Classifying Voronoi graphs of hex spheres

TL;DR

This paper classifies the Voronoi graphs of hex spheres, which are singular Euclidean spheres with four cone points, using elementary Euclidean geometry to understand their Voronoi decompositions.

Contribution

It provides a classification of Voronoi graphs of hex spheres, a new geometric object, up to graph isomorphism, based on their Euclidean properties.

Findings

Voronoi regions of hex spheres are characterized using elementary Euclidean geometry.

The paper classifies Voronoi graphs of hex spheres up to isomorphism.

A detailed description of Voronoi decompositions for hex spheres is provided.

Abstract

A hex sphere is a singular Euclidean sphere with four cones points whose cone angles are (integer) multiples of 2*pi/3 but less than 2*pi. Given a hex sphere M, we consider its Voronoi decomposition centered at the two cone points with greatest cone angles. In this paper we use elementary Euclidean geometry to describe the Voronoi regions of hex spheres and classify the Voronoi graphs of hex spheres (up to graph isomorphism).