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A Geometric Approach to Curvature Estimation on Triangulated 3D Shapes

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A Geometric Approach to Curvature Estimation on Triangulated 3D Shapes A GEOMETRIC APPROACH TO CURVATURE ESTIMATION ON TRIANGULATED 3D SHAPES Mohammed Mostefa Mesmoudi, Leila De Floriani and Paola Magillo Department of Computer Science and Information Science (DISI), University of Genova Via Dodecaneso 35, 16146 Genova, Italy Keywords: Gaussian and mean curvatures, Surface meshes. Abstract: We present a geometric approach to define discrete normal, principal, Gaussian and mean curvatures, that we call Ccurvature. Our approach is based on the notion of concentrated curvature of a polygonal line and a simulation of rotation of the normal plane of the surface at a point. The advantages of our approach is its simplicity and its natural meaning. A comparison with widely-used discrete methods is presented. 1 INTRODUCTION define a discrete normal curvature of a polygonal sur- face at a vertex. We simulate the rotation of normal Curvature is one of the main important notions used planes to define principal curvatures and, thus, obtain to study the geometry and the topology of a surface. new discrete estimators for Gaussian and mean curva- In combinatorial geometry, many attempts to define a tures. We call all such curvatures Ccurvatures, since discrete equivalent of Gaussian and mean curvatures they are obtained as generalization to surfaces of the have been developed for polyhedral surfaces (Gatzke concept of concentrated curvature for polygonal lines and Grimm, 2006; Surazhsky et al., 2003). Dis- just introduced. The major advantage of this method crete approaches include smooth approximations of is the use of intrinsic properties of a discrete mesh to the surface using interpolation techniques (Hahmann define geometric features that have the same proper- et al., 2007), and approaches that deal directly with ties as the analytic methods. In this work, we also the mesh (Meyer et al., 2003; Taubin, 1995; Watanabe compare Gaussian and mean Ccurvatures with widely and Belyaev, 2001). All the methods are not satisfac- used discrete curvature estimators for analytic Gaus- tory in what concerns approximation errors, control, sian and mean curvatures and with concentrated cur- and convergence when refining a mesh (Borrelli et al., vature (according to Aleksandrov’s definition). 2003; Surazhsky et al., 2003; Xu, 2006). In the fifties, Aleksandrov introduced concen- trated curvature as an intrinsic curvature measure for polygonal surfaces (Aleksandrov, 1957). This tech- 2 BACKGROUND NOTIONS nique has been used in the geometric modeling com- munity under the name of angle defect method (Al- In this Section, we briefly review some fundamental boul et al., 2005; Akleman and Chen, 2006). Con- notions on curvature in the analytic case and on con- centrated curvature does not suffer from the problems centrated curvature. linked to errors and and their control, and satisfies a Let C be a smooth curve having parametric repre- discrete analogous version of the well known Gauss- sentation (c(t))t2R. The curvature k(p) of C at a point Bonnet theorem. However, Concentrated curvature p = c(t0) is given by depends weakly on the local geometric shape of the surface. 1 jc0(t) ^ c”(t)j Here, we use Aleksandrov’s idea to define con- k(p) = = ; r jc0(t)j3 centrated curvature for polygonal lines. We prove in such a case that concentrated curvature is an intrin- where r is called the curvature radius. Number r cor- sic measure that is expressed using only the fracture responds to the radius of the osculatory circle tangent angle of the polygonal line at its vertices. We then to C at p. 90 Mesmoudi M., De Floriani L. and Magillo P. (2010). A GEOMETRIC APPROACH TO CURVATURE ESTIMATION ON TRIANGULATED 3D SHAPES. In Proceedings of the International Conference on Computer Graphics Theory and Applications, pages 90-95 DOI: 10.5220/0002825900900095 Copyright c SciTePress A GEOMETRIC APPROACH TO CURVATURE ESTIMATION ON TRIANGULATED 3D SHAPES Let S be a smooth surface and P be a plane which The above discrete definition of curvature can be −! contains the unit normal vector np at a point p 2 S. justified as follows. The surface is assumed to have Plane P intersects S through a smooth curve C con- a conical shape at each of its vertices. Each cone is taining p with curvature k (p) at the point p called then approximated from its interior with a sphere S2 C −! r normal curvature. When P turns around np, curves of radius r, as shown in Figure 1(a). Following C vary. There are two extremal curvature values k1(p) ≤ k2(p) which