An Obstruction to Delaunay Triangulations in Riemannian Manifolds

Authors

Jean Daniel Boissonnat, Université Côte d'Azur
Ramsay Dyer, Université Côte d'Azur
Arijit Ghosh, Indian Statistical Institute, Kolkata
Nikolay Martynchuk, Rijksuniversiteit Groningen

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

Recommended Citation

Boissonnat, Jean Daniel; Dyer, Ramsay; Ghosh, Arijit; and Martynchuk, Nikolay, "An Obstruction to Delaunay Triangulations in Riemannian Manifolds" (2018). Journal Articles. 1626.
https://digitalcommons.isical.ac.in/journal-articles/1626