Isotopic triangulation of a real algebraic surface

Authors:
Lionel Alberti;Bernard Mourrain;Jean-Pierre Técourt
Affiliations:
GALAAD, INRIA Méditerranée, 2004 route des Lucioles, B.P. 93, 06902 Sophia-Antipolis, Cedex, France;GALAAD, INRIA Méditerranée, 2004 route des Lucioles, B.P. 93, 06902 Sophia-Antipolis, Cedex, France;GALAAD, INRIA Méditerranée, 2004 route des Lucioles, B.P. 93, 06902 Sophia-Antipolis, Cedex, France
Venue:
Journal of Symbolic Computation
Year:
2009

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Abstract

We present a new algorithm for computing the topology of a real algebraic surface S in a ball B, even in singular cases. We use algorithms for 2D and 3D algebraic curves and show how one can compute a topological complex equivalent to S, and even a simplicial complex isotopic to S by exploiting properties of the contour curve of S. The correctness proof of the algorithm is based on results from stratification theory. We construct an explicit Whitney stratification of S, by resultant computation. Using Thom's isotopy lemma, we show how to deduce the topology of S from a finite number of characteristic points on the surface. An analysis of the complexity of the algorithm and effectiveness issues conclude the paper.