JCAD: Model pp (col. fig: NIL) ARTICLE IN PRESS. Computer-Aided Design xx (xxxx) xxx xxx. Dan Liu, Guoliang Xu

Transcription

1 JCAD: + Model pp. (col. fig: NIL) 0 Computer-Aided Design xx (xxxx) xxx xxx Angle deficit approximation of Gaussian curvature and its convergence over quadrilateral meshes Abstract Dan Liu, Guoliang Xu State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 0000, China Received July 00; accepted January 00 We propose a discrete approximation of Gaussian curvature over quadrilateral meshes using a linear combination of two angle deficits. Let g i j and b i j be the coefficients of the first and second fundamental forms of a smooth parametric surface F. Suppose F is sampled so that a surface mesh is obtained. Theoretically we show that for vertices of valence four, the considered two angle deficits are asymptotically equivalent to rational functions in g i j and b i j under some special conditions called the parallelogram criterion. Specifically, the numerators of the rational functions are homogenous polynomials of degree two in b i j with closed form coefficients, and the denominators are g g g. Our discrete approximation of the Gaussian curvature derived from the combination of the angle deficits has quadratic convergence rate under the parallelogram criterion. Numerical results which justify the theoretical analysis are also presented. c 00 Published by Elsevier Ltd Keywords: Angle deficit; Gaussian curvature; Quadrilateral mesh; Convergence. Introduction In many application areas such as image processing, surface processing, computer aided geometric design and computer graphics etc., the geometric data are usually available as polygonal meshes, typically triangular and quadrilateral meshes. Therefore, the estimation of several intrinsic geometric quantities of a surface, such as the normal vector, the mean curvature and the Gaussian curvature, has been a significant task. Our attention in this research is focused on the estimation of the Gaussian curvature over quadrilateral meshes based on the Gauss Bonnet theorem. Many approaches for estimating the Gaussian curvature of triangular surface meshes have been developed. Several of these approaches are established based on local surface fitting techniques, including paraboloid fitting (see,0,), quadratic fitting (see,,) and circular fitting (see, ). The approximate Gaussian curvature is calculated from Partially supported by Natural Science Foundation of China (00) and National Key Basic Research Project of China (00CB000). Corresponding author. Tel.: + 0 ; fax: address: xuguo@lsec.cc.ac.cn (G. Xu). the fitting function at the surface point. Some other approaches are based on differential geometric theorems and formulas such as the Gauss Bonnet theorem, Euler theorem and Meusnier 0 theorem. In,, a theorem about Gauss maps is used. The theorem states that the Gaussian curvature at a given point on a surface is the limit of the area of the spherical image of the region around the point divided by the area of the region as the region shrinks around the point. Using the Gauss Bonnet theorem, a discrete scheme called the Gauss Bonnet or angle deficit scheme is obtained in,,,,. With the aid of an eigenvalues/vectors analysis of matrices that is defined via an integral formula related to normal curvature, a discrete curvature scheme is presented in. In, Watanabe and 0 Belyaev propose a scheme based on the integral formulas of the normal curvature and its square derived from Euler s theorem. In,, the convergence of invariants is obtained in a different way, the geometric conditions are translated into algebraic properties of the metric distortion tensor. Surazhsky et al. compare five estimation methods of Gaussian curvature on triangular meshes and conclude that the Gauss Bonnet (angle deficit) scheme obtained from the Gauss Bonnet theorem is the best one among the five. Xu presents a sufficient condition (a parallelogram criterion for /$ - see front matter c 00 Published by Elsevier Ltd doi:0.0/j.cad Design (00), doi:0.0/j.cad

2 JCAD: D. Liu, G. Xu / Computer-Aided Design xx (xxxx) xxx xxx vertices of valence six) for the convergence of this scheme and proves that the approximation converges with a quadratic rate. Borrelli et al. prove that the angle deficit is asymptotically equivalent to a homogeneous polynomial of degree two in the principal curvatures. Moreover, for regular meshes, the angle deficit provides an estimation with linear convergence rate for Gaussian curvature only when the valence of the triangulation is six or the valence is four with the one-ring neighbors aligned with the principal directions (explanations can be found in 0 Remark.). The discrete schemes over triangular meshes can be easily applied to quadrilateral ones by subdividing each quadrilateral into two triangles. However, the computational results are often very different for the same discrete scheme applied to the same quad mesh with different subdivisions (see Section ). In this paper, we propose a discrete scheme that can be used to compute the Gaussian