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Delaunay Triangulations of Closed Euclidean d-Orbifolds

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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.

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Notes

  1. http://www.cgal.org/Events/PeriodicSpacesWorkshop/.

  2. http://www.lorentzcenter.nl/lc/web/2009/357/info.php3?wsid=357.

  3. The number of Bieberbach groups by dimension is assigned the id A006227 in the On-Line Encyclopedia of Integer Sequences [44]. The number of torsion-free Bieberbach groups is assigned the id A059104.

  4. We skip the technicalities of the initialization of the triangulation with the first point of \({\mathcal {P}}\).

  5. The incremental algorithm actually requires a slightly stronger result than Proposition 6.2. All details can be found in [12, Sect. 2.3.1], in particular Lemma 2.3.2. We omit them in this paper, since the precise setting and the proof are long and technical.

  6. Using the random point generator from Cgal to generate uniformly distributed points in a cube.

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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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Authors and Affiliations

  1. INRIA Sophia Antipolis – Méditerranée, 2004 route des Lucioles - BP 93, 06902, Sophia Antipolis, France

    Manuel Caroli

  2. INRIA Nancy – Grand Est, LORIA, 615 Rue du Jardin Botanique, 54600, Villers-lès-Nancy, France

    Monique Teillaud

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  1. Manuel Caroli
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Correspondence to Monique Teillaud.

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Editor in Charge: Günter M. Ziegler

Preliminary versions of this paper were presented in conferences: the first one gave some results in the case of the 3D flat torus [15], and the second generalized the results to closed Euclidean d-manifolds [16]. Most of the work was done while the authors were working at INRIA Sophia Antipolis – Méditerranée.

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Caroli, M., Teillaud, M. Delaunay Triangulations of Closed Euclidean d-Orbifolds. Discrete Comput Geom 55, 827–853 (2016). https://doi.org/10.1007/s00454-016-9782-6

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