Representing infinite hyperbolic periodic Delaunay triangulations using finitely many Dirichlet domains

Vincent Despré (1) , Benedikt Kolbe (1) , Monique Teillaud (1)

1 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm

The Delaunay triangulation of a set of points $P$ on a hyperbolicsurface is the projection of the Delaunay triangulation of the set$\tilde{P}$ of lifted points in the hyperbolic plane. Since$\tilde{P}$ is infinite, the algorithms to compute Delaunaytriangulations in the plane do not generalize naturally. Assuming thatthe surface comes with a Dirichlet domain, we exhibit a finite set of points that captures the full triangulation.Indeed, we prove that an edge of a Delaunay triangulation has acombinatorial length (a notion we define in the paper) smaller than$12g-6$ with respect to a Dirichlet domain. On the way, we prove thatboth the edges of a Delaunay triangulation and of a Dirichlet domainhave some kind of distance minimizing properties that are of intrinsicinterest.The bounds produced in this paper depend only on the topology of thesurface. They provide mathematical foundations for hyperbolic analogsof the algorithms to compute periodic Delaunay triangulations inEuclidean space.

Mots clés: Delaunay triangulations; Hyperbolic surfaces; Fuchsian group; Topology; Dirichlet domain

Domaines: Géométrie algorithmique [cs.CG]; Topologie géométrique [math.GT]

Vincent Despré, Benedikt Kolbe, Monique Teillaud. Representing infinite hyperbolic periodic Delaunay triangulations using finitely many Dirichlet domains. 2021. ⟨hal-03045921v3⟩