Tessellations of hyperbolic surfaces

2011/03/23 by Jason DeBlois, DeBlois, Jason
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT

paper · pdf · doi:10.48550/arxiv.1103.4604

56 pages, 8 figures

arxiv created 2011/03/23 · arxiv updated 2011/03/24

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.

Related

• The Delaunay tessellation in hyperbolic space
• Bounding the area of a centered dual two-cell below, given lower bounds on its side lengths
• Canonical tessellations of decorated hyperbolic surfaces
• Combinatorial optimization in geometry
• Minimal Delaunay triangulations of hyperbolic surfaces