer a = g(x) of the local ‘energy’ function E(a) = Z Z ρ(x,v) ∂ S 0 ∂r 2 F(x + rv) dr dv, where F : Ω → R is the radially linear function, F(x + rv) = ρ(x, v) − r r a+ f (p(x, v)), ρ(x, v) ρ(x, v) v ∈ S, 0 ≤ r ≤ ρ(x, v). c 2008 The Author(s) c 2008 The Eurographics Association and Blackwell Publishing Ltd. Journal compilation M. S. Floater & C. Schulz / Pointwise radial minimization: Hermite interpolation on arbitrary domains To see this observe that f (p(x, v)) − a ∂ F(x + rv) = , ∂r ρ(x, v) and therefore, Z E(a) = S ( f (p(x, v)) − a)2 dv. ρ(x, v) v1 Thus, setting the derivative of E with respect to a to zero gives S v2 x f (p(x, v)) − a dv = 0, ρ(x, v) Z and solving this for a gives the solution a = g(x). p 2(x ,v) p 3(x ,v) 2.2. Non-convex domains An interpretation in terms of functional minimization can also be made for non-convex domains. Let Ω ⊂ Rn be an open, not necessarily convex domain. Recall that the intersection between a line and a surface is said to be transversal if the line does not lie in the tangent space of the surface at the point of intersection. We will say that a unit vector v is transversal with respect to x in Ω if all the intersections between {x + rv : r ≥ 0} and ∂Ω are transversal. For example, in the R2 case of Figure 3 all unit vectors are transversal at x except for v1 and v2 . If v is transversal, let µ(x, v) be the number of intersections of {x + rv : r ≥ 0} with ∂Ω which will be an odd number, assumed finite, and let p j (x, v), j = 1, 2, . . . , µ(x, v), be the points of intersection, ordered so that their distances ρ j (x, v) = kp j (x, v) − xk are increasing, x Figure 3: (a) Two non-transversal vectors. (b) A transversal vector with three intersections. Indeed, similar to the convex case, we find ρ1 (x, v) < ρ2 (x, v) < · · · < ρµ(x,v) (x, v). For example, for v in Figure 3b, there are three such intersections and so µ(x, v) = 3. We make the assumption that the set {v ∈ S : v is non-transversal} has measure zero, so that non-transversal v can be ignored when integrating over S. The mean value interpolant [DF07, BF] is then Z µ(x,v) g(x) = ∑ S j=1 Z µ(x,v) φ(x) = ∑ S j=1 (−1) j−1 f (p j (x, v)) dv ρ j (x, v) Z µ(x,v) ∑ S j=1 Z ρ j (x,v) 0 2 q0v, j (r) dr dv, where, for v ∈ S, 0 ≤ r ≤ ρ j (x, v), and j = 1, . . . , µ(x, v), qv, j is the linear polynomial qv, j (r) = ∑ (−1) j−1 S j=1 ( f (p j (x, v)) − a)2 dv, ρ j (x, v) and setting E 0 (a) = 0 gives a = g(x). φ(x), (−1) j−1 dv. ρ j (x, v) (−1) j−1 Z µ(x,v) E(a) = 3. Hermite interpolation We claim that g(x) is now the unique minimizer a = g(x) of E(a) = p 1(x ,v) v ρ j (x, v) − r r a+ f (p j (x, v)). ρ j (x, v) ρ j (x, v) c 2008 The Author(s) c 2008 The Eurographics Association and Blackwell Publishing Ltd. Journal compilation Having now recast mean value interpolation as the minimization of a radial energy function, we are ready to explore a possible generalization in which the interpolant also matches derivative boundary data. Instead of minimizing over first derivatives of linear polynomials, we now minimize over (k + 1)-st derivatives of polynomials of degree 2k + 1, for any k = 0, 1, 2, . . .. Specifically, we want to interpolate in Ω a Ck continuous real function f , given its values and partial derivatives up to order k at the boundary ∂Ω. Fix x ∈ Ω and let τ be some polynomial in πk (Rn ), the linear space of n-variate polynomials of degree ≤ k. We think of τ as its own Taylor series at x because we are only interested in the derivatives of τ up to order k at x. Then, for each transversal v ∈ S and for j = 1, . . . , µ(x, v), let qv, j be the M. S. Floater & C. Schulz / Pointwise radial minimization: Hermite interpolation on arbitrary domains univariate polynomial of degree ≤ 2k + 1 such that (i) qv, j (0) = Div τ(x), Then, Ev, j and E can be viewed as functions