bound the curvature values of all curves C. The corresponding curves C1 and C2 are orthogonal at point p (Do Carno, 1976). These extremal curvatures are called principal normal cur- −! vatures. Note that, if the normal vector np is on the same side as the osculatory circle, then the curvature value ki of curve Ci has a negative sign. Definition 1. The Gaussian curvature Kp and the (a) (b) mean curvature Hp at point p = f(x;y) are defined, respectively, as Figure 1: In (a), spheres tangent to a cone from it interior; in (b), parameters for computation. k (p) + k (p) K = k (p) ∗ k (p); H = 1 2 (1) p 1 2 p 2 the Gauss-Bonnet theorem, the spherical cap approx- imating the cone has a total curvature which is equal The formula defining H turns out to be the mean p to 2p−Q, where Q is the angle of the cone at its apex. of all values of normal curvatures at point p. Gaus- This quantity does not depend on the radius of sphere sian and mean curvatures depend strongly on the (lo- S2 by which we approach the cone and hence 2p−Q cal) geometrical shape of the surface. We will see that r p is an intrinsic value for the surface at vertex p. This this property is relaxed for concentrated curvature. A remarkable property fully justifies the name concen- remarkable property of Gaussian curvature is given trated curvature. by the Gauss-Bonnet Theorem, which relates the ge- The local shape of the surface does not play a ometry of a surface, given by the Gaussian curvature, role here, unlike in the analytic case, where Gaus- to its topology, given by its Euler characteristic (see sian curvature is strongly dependent of the local sur- (Do Carno, 1976)). face shape. Concentrated curvature satisfies a discrete A singular flat surface is a surface endowed with a equivalent of the Gauss-Bonnet theorem (Akleman metric such that each point of the surface has a neigh- and Chen, 2006). borhood which is either isometric to a Euclidean disc or a cone of angle Q 6= 2p at its apex. Points satis- fying this latter property are called singular conical points. Any piecewise linear triangulated surface has 3 CCURVATURE FOR a structure of a singular flat surface. All vertices with POLYGONAL CURVES a total angle different from 2p (or p for boundary ver- tices) are singular conical points. As shown below, In this Section, we use the concentrated curvature the Gaussian curvature is accumulated at these points principle to define a concentrated curvature for polyg- so that the Gauss-Bonnet formula holds. onal curves. In this way, we can define principal con- Let S be a (piecewise linear) triangulated sur- centrated curvatures for a triangulated surface and fol- face and let p be a vertex of the triangle mesh. Let low the same construction, used in Section 2 for an- D1;··· ;Dn be the triangles incident at p such that Di alytic mean and Gaussian curvatures, to define simi- and Di+1 are edge-adjacent. If ai, bi are the vertices larly new discrete curvature estimators. We will call of triangle Di different from p, then the total angle Qp them Ccurvatures. The initial C is a shortcut for “con- n at p is given by Qp = ∑i=1 a[iPbi centrated”. We will show also that Ccurvature does Definition 2. (Aleksandrov, 1957; Troyanov, 1986) not suffer from convergence problems. Let C be a simple polygonal curve in the three- The concentrated Gaussian curvature KC2(p), at a vertex p of the triangulated surface, is the value dimensional Euclidean space. Let p be a vertex on C and a, b its two neighbors on C. Points a, b and p define a plane P. If the angle g = apbd is equal to p, 2p − Qp if p is an interior vertex, and KC2(p) = then the curvature value k(p) of C is 0. Otherwise, let p − Qp if p is a boundary vertex. Sr ⊂ P be a circle of any radius r > 0 tangent C at two 91 GRAPP 2010 - International Conference on Computer Graphics Theory and Applications points u 2 [p;a] and v 2 [p;b] (see Figure 2(a)). defined by the average of the normal vectors of the triangles incident in p. Let P be a plane passing by p and containing the normal vector −!n . This plane cuts surface S along a polygonal curve C = S \ P con- taining point p. We compute the Ccurvature kC(p) at point p of curve C as described in Section 3. Note that the position of the normal vector −!n with respect to the polygonal curve C should be taken into account. If the normal vector −!n and the polygonal curve C lie in two different half planes (or equivalently they are separated by the “tangent” plane Tp whose normal vector is −!n , see Figure 3), then the angle g of C at (a) (b) p is smaller than p and the Ccurvature value p − g is Figure 2: In (a), circles tangent to the sector from it interior.
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