curvature over quadrilateral meshes directly. This discrete scheme is also based on the Gauss Bonnet theorem, but a modification is made. Suppose 0 a surface mesh is obtained from a sampling of a smooth parametric surface F with g i j and b i j as the coefficients of the first and second fundamental forms of F, respectively; theoretically we show that for vertices of valence four, the considered two angle deficits are asymptotically equivalent to rational functions, which under some special conditions called a parallelogram criterion are homogenous polynomials of degree two in b i j divided by g g g. Considering different results obtained from different angle deficits applied to compute the Gaussian curvature, we use an arithmetic average of two kinds 0 of angles as a substitution in our scheme (details can be found in the next section). This modification yields a convergent result with quadratic rate. Furthermore, the condition of the convergence is more general than that of Borrelli s. The rest of the paper is organized as follows. In Section, we describe the discretization of the Gaussian curvature in detail. In Section, we show the main result that two angle deficits are asymptotically equivalent to homogenous polynomials of degree two in the coefficients of the second fundamental form divided by g g g and their average converges with a 0 quadratic rate under the parallelogram criterion. In Section, we compare some numerical results of the convergence property of the Gauss Bonnet scheme over triangular meshes and our discrete scheme over quadrilateral meshes. In Section, we describe several applications of the Gaussian curvature. Section concludes the paper. In the Appendix, we give the proof of the theorem of Section in detail. 0. Gaussian curvature and its discretization Gaussian curvature. Let F(ξ, ξ ) be a regular C continuous parametric surface in R. Let t i = F(ξ, ξ ), t i j = F(ξ, ξ ), i, j =,. ξ i ξ i ξ j Then the unit normal vector of F(ξ, ξ ) is given by n = t t t t. Let g i j = t T i t j, b i j = n T t i j, i, j =,, Fig... Definitions of region D and quadrilateral p i a j a j+ m j. g i j and b i j are coefficients of the first and the second fundamental forms, respectively. We know that the Gaussian curvature K (p) of F at a surface point p = F(ξ, ξ ) is defined by the following relation K (p) := b b b g g g. In this paper, we will use another expression for the Gaussian curvature (see 0) to prove our convergence result 0 K (p) = tt Qt t T Qt, (.) where G = g i j R, Q = I t, t G t, t T R. Discretization of Gaussian curvature. Let D be a region of a C parametric surface F, whose boundary consists of piecewise smooth curves Γ j. If all these Γ j are geodesic curves, the reduced local Gauss Bonnet theorem is as follows: K (p)da = π γ j, (.) D where γ j is the exterior angle at the jth corner point p j of the boundary (see Fig.. (left)). 0 Now we consider the discretization of the Gaussian curvature K based on the Gauss Bonnet theorem. Let M be a quadrilateral mesh of F in space R and {p i } be the vertex set of M. For the vertex p i, let N(i) = {,,..., n} denote the set of the vertex indices of one-ring neighbors of p i. Let p i p j p j+ p j+n be a neighbor quadrilateral of p i in M, where p j+n is the opposite vertex of p i in the quadrilateral p i p j p j+ p j+n. The integral region in (.) is approximated by the union of the quadrilateral p i a j a j+ m j as shown in Fig.. (right), where a j, a j+ are the midpoints of the 0 edges p i p j, p i p j+ respectively. a j m j and a j+ m j are parallel to p i p j+ and p i p j, respectively. The area of quadrilateral p i a j a j+ m j tends to a quarter of the area of quadrilateral p i p j p j+ p j+n, as the diameter of the mesh tends to zero (see the Remark. in Section ). Moreover, we assume that K (p) is a constant approximation in the local neighborhood of p i. By simple trigonometry, j γ j = j θ j, Eq. (.) is discretized as K () (p i ) := A(p i ) π n θ j, (.) Design (00), doi:0.0/j.cad

3 JCAD: D. Liu, G. Xu / Computer-Aided Design xx (xxxx) xxx xxx Fig... The definitions of angles θ j, α j, β j. where A(p i ) is the sum of the areas of the quadrilaterals p i p j p j+ p j+n, j =,..., n and θ j = p j p i p j+ ( j = i,..., n) (as shown in Fig..). Since points p i, p j, p j+, p j+n are not coplanar, the area of the quadrilateral p i p j p j+ p j+n denoted as A(p i p j p j+ ) can be expressed as the sum of the areas of two triangles p i p j p j+n and p i p j+ p j+n. This scheme can be regarded as a natural generalization of the Gauss Bonnet scheme from the triangular mesh to the quadrilateral mesh. However, no convergence property is observed in our numerical experiments. Therefore, a modification is considered. Approximating θ j by α j + β j with α j = p j+ p i p j+n and β j = p j+n p i p j, we obtain another approximation of the Gaussian curvature K () (p i ) := n π (α j + β j ). (.) A(p i ) It is regrettable that this discrete scheme is not convergent either. Since α j + β j θ j, K () (p i ) K () (p i ). Motivated by this inequality, we conjecture that K () (p