2008/03/09 by Steve Y. Oudot, Oudot, Steve Y.
Computer Science · #Computational Geometry (cs.CG) #FOS: Computer and information sciences #cs.CG
paper · pdf · doi:10.48550/arxiv.0803.1296
arxiv created 2008/03/09 · arxiv updated 2009/12/01
It is a well-known fact that, under mild sampling conditions, the restricted Delaunay triangulation provides good topological approximations of 1- and 2-manifolds. We show that this is not the case for higher-dimensional manifolds, even under stronger sampling conditions. Specifically, it is not true that, for any compact closed submanifold M of Rn, and any sufficiently dense uniform sampling L of M, the Delaunay triangulation of L restricted to M is homeomorphic to M, or even homotopy equivalent to it. Besides, it is not true either that, for any sufficiently dense set W of witnesses, the witness complex of L relative to M contains or is contained in the restricted Delaunay triangulation of L.