On the Topology of the Restricted Delaunay Triangulation and Witness Complex in Higher Dimensions

Details On the Topology of the Restricted Delaunay Triangulation and Witness Complex in Higher Dimensions On the Topology of the Restricted Delaunay Triangulation and Witness Complex in Higher Dimensions

2008-03-09

arxiv.org/abs/0803.1296v1

Computer Science

Computational Geometry

Scientific paper

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 R^n, 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.

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