i ) and K () (p i ) may be over- and under-estimations of K (p i ), respectively. Hence, their weighted average may yield a more accurate approximation. An amazing fact we find from the numerical experiments is that the errors of (.) and (.) are almost of the same magnitude but opposite sign under the condition of Theorem. (parallelogram criterion). This fact is also justified by theoretical analysis. Therefore, we take the arithmetic average of these two schemes as a better approximation of the Gaussian curvature K (p i ) n θ j + α j + β j π. (.) A(p i ) The obtained discrete scheme (.) can be used to compute the Gaussian curvature over quadrilateral meshes directly. Convergence analysis of this scheme is presented in Sections and. Remark.. As we have stated, discrete schemes over triangular meshes can be applied to quadrilateral ones by subdividing each quadrilateral into two triangles. Since each quadrilateral can be subdivided in two ways, there are n possibilities for subdividing a quad mesh with n quadrilaterals. Fig.. shows three of the eight subdivision cases of four quadrilaterals around q i j which make the valence of q i j even. The approximate Gaussian curvatures over these three types of triangular domains can be computed using the following 0 Gauss Bonnet scheme n K T (p i ) := π θ j, (.) A T (p i ) where A T (p i ) is the sum of the areas of triangles p i p j p j+ and θ j are the angles opposite to the edges p j p j+. The convergence property can be observed for the domain triangulation (a) which satisfies the convergent conditions presented in. For the domain triangulations (b) and (c), the scheme (.) is not convergent. Therefore, the convergence properties are different for the same discrete scheme applied to the same quad mesh with different subdivision strategies. 0. Convergence results Theorem.. Let p i be a vertex of a quadrilateral mesh M with valence, p j ( j =,..., ) be its neighbor vertices and p j+ ( j =,..., ) be its opposite vertices in the quadrilateral p i p j p j+ p j+. Suppose p i and p j ( j =,..., ) are on a sufficiently smooth parametric surface F(ξ, ξ ) R, and there exist q i, q j R such that (see Fig..) { q j+ q i = (q j q i ), j =,,,, (.) q j+ = q j + q ( j+) mod q i, j =,...,, p i = F(q i ), p j = F(q j ), j =,...,. 0 Then π θ j (h) = K (p i ) + (B + cot θ B ) + O(h ), (.) π (α j (h) + β j (h)) = K (p i ) (B + cot θ B ) + O(h ), (.) where B = b cos ψ + b sin ψ cos ψ + b sin ψ, B = (b b ) cos ψ sin ψ + b (cos ψ sin ψ), θ (0, π) is the angle between edges q q i and q q i, ψ 0, π) is the angle from the horizontal axis of the ξ ξ -plane to 0 q q i in the counter-clockwise direction, α j (h), β j (h), θ j (h) and are defined as before from the vertices p j (h), with p j (h) = F(q j (h)), q j (h) = q i + h(q j q i ), j =,...,. The outline of the proof. The detail of the proof can be found in Appendix. Here we only give an outline. Design (00), doi:0.0/j.cad

4 JCAD: D. Liu, G. Xu / Computer-Aided Design xx (xxxx) xxx xxx 0 Fig... Quadrilateral mesh on domain with s = q q i, s = q q i. First, we expand the surface point p j (h) into a power series at p i with respect to h. Therefore the distance between two surface points, the area of the quadrilateral on the surface and the cotangents of the angles θ j (h), α j (h), β j (h) can also be expressed as a power series with respect to h. Then using the parallelogram criterion, we find that for ε = 0,,, the coefficients of h ε in power series of the sum π θ j (h) and π (α j (h) + β j (h)) vanish, while the coefficient of h in the series is K (p i )+ (B +cot θ B ) and K (p i ) (B +cot θ B ), respectively. So the result is proved. Discussion. Because the proof of the theorem is rather complicated, one may ask why the convergence result is correct and how it can be achieved under the given conditions. To answer, we give some geometric explanations based on the result of. Given a triangle on the surface with geodesic boundaries (see Fig..), we know that the difference between the angle ϕ of geodesics and the angle ϕ of the chords can be expressed as ϕ ϕ = d d k k sin ϕ d k + d k tan ϕ + O(d ), (.) 0 where k and k are the normal curvatures of geodesics l and l, respectively, d, d are the lengths of the chords, d = max{d, d }. Consider a surface expressed as a graph of a bivariate function f (x, y) = b x + b xy + b y. Noticing that the tangent plane of this quadratic surface is parallel to the parametric plane. It is easy to calculate that the Gaussian curvature K = b b b at the origin. In this simple case, our derivation yields the following result, which is similar to (.), 0 θ j θ j (h) = s js j+ k n ( j)k n ( j + ) sin θ j h s j k n ( j) + s j+ k n ( j + ) h + O(h ), (.) tan θ j where θ j (h) is the angle between p j (h)p i and p j+ (h)p i, θ j is the corresponding angle on the parametric plane; s j = q j q i, k n ( j) is the normal curvature of the direction (l j, r j ) T =. It is easy to see that q j q i q j q i θ j (h) = ϕ, θ j = ϕ + O(h ). Hence relations (.) and (.) are equivalent within the error O(h ). Using the relation Fig... ϕ is the angle between geodesics l, l ; d, d denote the length of the chords; ϕ is the angle between the chords. k n ( j) = b l j + b l j r j + b r j, and summing (.) for j =,,,, we can derive that under 0 the parallelogram criterion, π θ j (h) = Ah K (p i ) + Ah R + O(h ), where A is the area of the quadrilateral q i q j q j+ q j+, R = (b + cot θb ). Moreover, the relationship between the area A and the area of the quadrilateral p i p j (h)p j+ (h)p j+ (h), denoted as, can be easily derived = Ah + O(h ). So we obtain (.). (.) can be derived similarly. Remark.. Readers may consider that the proof of Theo- rem. can be simplified by using formula (.). Since the 0 angle ϕ in (.) is defined on the tangent plane, a conclusion similar to (.) and (.) can be obtained by using (.) but its convergence rate is linear. However, in Theorem., the angle θ is defined on the parametric plane, so we can prove that (.) and (.) converge with quadratic rate. Eqs. (.) and (.) imply that K () (p i ) and K () (p i ) are truly over- and under-estimations of K (p i ) respectively, as h 0. Based on Theorem., we can obtain easily the following theorem. Theorem.. Under the conditions of Theorem., we have 0 θ j (h) + α j (h) + β j (h) π = K (p i ) + O(h ), as h 0. (.) This says that the discrete approximation (.) converges to K (p i ) at a quadratic rate as h 0. Multiplying both sides of (.) and (.) by /, we have the following corollary. Corollary.. Under the conditions of Theorem., we have π θ j (h) = h q q i q q i sin θ K (p i ) + (B + cot θ B ) + O(h ), (.) Design (00), doi:0.0/j.cad

5 JCAD: D. Liu, G. Xu / Computer-Aided Design xx (xxxx) xxx xxx π (α j (h) + β j (h)) = h q q i q q i sin θ K (p i ) (B + cot θ B ) + O(h ). (.) In practice, the sampling data may suffer from perturbations. The following corollary states how large a perturbation of q j (h) is allowed such that the convergence conclusion still holds. Corollary.. Under the condition of following conclusions hold Theorem., the (i) if q j (h) is perturbed as q j (h) = q i + h(q j q i ) + ε j (h), ε j (h) R, such that ε j (h) C j h α, then π α, C j is a constant, θ j (h) + α j (h) + β j (h) = K (p i ) + O(h min{α,} ) as h 0, (.) (ii) if q j (h) is perturbed as q j (h) = q i + (h + ε j (h))(q j q i ), ε j (h) R, such that ε j (h) C j h α, then π α, C j is a constant, θ j (h) + α j (h) + β j (h) = K (p i ) + O(h min{α,} ), as h 0. (.0) The proof of this corollary is similar to the proof of Theorem.. Remark.. Under the condition (.), the four quadrilaterals q i q j q j+ q j+n around q i are congruent parallelograms. Hence, we refer to this condition as the parallelogram criterion. The convergence result holds under this particular condition. However, this particular case is very useful and important. For example, many numerical simulations of geometric partial differential equations, which involve curvatures, are conducted over a domain grid formed by a uniform two-directional partition. This kind of domain grid is a special case of the condition. Another example is the curvature estimation of discretized four-sided parametric surface patches, such as B-spline, NURBS and Coons surfaces. These patches are often discretized by a uniform two-directional partition of their domains. Remark.. In this remark, we mention a related result from. Let p be a point of a smooth surface S and p i, i =,..., n be its one-ring neighbors. Point p is called a regular vertex if the p i s lie in normal sections two consecutive ones of which form an angle of θ(n) = π/n and the distance from p to p i denoted as η i all take the same value η. For n =, the sufficient condition that the angle deficit tends to the exact Gaussian curvature with linear rate is that the one-ring neighbors are regularly distributed and aligned with the principal directions. This is a rather restrictive condition, obviously. Moreover, for regular meshes, n = is the only value of n such that π i γ i depends upon the principal directions. So the curvature cannot be computed from the angle deficit of two meshes of valences n and n, 0 where n =, n. Compared with Borrelli s method, Eq. (.) is more applicable. The convergent condition (.) is more general than the regular condition. Condition (.) demands q i q j q j+ q j+, j =,..., are four congruent parallelograms, while the regular condition demands the angles θ and the distances η all take the same value. Furthermore, Eq. (.) converges with quadratic rate, while under the regular condition, Borrelli s method converges with linear rate. Let us consider a special case of Corollary.. Take θ = π, q q i h = q q i h = η. Suppose surface 0 F is parameterized such that t and t are the unit principal directions, respectively. Then we know that g = g =, g = b = 0. Hence, (.) becomes π θ j (h) = K (p i ) + sin ψ cos ψ(k k ) η + O(η ), where k, k are the principal curvatures of the surface F. This result is the same as Borrelli et al. s, except for the remainder which is O(η ) instead of o(η ). Remark.. Now we further assume the surface considered above is a graph of a bivariate function z = f (x, y). This is the 0 case considered by Meek and Walton. But their conclusion for the angle deficit scheme (.) has the approximation accuracy of O(). Moreover, the combination of / (angle deficit scheme (.) with points) + / (angle deficit scheme (.) with points) approximates the Gaussian curvature with accuracy of O(h). Our arithmetic averaging strategy yields a quadratic convergence scheme.. Numerical experiments In this section, we test the convergence property of the Gauss Bonnet scheme over triangular meshes and demonstrate 0 numerical behaviors of our scheme on quadrilateral meshes as well. We take four bivariate functions: F = { exp } (x 0.) + (y 0.),. + cos(.y) F = + (x ), F = / (x 0.) (y 0.) 