A system of of Hamilton-Jacobi equations characterizing geodesic centroidal tessellations

TL;DR

This paper introduces Hamilton-Jacobi equations to characterize geodesic centroidal tessellations, enabling efficient computation of such tessellations using a Fast Marching method on unstructured grids.

Contribution

It presents a novel system of Hamilton-Jacobi equations for geodesic centroidal tessellations and adapts the Fast Marching method for their numerical solution.

Findings

Effective computation of geodesic centroidal tessellations demonstrated

The method applies to Voronoi and power diagram tessellations

Numerical examples illustrate the technique's features

Abstract

We introduce a class of systems of Hamilton-Jacobi equations characterizing geodesic centroidal tessellations, i.e. tessellations of domains with respect to geodesic distances where generators and centroids coincide. Typical examples are given by geodesic centroidal Voronoi tessellations and geodesic centroidal power diagrams. An appropriate version of the Fast Marching method on unstructured grids allows computing the solution of the Hamilton-Jacobi system and therefore the associated tessellations. We propose various numerical examples to illustrate the features of the technique.