Page 140 - Geometric Modeling and Algebraic Geometry

8

Ridges and Umbilics of Polynomial Parametric
Surfaces

Fr´ ed´ eric Cazals , Jean-Charles Faug` ere , Marc Pouget , and Fabrice Rouillier 2
2
1
3
INRIA Sophia-Antipolis, Geometrica project,
1
2004 route des Lucioles, BP 93,
F-06902 Sophia-Antipolis, FRANCE.
Frederic.Cazals@sophia.inria.fr
INRIA Rocquencourt and
2
Universit Pierre et Marie Curie-Paris6, UMR 7606, LIP6, Salsa project,
Domaine de Voluceau, BP 105,
F-78153 Le Chesnay Cedex, FRANCE.
Jean-Charles.Faugere@inria.fr
Fabrice.Rouillier@inria.fr
LORIA, INRIA Nancy - Grand Est,
3
Villers-l` es-Nancy, F-54602 FRANCE.
Marc.Pouget@loria.fr
Summary. 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 sur-
face, 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 B´ ezier patches.

8.1 Introduction

8.1.1 Ridges
Originating with the parabolic lines drawn by Felix Klein on the Apollo of Belvedere
[10], curves on surfaces have been a natural way to apprehend the aesthetics of
shapes [12]. Aside these artistic concerns, applications such as surface segmenta-
tion, analysis, registration or matching [11, 16] are concerned with the curves of
extremal curvature of a surface, which are its so-called ridges. (We note in passing