0., F = e x+y, over the xy-plane as three dimensional surfaces. Both the exact and discrete Gaussian curvatures are computed at the selected Design (00), doi:0.0/j.cad

6 JCAD: D. Liu, G. Xu / Computer-Aided Design xx (xxxx) xxx xxx Fig... Subdivision of quadrilaterals over xy-plane. Domain (a): Triangular mesh with valence ; Domain (b): Triangular mesh with valence ; Domain (c): Triangular mesh with valence. Table. The asymptotic maximal errors of (.) for functions F,..., F n F F F F (a).e+0 r.e+0 r.e+0 r.e 0 r (b).0e+0.e+0.0e+00.00e 0 (c).e+0.e+0.0e+00.0e 0 points q i j = (x i, y j ) defined as (x i, y j ) = ( i 0 ) for i =,...,, j =,...,. In order to get the polygonal meshes of these surfaces, we subdivide the domain around q i j into polygonal meshes, and then map the planner polygonal meshes onto the surface by theses bivariate functions. First of all, we intend to illustrate the convergence property of the Gauss Bonnet scheme (.) over triangular meshes as shown in Fig... The vertex q i j has four adjacent parallel quadrilaterals whose edge length is r, acute angle is θ, where the edge length r is selected as n, n =,,,.... Table. shows the 0 asymptotic maximal errors using the Gauss Bonnet scheme (.) to compute the Gaussian curvatures of the surfaces defined by functions F,..., F over domain triangulations (a) (c). The first row exhibits a quadratic convergence rate. The last two rows show the non-convergence property. In the following, we use discrete schemes (.), (.) and (.) to compute the Gaussian curvature over quadrilateral meshes. The quadrilateral mesh around q i j is defined as shown in Fig.., where 0 θ = π, q = q i j + (r, 0), q = q i j + r(cos θ, sin θ), q = q i j + (r + r cos θ, r sin θ), q = q i j + (r cos θ r, r sin θ), q j+ q i j = (q j q i j ), j =,,,, 0, j and the edge length r is also selected as n with n =,,,,,.... Fig.. shows the errors of (.) and (.), respectively. The Gaussian curvatures of surfaces defined by functions F,..., F are computed over quadrilateral meshes as shown in Fig.. with r = /. In these figures, 0 (.) is used for the left column, (.) is used for the mid column, and the right column shows the sum of the errors of these two cases. We can see easily from Fig.. that the errors of (.) and (.) are of the same magnitude and opposite sign. The numerical experiments show that as r 0, the maximal error of the approximated Gaussian curvature computed by (.) over the quadrilateral meshes and the exact Gaussian curvature computed from the continuous surfaces defined by F j tend asymptotically to zero. Fig.. shows that the maximal errors of the approximate Gaussian curvature for the functions F,..., F converge to zero with a quadratic rate. 0. Applications It is well-known that the Gaussian curvature K for a regular surface is an important intrinsic geometric quantity. It tells us whether a surface is elliptic (K > 0), hyperbolic (K < 0), or parabolic (K = 0). When K = 0 everywhere, the surface is also known as developable. Hence the Gaussian curvature has been widely used as a useful tool for describing shapes of surfaces in many applications, such as surface feature extraction (see 0, ), surface parametrization (see,), surface matching or alignment (see,), segmentation and flattening (see ), 0 etc. Below we describe some of them. Matching or alignment. A key issue of three dimensional objects matching is to search the correspondence between two surfaces. So the best transform which aligns these two surfaces as closely as possible can be found. By calculating distinct features of the surface s to be matched, the correspondence can be located, then the transform making the distance of the feature points minimal can be determined. So intrinsic geometric features which are invariant to the orientation and the translation of the coordinate system have been used for 0 matching purposes. In, two intrinsic surface properties, the Gaussian and the mean curvatures, are used as features for matching D free form objects. In, an algorithm aligning the meshes which are created by triangulating the point cloud that results from scanning an object is presented. The Gaussian curvatures of points on two meshes are compared to find the correspondence on the pair. If the difference between the Gaussian curvature of any two points on the mesh pair is less than a threshold, these two points are accepted as the corresponding points. In this application, an accurate estimation 0 of Gaussian curvature is crucial for achieving a trustable matching result. Surface segmentation and flattening. Segmenting a polygonal mesh into charts is an important problem in computer graphics. Many applications need segmentation techniques such as surface parametrization, compression, surface flattening and morphing. The criterion presented in for mesh segmentation is based on the even distribution of the Gaussian curvature over the resulting charts. This method generates low-distortion Design (00), doi:0.0/j.cad

