Robust construction of the three-dimensional flow complex

The Delaunay triangulation and its dual the Voronoi diagram are ubiquitous geometric complexes. From a topological standpoint, the connection has recently been made between these cell complexes and the Morse theory of distance functions. In particular, in the generic setting, algorithms have been proposed to compute the flow complex --the stable and unstable manifolds associated to the critical points of the distance function to a point set. As algorithms ignoring degenerate cases and numerical issues are bound to fail on general inputs, this paper develops the first complete and robust algorithm to compute the flow complex. First, we present complete algorithms for the flow operator, unraveling a delicate interplay between the degenerate cases of Delaunay and those which are flow specific. Second, we sketch how the flow operator unifies the construction of stable and unstable manifolds. Third, we discuss numerical issues related to predicates on cascaded constructions. Finally, we re...

• Type: Conference
• Year: 2008
• Where: COMPGEOM
• Authors: Frédéric Cazals, Aditya G. Parameswaran, Sylvain Pion
• Keywords: COMPGEOM 2008, Delaunay Triangulation, Discrete Geometry, Flow Complex, Flow Operator