Approximation of Ridges and Umbilics on Triangulated Surface Meshes

Abstract This paper describes the CGAL package for the approximating the ridges and umbilics of a smooth surface discretized by a triangle mesh. Given a smooth surface, a ridge is a curve along which one of the principal curvatures has an extremum,along its curvature line. An umbilic is a point at which both principal curvatures are equal. Ridges define a singular curve —i.e. a self-intersecting curve, and umbilics are are special points on this curve. Ridges are curves of extremal curvature and therefore encode important informations used in segmentation, registration, matching and surface analysis. Based on the results of the article [CP05b], we propose algorithms to identify and extract dierent parts of this singular ridge curve as well as umbilics on a surface given as a triangulated surface mesh. Dierential quantities associated to the mesh vertices are assumed to be given for these algorithms; such quantities may be computed by the package Estimation of Local Dierential,Properties of Sampled Surfaces via Polynomial Fitting. Notice this package needs the third party libraries LAPACK and BLAS for linear algebra operations. 1 Overview Section 2 presents the basics of the theory of ridges and umbilics on smooth surfaces. Sections 3