of a and b, and the task is to find a ∈ R and b ∈ Rn to minimize i = 0, 1, . . . , k, (i) qv, j (ρ j (x, v)) = Div f (p j (x, v)), Z µ(x,v) E(a, b) = i = 0, 1, . . . , k, where Dv f denotes the directional derivative of f in the direction v. Then, defining Z ρ j (x,v) Ev, j (τ) = 0 Z ρ j (x,v) Ev, j (a, b) = ∑ (−1) j−1 Ev, j (τ) dv, q(0) = f0 , (1) This procedure defines, in a pointwise fashion, a function g : Ω → R. In general, the minimization will yield a different polynomial τ at each x and so g will not itself be a polynomial. Several questions arise, the main ones being 1. does the minimization have a unique solution τ, so that g is well-defined, and 2. do the derivatives of g up to order k match those of f on ∂Ω? Considering first the univariate case, n = 1, it is well known that among all piecewise Ck+1 functions q on a real interval (a, b) that match derivatives up to order k of a given function f at a and b, the unique minimizer of the energy Z b (k+1) (q 2 q00 v, j (r) dr, (3) Lemma 1 Let q be the cubic polynomial such that and then set g(x) = τ(x). 0 and set g(x) = a. In turns out that E has a unique minimizing pair (a, b). In order to show this and to compute a we begin with a lemma. Z µ(x,v) S j=1 (2) where 2 (k+1) qv, j (r) dr, we propose to choose τ in πk (Rn ) to minimize E(τ) = (−1) j−1 Ev, j (a, b) dv, ∑ S j=1 2 (x)) dx a is the polynomial Hermite interpolant to f of degree at most 2k + 1. Thus when n = 1 and Ω = (a, b), we see that g, defined by (1), will simply be the Hermite polynomial interpolant to f of degree ≤ 2k + 1. Thus the answer to both questions is ‘yes’ when n = 1. We are not able to answer them for general n, however. Instead, in the rest of the paper, we focus on the case k = 1 and we show that there is a unique minimizer for all n. We cannot yet show that g really interpolates the derivatives of f of order 0 and 1, but we can show that if f is a cubic then g = f , in which case it trivially interpolates, and numerical examples in R2 strongly suggest that it interpolates any f when n = 2. In the case k = 1, we can express any polynomial τ ∈ π1 (Rn ) in the form τ(y) = a + (y − x) · b, q0 (0) = m0 , q(h) = f1 , q0 (h) = m1 , for some h > 0, and let Z h E= (q00 (x))2 dx. (4) 0 Then, with ∆ f0 = f1 − f0 , m20 + m0 m1 + m21 (∆ f0 )2 (∆ f0 )(m0 + m1 ) − 12 + 4 . h h3 h2 Proof Using the Bernstein polynomials ! k i k u (1 − u)k−i , k ≥ 0, 0 ≤ i ≤ k, Bi (u) = i E = 12 we can express q in the Bernstein form 3 q(x) = ∑ ci B3i (x/h) i=0 where c0 = f 0 , c2 = f 1 − M 1 , c1 = f 0 + M 0 , c3 = f 1 , and Mi = hmi /3. Then q00 (x) = 6 h2 1 ∑ ∆2 ci B1i (x/h), i=0 where ∆2 ci = ci+2 − 2ci+1 + ci , and squaring and integrating over [0, h] yields E= 12 (∆2 c0 )2 + ∆2 c0 ∆2 c1 + (∆2 c1 )2 . h3 Then, since ∆2 c0 = ∆ f0 − 2M0 − M1 , ∆2 c1 = −∆ f0 + M0 + 2M1 , qv, j (0) = a, qv, j (ρ j (x, v)) = f (p j (x, v)), a simple calculation gives 12 E = 3 (∆ f0 )2 − 3(∆ f0 )(M0 + M1 )+ h 3(M02 + M0 M1 + M12 ) , q0v, j (0) = v · b, q0v, j (ρ j (x, v)) = Dv f (p j (x, v)). which gives the result. for some a ∈ R and b = (b1 , . . . , bn )T ∈ Rn and then qv, j is the cubic polynomial such that c 2008 The Author(s) c 2008 The Eurographics Association and Blackwell Publishing Ltd. Journal compilation M. S. Floater & C. Schulz / Pointwise radial minimization: Hermite interpolation on arbitrary domains We can now apply Lemma 1 to give a formula for Ev, j (a, b) in (3), using matrix notation. Setting f0 = a, m0 = v · b, f1 = f (p(x, v)), m1 = Dv f (p(x, v)), and h = ρ(x, v), gives T Ev, j (a) = aT Mv, j a + Nv, j a + Pv, j , (5) where, with the shorthand p := p j (x, v) and ρ := ρ j (x, v), 2 6 3ρvT a a= , Mv, j = 3 , b ρ 3ρv 2ρ2 vvT Nv, j = 4 ρ3 aT Ma = E(a). Since qv,µ (ρµ (x, v)) = q0v,µ (ρµ (x, v)) = 0, E(a) ≥ −6 f (p) + 3ρDv f (p) , −3ρ f (p)v + ρ2 Dv f (p)v Substituting (5) into (2) now gives Z Ev,µ(x,v) (a, b) dv = 0, and hence for each v ∈ S, with µ = µ(x, v), we must have (6) Z ρµ (x,v) Ev,µ (a, b) = where M is the matrix 0 Z µ(x,v) M= Ev,µ(x,v) (a, b) dv ≥ 0. Thus aT Ma ≥ 0 for all a ∈ Rn+1 . Further, suppose that aT Ma = 0. Then, letting f = 0 again, we must have S E(a) = aT Ma + N T a + P, Z S 4 3( f (p))2 − 3ρ f (p)Dv f (p) + ρ2 (Dv f (p))2 . 