Subdivision Methods for the Topology of 2d and 3d Implicit Curves

Chen Liang 1 Bernard Mourrain 1 Jean-Pascal Pavone 1
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : In this paper, we describe a subdivision method for handling algebraic implicit curves in 2d and 3d. We use the representation of polynomials in the Bernstein basis associated with a given box, to check if the topology of the curve is determined inside this box, from its points on the border of the box. Subdivision solvers are used for computing these points on the faces of the box, and segments joining these points are deduced to get a graph isotopic to the curve. Using envelop of polynomials, we show how this method allow to handle efficiently and accurately implicit curves with large coefficients. We report on implementation aspects and experimentations on 2d curves such as ridge curves or self intersection curves of parameterized surfaces, and on silhouette curves of implicit surfaces, showing the interesting practical behavior of this approach.
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Chapitre d'ouvrage
Bert Juetller and Ragni Piene. Geometric Modeling and Algebraic Geometry, Springer, pp.199-214, 2007, 978-3-540-72184-0
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Chen Liang, Bernard Mourrain, Jean-Pascal Pavone. Subdivision Methods for the Topology of 2d and 3d Implicit Curves. Bert Juetller and Ragni Piene. Geometric Modeling and Algebraic Geometry, Springer, pp.199-214, 2007, 978-3-540-72184-0. 〈inria-00130216〉

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