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Discrete & Computational Geometry

, Volume 55, Issue 4, pp 827–853 | Cite as

Delaunay Triangulations of Closed Euclidean d-Orbifolds

  • Manuel Caroli
  • Monique Teillaud
Article

Abstract

We give a definition of the Delaunay triangulation of a point set in a closed Euclidean d-manifold, i.e. a compact quotient space of the Euclidean space for a discrete group of isometries (a so-called Bieberbach group or crystallographic group). We describe a geometric criterion to check whether a partition of the manifold actually forms a triangulation (which subsumes that it is a simplicial complex). We provide an incremental algorithm to compute the Delaunay triangulation of the manifold defined by a given set of input points, if it exists. Otherwise, the algorithm returns the Delaunay triangulation of a finite-sheeted covering space of the manifold. The algorithm has optimal randomized worst-case time and space complexity. It extends to closed Euclidean orbifolds. An implementation for the special case of the 3D flat torus has been released in Cgal 3.5. To the best of our knowledge, this is the first general result on this topic.

Keywords

Delaunay triangulation Orbit space Crystallographic groups Covering space Incremental algorithm Implementation 

Notes

Acknowledgments

The authors wish to thank Olivier Devillers for contributions to Sects. 5.3 and 6.3.2, Ramsay Dyer for discussions about the hypothesis in Proposition 2.1, Nico Kruithof for initial work on 3D periodic triangulations, Günter Rote for his comments on a preliminary version of [15], Jean-Marc Schlenker for helpful discussions, and Rien van de Weijgaert for providing us with data sets from cosmology research projects. We also acknowledge reviewers of a first version of this paper for their useful comments. This work was partially supported by the ANR (Agence Nationale de la Recherche) under the “Triangles” Project of the Programme blanc (No BLAN07-2_194137) http://www-sop.inria.fr/geometrica/collaborations/triangles/.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.INRIA Sophia Antipolis – MéditerranéeSophia AntipolisFrance
  2. 2.INRIA Nancy – Grand Est, LORIAVillers-lès-NancyFrance

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