3 ρ ∑ S j=1 (−1) j−1 Mv, j dv, (9) Lemma 2 shows that Ev, j (a) in (3) is non-increasing in j. Therefore Ev, j (a, b) − Ev, j+1 (a, b) ≥ 0 in (2) and so and Pv, j = Proof We first show that aT Ma ≥ 0. To do this, let f = 0. Then since N = 0 and P = 0, we have (7) and similarly for N and P, and integrating a matrix means integrating each element. As the set of non-transversal v has measure zero for each x ∈ Ω, the integration can be achieved by splitting the integral into a sum over integrals over the regions of S where the number of intersections µ(x, v) is constant. The matrix M is clearly symmetric and if it is also positive definite then it is non-singular and the unique minimum of E is the unique solution a to the equation 1 (8) Ma = − N, 2 and a can be computed directly from this using Cramer’s rule. To show that M is positive definite we use the following lemma. Lemma 2 If f1 = m1 = 0 in Lemma 1 then E in (4) is nonincreasing in h. Proof From Lemma 1 we know that f02 m2 f m + 12 0 2 0 + 4 0 , 3 h h h and computing the derivative with respect to h yields f2 m2 dE f m = − 36 04 + 24 0 3 0 + 4 20 dh h h h 2 f0 m0 = − 6 2 +2 ≤ 0. h h 2 q00 v,µ (r) dr = 0. Therefore, for each v ∈ S, qv,µ is linear and by (9), qv,µ = 0. Hence a = 0. 4. Boundary integrals If a parametric representation of ∂Ω is available, the integrals in M and N in (6) can be converted to integrals over the parameters of ∂Ω, in the spirit of [JSW05]. To this end, suppose that s = (s1 , . . . , sn−1 ) : D → ∂Ω is a parameterization of the curve or surface ∂Ω with parameter domain D ⊂ Rn−1 . We will assume that s is a regular parameterization, by which we mean that s is piecewise C1 and that at every point of differentiability t = (t1 , . . . ,tn−1 ) ∈ D, the first order partial derivatives Di s := ∂s/∂ti , i = 1, 2, . . . , n−1, are linearly independent. Thus, following the notation of [JSW05] and [BF], their cross product, s⊥ (t) := det(D1 s(t), . . . , Dn−1 s(t)) is non-zero, and is orthogonal to the tangent space at s(t). We make the convention that s⊥ points outwards from Ω. Then, considering for example the integrals in M in (7), we make the substitution p j (x, v) = s(t) in (7), so that E = 12 v= s(t) − x , ks(t) − xk and ρ j (x, v) = ks(t) − xk, and then (see [BF]), dv = (s(t) − x) · s⊥ (t) dt. ks(t) − xkn Substituting these expressions into (7), and similarly for N, gives the boundary representations Z Theorem 1 The matrix M is positive definite. c 2008 The Author(s) c 2008 The Eurographics Association and Blackwell Publishing Ltd. Journal compilation M= Z M̂w(x, t) dt, D N= N̂w(x, t) dt, D (10) M. S. Floater & C. Schulz / Pointwise radial minimization: Hermite interpolation on arbitrary domains where (s(t) − x) · s⊥ (t) , ks(t) − xkn+3 w(x, t) = and, with the shorthand s := s(t), 6 3(s − x)T M̂ = 2 , 3(s − x) 2(s − x)(s − x)T −6 f (s) + 3Ds−x f (s) N̂ = 4 . (−3 f (s) + Ds−x f (s))(s − x) This provides a way of numerically computing the value of g at a point x by sampling the surface s and its first derivatives and applying numerical integration. It also shows that g is C∞ smooth in Ω, due to differentiation under the integral sign. 5. Cubic precision We now establish cubic precision (for k = 1 and all n): Theorem 2 Suppose that f : Ω → R is a cubic polynomial. Then g = f in Ω. Proof Fix an arbitrary x ∈ Ω and let â := f (x) and b̂ := ∇ f (x). We will show that E(â, b̂) ≤ E(a, b) for all