Minimal Delaunay Triangulations of Hyperbolic Surfaces

Authors Matthijs Ebbens, Hugo Parlier, Gert Vegter

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Matthijs Ebbens
  • Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, The Netherlands
Hugo Parlier
  • Mathematics Research Unit, University of Luxembourg, Luxembourg
Gert Vegter
  • Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, The Netherlands


The authors warmly thank Monique Teillaud and Vincent Despré for fruitful discussions. We would also like to thank the referees for their helpful comments.

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Matthijs Ebbens, Hugo Parlier, and Gert Vegter. Minimal Delaunay Triangulations of Hyperbolic Surfaces. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Motivated by recent work on Delaunay triangulations of hyperbolic surfaces, we consider the minimal number of vertices of such triangulations. First, we show that every hyperbolic surface of genus g has a simplicial Delaunay triangulation with O(g) vertices, where edges are given by distance paths. Then, we construct a class of hyperbolic surfaces for which the order of this bound is optimal. Finally, to give a general lower bound, we show that the Ω(√g) lower bound for the number of vertices of a simplicial triangulation of a topological surface of genus g is tight for hyperbolic surfaces as well.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Geometric topology
  • Mathematics of computing → Graphs and surfaces
  • Delaunay triangulations
  • hyperbolic surfaces
  • metric graph embeddings
  • moduli spaces


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