7 JCAD: D. Liu, G. Xu / Computer-Aided Design xx (xxxx) xxx xxx charts suited better for surface parametrization. Vivodtzev et al. describe a method to segment a surface based on the discrete mean curvature and the Gaussian curvature. They do not use the magnitude of these curvatures but only their signs. Hence, a correct sign for the Gaussian curvature is important. Wang et al. present a method for finding cutting paths on a D triangular mesh surface to reduce the stretch in the flattened surface. Since the stretch depends directly on the Gaussian Fig... From the top down: Errors of functions F,..., F for quadrilateral mesh. curvature, the cutting of the surface along the paths linking the surface boundary and the nodes with high Gaussian curvature 0 can reduce the stretch in the flattened surface. From the above applications, we note that poorly estimated Gaussian curvature may lead to unacceptable results. So an accurate estimation of the Gaussian curvature is important for achieving a desirable performance. Now we show an example to illustrate that even for a very regular mesh the widely used Design (00), doi:0.0/j.cad

8 JCAD: D. Liu, G. Xu / Computer-Aided Design xx (xxxx) xxx xxx 0 Fig... Maximal errors of functions F,..., F. Fig... The exact Gaussian curvature and the discrete Gaussian curvatures computed by (.), (.) and (.), respectively. angle deficit schemes result in very poor estimation. Consider an analytic surface defined as the graph of the bivariate function F(x, y) = sin(x) sin(y)e (x +y ), (x, y).,.. We discretize the surface by uniformly partitioning the domain.,. into rectangles. Then the quadrilateral mesh is generated by mapping the domain partition onto the surface. Fig.. shows the exact Gaussian curvature and the discrete Gaussian curvatures computed by (.), (.) and (.), respectively. The left figure in the first row shows the plot of the exact Gaussian curvature of the function F. The right one in this row shows the plot of the discrete Gaussian curvature computed by (.). It can be easily observed that (.) gives a good approximation of the exact Gaussian curvature. The two figures in the second row show the plots of the discrete Gaussian curvature computed by (.) and (.), respectively. Both of them give very poor approximations. Also notice that in most of the region, the sign of the estimated Gaussian curvature is incorrect. It could be imagined that in using these estimated Gaussian curvatures to handle the practical problems mentioned 0 above, the outcomes would surely be very misleading. Design (00), doi:0.0/j.cad

9 . Conclusions JCAD: D. Liu, G. Xu / Computer-Aided Design xx (xxxx) xxx xxx So we have 0 Based on the Gauss Bonnet theorem, a discretized scheme for Gaussian curvature has been derived for quadrilateral meshes. We show that the traditional angle deficit schemes could not result in convergent conclusion for vertices with valence (see Theorem.), while their arithmetic averaging yields surprisingly convergent results. Theoretical analysis shows that the proposed combination scheme converges to the exact one as the mesh size tends to zero with a quadratic rate under some special conditions (see Theorem.). The given numerical results which justify the theoretical analysis illustrate that the new scheme leads to a good approximation of Gaussian curvature under the given conditions. Appendix. Proof of Theorem. In this section, we give the proofs in detail of the convergent results. The proof requires a lot of fine derivations, readers who are not interested in the details may skip this section. We first outline the main idea of the proof. The key step of the proof is to compute π φ by arctan(cot(φ)) for ψ = 0. We 0 expand arctan(cot(θ j (h))) and arctan(cot(α j (h) + β j (h))) by Taylor expansion at h = 0. Then we prove that the coefficient of h divided by is just K (p i ) ± (b +cot θb ) det G and the other coefficients of h ε with orders ε less than are all zero. Hence, the main task is to compute, cot θ j (h) and cot(α j (h) + β j (h)). Without loss of generality, we may assume that q q i =. Suppose cos ψ sin ψ S = sin ψ cos ψ 0 0 is a rotation transform such that (q q i ) = S(, 0) T. First we prove the theorem on the domain rotated. We still denote q q i = (, 0) T, then there exists a constant a > 0 and an angle θ > 0 (see Fig..) such that q q i = a(cos θ, sin θ) T, q q i = ( + a cos θ, a sin θ) T, (A.) q q i = (a cos θ, a sin θ) T, and q j+ q i = (q j q i ), j =,,,. Let q j q i = s j d j with