a ∈ R and b ∈ Rn and hence g(x) = â = f (x). Let ev, j : [0, ρ j (x, v)] → R be the cubic polynomial defined by ev, j (r) = qv, j (r) − f (x + rv). Then 2 ∂ f (x + rv) + e00 v, j (r) ∂r2 q00 v, j (r) = and so Z ρj Ev, j (a, b) = (x,v) 0 Z ρ j (x,v) 2 0 Z ρ j (x,v) 0 2 ∂2 f (x + rv) dr+ ∂r2 ∂2 f (x + rv) dr+ ∂r2 2 e00 v, j (r) dr. e00 v, j (r) Figure 4: Reproduction of a quadratic polynomial (maximal absolute error due to numerical integration: 3.4 · 10−8 ). We will show that K = 0, which will complete the proof. We apply integration by parts to the inner integral, giving Z ρ j (x,v) 0 e00 v, j (r) ∂2 f (x + rv) dr ∂r2 = ev, j (0)D3v f (x) − e0v, j (0)D2v f (x) = (a − â)D3v f (x) − (v · (b − b̂))D2v f (x). Then, since this expression is independent of j, and recalling that µ(x, v) is odd we deduce that Z Z K = 2 (a − â) D3v f (x) dv − (b − b̂) · D2v f (x)v dv . S S But both integrals in this expression are zero since D2−v f (x) = D2v f (x) and D3−v f (x) = −D3v f (x). Now Z ρ j (x,v) 2 ∂ ∂r2 0 2 f (x + rv) dr = Ev, j (â, b̂), and as ev, j is a cubic polynomial and ev, j (ρ j (x, v)) = e0v, j (ρ j (x, v)) = 0, it follows from Lemma 2 that µ(x,v) ∑ (−1) j−1 Z ρ j (x,v) 0 j=1 2 e00 v, j (r) dr ≥ 0. Thus, E(a, b) ≥ E(â, b̂) + K, where Z µ(x,v) K=2 ∑ S j=1 (−1) j−1 Z ρ j (x,v) 0 e00 v, j (r) ∂2 f (x + rv) dr dv. ∂r2 6. Numerical Examples We conclude the paper with some numerical examples of the minimizing C1 Hermite interpolant g in R2 . All of these examples were computed using Cramer’s rule to find the value of a in (8). The elements of M and N were found using the boundary formula (10) and adaptive numerical integration with a relative tolerance. Figure 4 shows g on the unit disk in the case that f (x, y) = r2 sin 2θ, with polar coordinates x = r cos θ and y = r sin θ. As predicted by Theorem 2, g is numerically equal to f in this case, since f is the quadratic polynomial f (x, y) = 2xy. In this example, the Hermite interpolant g is only slightly different to the mean value interpolant to the Lagrange data from f shown in Figure 2. Figure 5 shows g on a non-convex domain with a hole, c 2008 The Author(s) c 2008 The Eurographics Association and Blackwell Publishing Ltd. Journal compilation M. S. Floater & C. Schulz / Pointwise radial minimization: Hermite interpolation on arbitrary domains Figure 6: Comparison between interpolants (top) and errors (bottom) of the interpolant of [DF07] (left) and the minimizing interpolant g (right). Figure 5: Interpolant (top) and part of the error function (bottom) for a non-convex domain with hole and cusp. with f given by f (x, y) = sin(y) cos(2x). The interpolant g behaves nicely even though there is a cusp at the leftmost point of the boundary. The error between g and f near the boundary of the hole is shown in the lower part of Figure 5. The error function is cut open to demonstrate numerically that both the error and its gradient are zero at the boundary. Figure 9 shows further numerical evidence of the Hermite interpolation property of g. The errors in g along the x-axis are plotted for three different functions f on the unit circle. Figure 6 compares the minimizing Hermite interpolant g with the Hermite interpolant of [DF07] on the unit circle with f (x, y) = r2 sin 4θ. The top left shows the interpolant of [DF07] and the bottom left its error function. The right part of the figure shows g on the top and its error function at the bottom. Note that both error functions are tangentially zero at the boundary. In Figure 7 we use the interpolant for hole filling, with a circular hole, and with f (x, y) = cos r. In the top f is shown together with the boundary of the hole that is cut out. Then the Hermite interpolant of [DF07] is shown, and at the bottom the minimizing interpolant g. Figure 8 shows another example of the nice behaviour of c 2008 The Author(s) c 2008 The Eurographics Association and Blackwell Publishing Ltd. Journal compilation Figure 7: Hole filling with circular boundary. From top to bottom: original function, interpolant of [DF07], minimizing interpolant. M. S. Floater & C. Schulz / Pointwise radial minimization: Hermite interpolation on arbitrary domains [FHK06] F LOATER M. S., H ORMANN K., KÓS G.: A general construction of barycentric coordinates over convex polygons. Advances in Computational Mathematics 24, 1–4 (January 2006), 311–331. 