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Computing a Dirichlet Domain for a Hyperbolic Surface

Authors: Vincent Despré, Benedikt Kolbe, Hugo Parlier, and Monique Teillaud

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)


Abstract
This paper exhibits and analyzes an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in topological considerations, the algorithm makes key use of the geometry of the surface. We introduce data structures that reflect this interplay between geometry and topology and show that the algorithm runs in polynomial time, in terms of the initial perimeter and the genus of the surface.

Cite as

Vincent Despré, Benedikt Kolbe, Hugo Parlier, and Monique Teillaud. Computing a Dirichlet Domain for a Hyperbolic Surface. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 27:1-27:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{despre_et_al:LIPIcs.SoCG.2023.27,
  author =	{Despr\'{e}, Vincent and Kolbe, Benedikt and Parlier, Hugo and Teillaud, Monique},
  title =	{{Computing a Dirichlet Domain for a Hyperbolic Surface}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{27:1--27:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.27},
  URN =		{urn:nbn:de:0030-drops-178771},
  doi =		{10.4230/LIPIcs.SoCG.2023.27},
  annote =	{Keywords: Hyperbolic geometry, Topology, Voronoi diagram, Algorithm}
}
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