Ridges and Umbilics of Polynomial Parametric Surfaces

Given a smooth surface, a blue (red) ridge is a curve such that at each of its point, the maximum (minimum) principal curvature has an extremum along its curvature line. As curves of extremal curvature, ridges are relevant in a number of applications including surface segmentation, analysis, registration, matching. In spite of these interests, given a smooth surface, no algorithm reporting a certified approximation of its ridges was known so far, even for restricted classes of generic surfaces. This paper partly fills this gap by developing the first algorithm for polynomial parametric surfaces — a class of surfaces ubiquitous in CAGD. The algorithm consists of two stages. First, a polynomial bivariate implicit characterization of ridges P = 0 is computed using an implicitization theorem for ridges of a parametric surface. Second, the singular structure of P = 0 is exploited, and the approximation problem is reduced to solving zero dimensional systems using Rational Univariate Representations. An experimental section illustrates the efficiency of the algorithm on Bézier patches.