Ridges and umbilics of polynomial parametric surfaces

Given a smooth surface, a blue (red) ridge is a curve along which the maximum (minimum) principal curvature has an extremum along its curvature line. Ridges are curves of extremal curvature and therefore encode important informations used in segmentation, registration, matching and surface analysis. State of the art methods for ridge extraction either report red and blue ridges simultaneously or separately —in which case a local orientation procedure of principal directions is needed, but no method developed so far topologically certifies the curves reported. In this context, we make two contributions. First, for any smooth parametric surface, we exhibit the implicit equation P = 0 of the singular curve P encoding all ridges of the surface (blue and red), we analyze its singularities and we explain how colors can be recovered. Second, we instantiate to the algebraic setting the implicit equation P = 0. For a polynomial surface, this equation defines an algebraic curve, and we develop the first certified algorithm to produce a topologically certified approximation of it. The algorithm exploits the singular structure of P —umbilics and purple points, and reduces the problem to solving zero dimensional systems using Rational Univariate Representations and isolate roots of univariate rational polynomials. An experimental section illustrates the efficiency of the algorithm on a Bezier patch.