s j = q j q i and d j = q j q i q j q i = (l j, r j ) T, j =,...,. Let Fd k j := Fd k j (q i ) denote the directional derivative of F at q i of order k and in the direction d j, then we can expand p j (h), j =,...,, into the following form p j (h) = p i + l= h l l! sl j Fl d j + O(h ). p j (h) p i, p k (h) p i m+n=l h l = m!n! sm j sn k Fm d j, Fd n k + O(h ) l= m,n = h jk + h jk + h jk + h jk + O(h ), and the area of the quadrilateral with the adjacent edges p j (h) p i and p k (h) p i is as follows ( p j (h) p i p k (h) p i p j (h) p i, p k (h) p i ) = h δ (0) jk + h δ () jk + h δ () jk + h δ () jk + O(h ) 0 = h (δ (0) jk ) + h δ () jk (δ(0) jk ) + h δ (0) jk δ() jk (δ () jk ) (δ (0) jk ) + h (δ (0) jk ) δ () jk δ (0) jk δ() jk δ() jk + (δ() jk ) (δ (0) jk ) + O(h ), (A.) where δ (l) m+n=l+ jk = m,n a+b=m c+d=n a,b,c,d s m j sn k a!b!c!d! ( Fa d j, F b d j F c d k, F d d k F a d j, F c d k F b d j, F d d k ) l = 0,...,. Let g i j = t T i t j, g i jk = t T i t jk, g i jkl = t T i j t kl, e i jkl = t T i t jkl, e i jklm = t T i t jklm, f i jklm = t T i j t klm, where t i = ξ F i, t i j = F ξ i ξ j, t i jk = F ξ i ξ j ξ k, t i jkl = 0 F ξ i ξ j ξ k ξ l, i, j, k, l, m =,, then we have F d j = l j t + r j t, F d j = l j t + l j r j t + r j t, F d j = l j t + l j r jt + l j r j t + r j t, F d j = l j t + l j r jt + l j r j t + l j r j t + r j t. Following the fact s j+ = s j, d j+ = d j, j =,,,, we obtain for j =,,,, k =,...,, F k d j+ = ( ) k F k d j, (A.) and for j, k =,,,, m = 0,,,, 0 δ (m) j+,k+ = ( )m δ (m) jk, m+ j+,k+ = ( )m m+ jk. (A.) Hence, we can derive δ (0) j,( j+) mod = δ(0) ( j+) mod, j+ = δ(0) j, j+ = a sin θ = δ, j =,...,. (A.) Design (00), doi:0.0/j.cad

10 JCAD: 0 D. Liu, G. Xu / Computer-Aided Design xx (xxxx) xxx xxx It is easy to see that δ is the area of the quadrilateral q i From (A.) and (A.), we have the sum of areas of quadrilaterals p i p j (h)p j+ (h)p j+ (h) = ( p j (h) p i p j+ (h) p i p j (h) p i, p j+ (h) p i ) / = h δ + O(h ). (A.) With this preparatory work, we are ready to prove the theorem. In the following, we intend to compute π θ j(h) and π α j (h) β j (h) respectively. Noticing that the cotangent of the angle between p j (h) p i and p k (h) p i can be expressed as 0 0 p j (h) p i, p k (h) p i p j (h) p i p k (h) p i p j (h) p i, p k (h) p i + δ jk jk δ() jk δ δ h + h δ δ δ jk δ δδ() jk jk + jk (δ() jk ) δδ() jk jk + O(h ) { + h δ δ jk δ jk δ δ () jk δδ() jk δ() jk = jk + (δ () jk ) jk δ δ () jk δ(δ() δ δ () } jk jk jk ) = A jk + h A jk + h A jk + h A jk + O(h ), we have cot θ j (h) = A j, j+ + h A j, j+ + h A j, j+ + h A j, j+ + O(h ), cot α j (h) = A j+, j+ + h A j+, j+ + h A j+, j+ + h A j+, j+ + O(h ), cot β j (h) = A j, j+ + h A j, j+ + h A j, j+ + h A j, j+ + O(h ), and cot(α j (h) + β j (h)) = cot α j(h) cot β j (h) cot α j (h) + cot β j (h) where B j = A j+, j+ A j, j+ A j+, j+ +, A j, j+ = B j + h B j + h B j + h B j + O(h ), B k j = k (A m j, j+ Ak m+ j+, j+ ) m= A j+, j+ + A j, j+ k B n j (Ak n+ j+, j+ + Ak n+ j, j+ ) n= A j+, j+ + A j, j+, k =,,. From the formula arccot x + arctan x = π, for all x R, 0 we have π θ j(h) = arctan(a j, j+ + h A j, j+ + h A j, j+ + h A j, j+ + O(h )), (A.) π (α j(h) + β j (h)) = arctan(b j + h B j + h B j + h B j + O(h )). (A.) It follows from (A.), that for j =,,,, m =,,,, A m, = ( )m A m,, Am, = ( )m A m, A m, = ( )m A m,, Am, = ( )m A m, B m j+ = ( )m B m j, (A.) and for j =, A j+,( j+) mod = A j, j+, B j+ = B j. (A.0) Since (A.) and (A.) can be expanded into the following form by a Taylor series expansion π θ j(h) = arctan A j, j+ + h A j, j+ + (A j, j+ ) A j, j+ + + (A A j, j+ (A j, j+ ) j, j+ ) + (A j, j+ ) h A j, j+ + + (A A j, j+ A j, j+ A j, j+ j, j+ ) + (A j, j+ ) + (A j, j+ ) + (A j, j+ ) (A j, j+ ) h + O(h ), 0 π (α j(h) + β j (h)) = arctan B j + h B j + (B j ) B j + + (B B j (B j ) j ) + (B j ) h B j + + (B B j B j B j j ) + (B j ) + (B j ) + (B j ) (B j ) h + O(h ), Design (00), doi:0.0/j.cad

11 JCAD: D. Liu, G. Xu / Computer-Aided Design xx (xxxx) xxx xxx we obtain the following results by using (A.), (A.0) and (.), π π θ j (h) = j(h) θ = h δ A j, j+ + (A j, j+ ) A j, j+ (A j, j+ ) + (A j, j+ ) h + O(h ) = tt Qt + t T Qt cot θ + t T Qt cot θ = K (p i ) + R + O(h ), + O(h ) (A.) where R = t T Qt +t T Qt cot θ +t T Qt cot θ. Similarly, we have π (α j (h) + β j (h)) = = h δ π α j(h) β j (h) B j + (B B j (B j ) j ) + (B j ) h + O(h ) = tt Qt t T Qt t T Qt cot θ t T Qt cot θ + O(h ) = K (p i ) R + O(h ). (A.) Let B i j (i, j =, ) be the coefficients of the second fundamental form of surface F(ξ, ξ ) for ψ = 0. Then it is easy to derive that ti T j Qt kl = B i j B kl, hence, R = (B + cot θ B ). (A.) Therefore, (.) and (.) hold for ψ = 0. For the general domain sampling q q i = S(, 0) T, let b i j (i, j =, ) be the coefficients of the second fundamental form of surface F(ξ, ξ ), we have B = b cos ψ + b sin ψ cos ψ + b sin ψ, B = (b b ) cos ψ sin ψ + b (cos ψ sin ψ). Moreover K (p i ) and are invariant under the rotation transform. Hence we can derive the results of the theorem. Remark A.. The area of the quadrilateral p i p j