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Topologically certified approximation of umbilics and ridges on polynomial parametric surface

Abstract : 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 encode important informations used in surface analysis or segmentation. But reporting the ridges of a surface requires manipulating third and fourth order derivatives whence numerical difficulties. Additionally, ridges have self-intersections and complex interactions with the umbilics of the surface whence topological difficulties. In this context, we make two contributions for the computation of ridges of polynomial parametric surfaces. First, by instantiating to the polynomial setting a global structure theorem of ridge curves proved in a companion paper, we develop the first certified algorithm to produce a topological approximation of the curve P encoding all the ridges of the surface. The algorithm exploits the singular structure of P umbilics and purple points, and reduces the problem to solving zero dimensional systems using Gröbner basis. Second, for cases where the zero-dimensional systems cannot be practically solved, we develop a certified plot algorithm at any fixed resolution. These contributions are respectively illustrated for Bezier surfaces of degree four and five.
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https://hal.inria.fr/inria-00071225
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Submitted on : Tuesday, May 23, 2006 - 2:43:10 PM
Last modification on : Friday, July 10, 2020 - 4:18:00 PM
Long-term archiving on: : Sunday, April 4, 2010 - 10:01:58 PM

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  • HAL Id : inria-00071225, version 1

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Frédéric Cazals, Jean-Charles Faugère, Marc Pouget, Fabrice Rouillier. Topologically certified approximation of umbilics and ridges on polynomial parametric surface. [Research Report] RR-5674, INRIA. 2005, pp.36. ⟨inria-00071225⟩

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