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 hyperbolic surface 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 Delaunay triangulations in the plane do not generalize naturally. Assuming that the 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 a combinatorial length (a notion we define in the paper) smaller than $12g-6$ with respect to a Dirichlet domain. On the way, we prove that both the edges of a Delaunay triangulation and of a Dirichlet domain have some kind of distance minimizing properties that are of intrinsic interest. The bounds produced in this paper depend only on the topology of the surface. They provide mathematical foundations for hyperbolic analogs of the algorithms to compute periodic Delaunay triangulations in Euclidean space.

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

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-03045921v2⟩