Isotopic meshing of a real algebraic surface

Bernard Mourrain (1) , Jean-Pierre Técourt (1)

1 GALAAD - Geometry, algebra, algorithms

We present a new algorithm for computing the topology of a real algebraic surface $S$, even in singular cases. We use previous algorithms for 2D and 3D algebraic curves and show how properties of the polar variety of $S$ yield a topological complex equivalent to $S$, or even a simplicial complex isotopic to $S$. The proof of correctness of the algorithm is detailed. It is based on tools 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 effectivity issues conclude the paper.

Mots clés: MESHING; IMPLICIT ALGEBRAIC SURFACE; ISOTOPY; THOM'S LEMMA; WHITNEY STRATIFICATION; SINGULARITY

Domaines: Autre [cs.OH]

Bernard Mourrain, Jean-Pierre Técourt. Isotopic meshing of a real algebraic surface. [Research Report] RR-5508, INRIA. 2006, pp.21. ⟨inria-00070499⟩