Ridges and Umbilics of Polynomial Parametric Surfaces

Frédéric Cazals (), Jean-Charles Faugère (), Marc Pouget () and Fabrice Rouillier ()
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Frédéric Cazals: INRIA Sophia-Antipolis, Geometrica project
Jean-Charles Faugère: INRIA Rocquencourt and Universit Pierre et Marie Curie-Paris6, UMR 7606, LIP6, Salsa project
Marc Pouget: LORIA, INRIA Nancy — Grand Est
Fabrice Rouillier: INRIA Rocquencourt and Universit Pierre et Marie Curie-Paris6, UMR 7606, LIP6, Salsa project

Chapter 8 in Geometric Modeling and Algebraic Geometry, 2008, pp 141-159 from Springer

Abstract: 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.

Keywords: Principal Curvature; Principal Direction; Study Point; Umbilical Point; Blue Ridge (search for similar items in EconPapers)
Date: 2008
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DOI: 10.1007/978-3-540-72185-7_8

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