p j+ p j+, denoted as A(p i p j p j+ ), can be computed as the sum of areas of two triangles p i p j p j+ and p i p j+ p j+. Let A (p i p j p j+ ) be the area of the parallel quadrilateral with the adjacent edges p i p j, p i p j+. From (A.) and (A.), we can see that A(p i p j p j+ ) = A (p i p j p j+ ) + O(h). Moreover, under the condition of Theorem., we have A(p i p j p j+ ) = A (p i p j p j+ ) + O(h ). Therefore, the area of the quadrilateral p i a j a j+ m j, which is obviously a quarter of A (p i p j p j+ ), is a quarter of A(p i p j p j+ ), as h 0. References Ko KK, Maekawa T, Patrikalakis NM. An algorithm for optimal free-form 0 object matching. Computer-Aided Design 00;:. Alboul L, van Damme R. Polyhedral metrics in surface reconstruction: Tight triangulation. Technical report. University of Twente, Dep. of Applied Mathematics;. Vadde S, Kamarthi SV, Gupta SM. Alignment of meshes using Gaussian curvature. In: Proceedings of industral engineering research conference. 00. Yamauchi H, Gumhold S, Zayer R, Seidel H-P. Mesh segmentation driven by Gaussian curvature. The Visual Computer 00;:. Wang CCL, Wang Y, Tang K, Yuen MMF. Reduce the strech in surface 0 flattening by finding cutting paths to the surface boundary. Computer- Aided Design 00;:. Vivodtzev F, Linsen L, Bonneau G-P, Hamann B, Joy KI, Olshausen BA. Hierarchical isosurface segmentation based on discrete curvature. In: Proceedings of vissym 0, Eurographics-IEEE TVCG symposium on visualization. 00. Razdan A, Bae MS. Curvature estimation scheme for triangle meshes using biquadratic Bézier patches. Computer-Aided Design 00;:. Hildebrande K, Polthier K, Wardetzky M. On the convergence of metric 0 and geometric properties of polyhedral surfaces. Geometria Dedicata 00 (in press). Dai J, Luo W, Yau S-T, Gu X. Geometric accuracy analysis for discrete surface approximation. In: Proceedings of geometric modeling and processing Rössl C, Kobbelt L, Seidel HP. Extraction of feature lines on triangulated surfaces using morphological operators. In: Proceedings of smart graphics 000, AAAI spring symposium p. Kohout J, Hlavaty T, Kolingerova I, Skala V. Feature Extraction of - manifold using Delaunay triangulation. In: Proceedings of algoritmy p. 0. Yarger RWI, Quek KH. Surface parameterization in volumetric images for feature classification. In: IEEE inter. symposium of bio-ehg., BIBE p. 0. Quek KH, Yarger RWI, Kirbas C. Surface parameterization in volumetric images for curvature-based feature classification. IEEE Transaction on Systems, Man and Cybernetics 00. Maltret JL, Daniel M. Discrete curvatures and applications : A survey. Rapport de recherche. Laboratoire des Sciences de l Information et des Systèmes no LSIS.RR Borrelli V, Cazals F, Morvan J-M. On the angular defect of triangulations and the point-wise approximation of curvatures. Computer Aided Geometric Design 00;0():. Chen X, Schmitt F. Intrinsic surfaces properties from surface triangulation. In: Sandini G, editor. Proc nd European conf. on computer vision.. p.. Dyn N, Hormann K, Kim SJ, Levin D. Optimizing d triangulation using discrete curvature analysis. In: Lyche T, Schmaker LL, editors. Mathematical methods in CAGD. Vanderbilt University Press; 00. p.. 0 Hamann B. Curvature approximation for triangulated surfaces. Comput- ing Supplement ;():. Kim SJ, Kim CH, Levin D. Surface simplification using a discrete curvature norm. Computers and Graphics 00;():. Design (00), doi:0.0/j.cad

12 JCAD: D. Liu, G. Xu / Computer-Aided Design xx (xxxx) xxx xxx 0 0 Krsek P, Pajdla T, Hlavac V. Estimation of differential parameters on triangulated surfaces. In: st workshop of the Austrian association for pattern recognition.. Martin R. Estimation of principal curvatures from range data. International Journal of Shape Modeling ;:. Meek D, Walton D. On surface normal and Gaussian curvature approximations given data sampled from a smooth surface. Computer Aided Geometric Design 000;:. Peng J, Li Q, Kuo CJ, Zhou M. Estimating Gaussian curvature from D meshes. Proceedings of SPIE 00;00:0 0. Stokely E, Wu SY. Surface parameterization and curvature measurement of arbitrary d-objects: Five practical methods. IEEE Transactions on Pattern Analysis and Machine Intelligence ;(): 0. Surazhsky T, Magid E, Soldea O, Elber G, Rivlin E. A comparison of Gaussian and mean curvatures estimation methods on triangular meshes. In: 00 IEEE international conference on robotics & automation. 00 p. 0. Taubin G. Estimating the tensor of curvatures of a surface from a polyhedral approximation. In: Proceedings th intl. conf. on computer vision.. p Watanabe K, Belyaev AG. Meshes: Detection of salient curvature features on polygonal surfaces. Computer Graphics Forum 00;0(). Xu G. Discrete Laplace Beltrami operators and their convergence. Computer Aided Geometric Design 00;:. Xu G. Convergence analysis of a discretization scheme for Gaussian curveture over triangular surfaces. Computer Aided Geometric Design 00;(): 0. 0 Xu G, Bajaj C. Curvature computations of -manifold in R k. Journal of Computational Mathematics 00;():. Design (00), doi:0.0/j.cad