Volume 1 (2005), Number 1 Parabola-Based Discrete Curvature Estimation † Hyoungseok Kim 1 and Jarek Rossignac 2 1 Dept. of Multimedia Engineering, Dongeui University San 21, Gaya-Dong, Pusanjin-Gu, Pusan, 614-714, KOREA 2 College of Computing and GVU Center, Georgia Institute of Technology Atlanta, Georgia, 30332-0280, USA Abstract The local geometric properties such as curvatures and normal vectors play important roles in analyzing the local shape of objects. The result of the geometric operations such as mesh simplification and mesh smoothing is depen- dent on how to compute the curvatures of meshes, because there is no exact definition of the discrete curvature in meshes. In this paper, we indicate the fatal error in computing sectional curvatures of the most previous discrete curvature estimations. Moreover, we present a new discrete sectional-curvature estimation to overcome the error, which is based on the parabola interpolation and the geometric properties of Bezier curve. Keywords: Discrete Curvature and Parabolic Interpolation 1. Introduction The problem of estimating the geometric properties such as normal vectors and curvatures in triangular meshes plays im- portant role in many applications such as surface segmenta- tion, anisotropic remeshing or non-photorealistic rendering. A lot of efforts have been devoted to this problem, but there is no consensus on the most appropriate way[1,3,4,5,8,9]. Popular methods include quadratic fitting, where the esti- mated curvature is the one of the quadratic that best fits a cer- tain neighborhood of a vertex locally. Most recently, Gold- feather propose the use of a cubic approximation scheme which takes into account vertex normals in the 1-ring. The accuracy of these curvature estimations is dependent of that of fitting. If the one-ring neighborhood has many vertices or has a oscillated shape, then the approximated surface does not resemble the local shape and these estimations may yield a high error. Other methods typically consider some defini- tion of curvature that can be extended to the polyhedral set- ting. These methods compute Gaussian curvature and Mean curvature based on the Gauss-Bonnet theory and Euler the- ory. Taubin presented a method to estimate the tensor of cur- † This work was supported by the Post-doctoral Fellowship Pro- gram of Korea Science & Engineering Foundation(KOSEF) vature of a surface at vertices of a mesh[6]. Watanabe pro- posed a simple method of estimating the principal curvatures of a discrete surface[7]. Meyer et. al proposed a discrete ana- log of the Laplace-Beltrami operator to estimate the discrete curvature[2]. Most of these methods compute directly the sectional curvatures for each adjacent edge of a vertex. They assume the normal curve interpolate both the given vertex and an adjacent vertex and the curve is represented by Taylor series. However, they make the same mistake that they adopt the distance between the given vertex and its adjacent neigh- bor vertex as the parameter of the series. Figure 1 shows their drawback. There are several polygons of different in- terior angles, all of which are circumscribed by circles of the same radius. The discrete curvatures estimated by those methods are the same as that of the circle. It is quite alien to universal concepts. Figure 1: Several Polygons with the same discrete curvature