Bounds on the k-Neighborhood for Locally Uniformly Sampled Surfaces

Given a locally uniform sample set P of a smooth surface S. We derive upper and lower bounds on the number k of nearest neighbors of a sample point p that have to be chosen from P such that this neighborhood contains all restricted Delaunay neighbors of p. In contrast to the trivial lower bound, the upper bound indicates that a sampling condition that is used in many computational geometry proofs is quite reasonable from a practical point of view.