Isotopic triangulation of a real algebraic surface

Lionel Alberti 1 Bernard Mourrain 1 Jean-Pierre Técourt 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 : 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.
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Lionel Alberti, Bernard Mourrain, Jean-Pierre Técourt. Isotopic triangulation of a real algebraic surface. Journal of Symbolic Computation, Elsevier, 2009, 44 (9), pp.1291-1310. ⟨10.1016/j.jsc.2008.02.007⟩. ⟨inria-00433141⟩

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