An efficient algorithm for the stratification and triangulation of an algebraic surface

Authors:
Eric Berberich;Michael Kerber;Michael Sagraloff
Affiliations:
Max-Planck-Institut für Informatik, 66123 Saarbrücken, Germany and School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel;Max-Planck-Institut für Informatik, 66123 Saarbrücken, Germany;Max-Planck-Institut für Informatik, 66123 Saarbrücken, Germany
Venue:
Computational Geometry: Theory and Applications
Year:
2010

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Abstract

We present a method to compute the exact topology of a real algebraic surface S, implicitly given by a polynomial f@?Q[x,y,z] of arbitrary total degree N. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of S including critical points. We compute a stratification @W"S of S into O(N^5) non-singular cells, including the complete adjacency information between these cells. This is done by a projection approach. We construct a special planar arrangement A"S with fewer cells than a cad in the projection plane. Furthermore, our approach applies numerical and combinatorial methods to minimize costly symbolic computations. The algorithm handles all sorts of degeneracies without transforming the surface into a generic position. Based on @W"S we also compute a simplicial complex which is isotopic to S. A complete C++-implementation of the stratification algorithm is presented. It shows good performance for many well-known examples from algebraic geometry.