An Obstruction to Delaunay Triangulations in Riemannian Manifolds

Article Type

Research Article

Publication Title

Discrete and Computational Geometry

Abstract

Delaunay has shown that the Delaunay complex of a finite set of points P of Euclidean space Rm triangulates the convex hull of P, provided that P satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds. However, Delaunay’s genericity assumption no longer guarantees that the Delaunay complex will yield a triangulation; stronger assumptions on P are required. A natural one is to assume that P is sufficiently dense. Although results in this direction have been claimed, we show that sample density alone is insufficient to ensure that the Delaunay complex triangulates a manifold of dimension greater than 2.

First Page

226

Last Page

237

DOI

10.1007/s00454-017-9908-5

Publication Date

1-1-2018

Comments

All Open Access, Green

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