Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-30T22:02:39.705Z Has data issue: false hasContentIssue false
  • English
  • Français

The Delaunay tessellation in hyperbolic space

Published online by Cambridge University Press:  27 September 2016

JASON DEBLOIS
JASON DEBLOIS*
Affiliation:
Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall Pittsburgh, PA 15260 e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The Delaunay tessellation of a locally finite subset of the hyperbolic space ℍn is constructed via convex hulls in ℝn+1. For finite and lattice-invariant sets it is proven to be a polyhedral decomposition, and versions (necessarily modified from the Euclidean setting) of the empty circumspheres condition and geometric duality with the Voronoi tessellation are proved. Some pathological examples of infinite, non lattice-invariant sets are exhibited.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 1 , January 2018 , pp. 15 - 46
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Akiyoshi, H. and Sakuma, M. Comparing two convex hull constructions for cusped hyperbolic manifolds. In Kleinian Groups and Hyperbolic 3-Manifolds (Warwick, 2001), London Math. Soc. Lecture Note Ser., vol. 299 (Cambridge University Press, Cambridge, 2003), pp. 209–246.Google Scholar
[2] Benedetti, R. and Petronio, C. Lectures on Hyperbolic Geometry. Universitext (Springer-Verlag, Berlin, 1992).Google Scholar
[3] Berger, M. Geometry. II. Universitext (Springer-Verlag, Berlin, 1987). Translated from the French by M. Cole and S. Levy.Google Scholar
[4] Berger, M. Geometry I. Universitext (Springer-Verlag, Berlin, 2009). Translated from the 1977 French original by M. Cole and S. Levy, Fourth printing of the 1987 English translation [MR0882541].Google Scholar
[5] Bogdanov, M., Devillers, O. and Teillaud, M. Hyperbolic Delaunay complexes and Voronoi diagrams made practical. INRIA Research Report n○ 8146 (December 2012).Google Scholar
[6] Bridson, M. R. and Haefliger, A. Metric spaces of non-positive curvature. Grundlehren Math. Wiss. vol. 319 [Fundamental Principles of Mathematical Sciences] (Springer-Verlag, Berlin, 1999).Google Scholar
[7] Charney, R., Davis, M. and Moussong, G. Nonpositively curved, piecewise Euclidean structures on hyperbolic manifolds. Michigan Math. J. 44 (1) (1997), 201–208.CrossRefGoogle Scholar
[8] Cooper, D. and Long, D. A generalization of the Epstein–Penner construction to projective manifolds. Preprint. arXiv:1307.5016v1 (July 2013).Google Scholar
[9] De Loera, J. A., Rambau, J. and Santos, F. Triangulations Algorithms Comput. Math. vol. 25 (Springer-Verlag, Berlin, 2010) Structures for algorithms and applications.Google Scholar
[10] DeBlois, J. The centered dual and the maximal injectivity radius of hyperbolic surfaces. Geom. Topol. 19 (2) (2015), 953–1014.CrossRefGoogle Scholar
[11] DeBlois, J. The local maxima of maximal injectivity radius among hyperbolic surfaces. Preprint. arXiv:1506.08080 (June 2015).CrossRefGoogle Scholar
[12] Devillers, O., Meiser, S. and Teillaud, M. The space of spheres, a geometric tool to unify results on Voronoi diagrams. INRIA Research Report 1620 (1992).Google Scholar
[13] Epstein, D. B. A. and Penner, R. C. Euclidean decompositions of noncompact hyperbolic manifolds. J. Differential Geom. 27 (1) (1988), 67–80.Google Scholar
[14] Graham, R. and Yao, F. A whirlwind tour of computational geometry. Amer. Math. Monthly 97 (8) (1990), 687–701.Google Scholar
[15] Näätänen, M. and Penner, R. C. The convex hull construction for compact surfaces and the Dirichlet polygon. Bull. London Math. Soc. 23 (6) (1991), 568–574.Google Scholar
[16] Ratcliffe, J. G. Foundations of hyperbolic manifolds. Graduate Texts in Math. vol. 149 (Springer, New York, second edition, 2006).Google Scholar