1 [FKR05] F LOATER M. S., KÓS G., R EIMERS M.: Mean value coordinates in 3d. Comp. Aided Geom. Design 22, 7 (October 2005), 623–631. 1 [Flo03] F LOATER M. S.: Mean value coordinates. Comp. Aided Geom. Design 20, 1 (March 2003), 19–27. 1, 2 Figure 8: Hole filling with a piecewise smooth boundary. g(x) − f (x) [GW74] G ORDON W. J., W IXOM J. A.: Pseudoharmonic interpolation on convex domains. SIAM Journal on Numerical Analysis 11, 5 (October 1974), 909–933. 1 [HF06] H ORMANN K., F LOATER M. S.: Mean value coordinates for arbitrary planar polygons. ACM Transactions on Graphics 25, 4 (2006), 1424–1441. 1 .1 [JSW05] J U T., S CHAEFER S., WARREN J.: Mean value coordinates for closed triangular meshes. ACM Transactions on Graphics 24, 3 (July 2005), 561–566. 1, 2, 5 0 −.1 −.2 −1.0 −0.5 0.0 0.5 1.0 x Figure 9: Errors along the x-axis (y = 0) for different f , with the unit circle as domain. the minimizing interpolant g for hole filling with the same function f but with a more complicated boundary. Acknowledgement We would like to thank Christopher Dyken for giving us access to his code for the mean value and Hermite interpolants described in [DF07]. References [Bel06] B ELYAEV A.: On transfinite barycentric coordinates. In SGP ’06: Proceedings of the fourth Eurographics symposium on Geometry processing (2006), Polthier K., Sheffer A., (Eds.), Eurographics Association, pp. 89– 99. 1 [BF] B RUVOLL S., F LOATER M. S.: Transfinite mean value interpolation in general dimension. preprint. 1, 2, 3, 5 [DF07] DYKEN C., F LOATER M. S.: Transfinite mean value interpolation. Comp. Aided Geom. Design (2007), doi:10.1016/j.cagd.2007.12.003. 1, 2, 3, 7, 8 [JSWD05] J U T., S CHAEFER S., WARREN J., D ESBRUN M.: A geometric construction of coordinates for convex polyhedra using polar duals. In SGP ’05: Proceedings of the third Eurographics symposium on Geometry processing (2005), Desbrun M., Pottman H., (Eds.), Eurographics Association, pp. 181–186. 1 [LBS06] L ANGER T., B ELYAEV A., S EIDEL H.-P.: Spherical barycentric coordinates. In SGP ’06: Proceedings of the fourth Eurographics symposium on Geometry processing (2006), Polthier K., Sheffer A., (Eds.), Eurographics Association, pp. 81–88. 1 [LKCOL07] L IPMAN Y., KOPF J., C OHEN -O R D., L EVIN D.: GPU-assisted positive mean value coordinates for mesh deformation. to appear in SGP ’07: Proceedings of the fifth Eurographics symposium on Geometry processing, 2007. 1 [LS08] L ANGER T., S EIDEL H.-P.: Higher order barycentric coordinates. In Eurographics 2008 (Crete, Greece, 2008), Drettakis G., Scopigno R., (Eds.), vol. 27, Blackwell. 1 [SJW07] S CHAEFER S., J U T., WARREN J.: A unified, integral construction for coordinates over closed curves. Comp. Aided Geom. Design 24, 8-9 (2007), 481–493. 1 [WSHD07] WARREN J., S CHAEFER S., H IRANI A. H., D ESBRUN M.: Barycentric coordinates for convex sets. Advances in Computational Mathematics 27, 3 (October 2007), 319–338. 1 [FH07] F LOATER M. S., H ORMANN K.: Barycentric rational interpolation with no poles and high rates of approximation. Numerische Mathematik 107, 2 (August 2007), 315–331. 1 c 2008 The Author(s) c 2008 The Eurographics Association and Blackwell Publishing Ltd. Journal compilation

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