B. Mourrain, Jean-Pierre Técourt: Isotopic meshing of real algebraic surfaces

Abstract: We present a new algorithm for computing the topology of compact real algebraic surfaces even singular surfaces i.e we provide an isotopic meshing of the surface. Assume S is an algebraic surface defined by the equation f (x, y, z) = 0 where f belomgs to R[x, y, z].

First, we detail algorithms for computing the topology of 2d and 3d algebraic curves.

After, we detail our algorithm : from the study of the polar variety for the projection into a generic direction, we compute a whitney stratification of S and we compute a list of x-values C for which the projection onto the x-axis is not a submersion (stratum by stratum). From Thom's isotopy lemma we deduce that the topology of the sections can change only for the sections corresponding to the x-values of C. For each section corresponding to a value of C, we compute a graph of points isotopic to the section of S. We detail an algorithm of connection of the graphs of the different sections according to the polar variety. At last, we explain the isotopic transformation between S and the triangulation.