Recognition of the Spherical Laguerre Voronoi Diagram

TL;DR

This paper presents an algorithm to identify spherical Laguerre Voronoi diagrams by linking tessellations to polyhedra, addressing the challenge of non-uniqueness of generators and their weights.

Contribution

It introduces a novel recognition algorithm based on polyhedral central projection, enabling the determination of spherical Laguerre Voronoi diagrams.

Findings

Algorithm successfully recognizes spherical Laguerre Voronoi diagrams.

Provides a method to recover generator locations and weights.

Addresses non-uniqueness issue in spherical Laguerre tessellations.

Abstract

In this paper, we construct an algorithm for determining whether a given tessellation on a sphere is a spherical Laguerre Voronoi diagram or not. For spherical Laguerre tessellations, not only the locations of the Voronoi generators, but also their weights are required to recover. However, unlike the ordinary spherical Voronoi diagram, the generator set is not unique, which makes the problem difficult. To solve the problem, we use the property that a tessellation is a spherical Laguerre Voronoi diagram if and only if there is a polyhedron whose central projection coincides with the tessellation. We determine the degrees of freedom for the polyhedron, and then construct an algorithm for recognizing Laguerre tessellations.