Tessellations of hyperbolic surfaces

TL;DR

This paper introduces a new centered dual decomposition method for hyperbolic surfaces, enabling detailed analysis of Delaunay tessellations and geometric properties like injectivity and covering radii.

Contribution

It develops the centered dual decomposition framework, linking combinatorics and geometry of Delaunay cells on hyperbolic surfaces, and applies it to genus-2 surfaces.

Findings

Extracted combinatorial information about Delaunay tessellations

Related injectivity radius to covering radius on genus-2 surfaces

Provided a parametrization of Delaunay cell possibilities

Abstract

A finite subset S of a closed hyperbolic surface F canonically determines a "centered dual decomposition" of F: a cell structure with vertex set S, geodesic edges, and 2-cells that are unions of the corresponding Delaunay polygons. Unlike a Delaunay polygon, a centered dual 2-cell Q is not determined by its collection of edge lengths; but together with its combinatorics, these determine an "admissible space" parametrizing geometric possibilities for the Delaunay cells comprising Q. We illustrate its application by using the centered dual decomposition to extract combinatorial information about the Delaunay tessellation among certain genus-2 surfaces, and with this relate injectivity radius to covering radius here.