Numerical computation of the cut locus via a variational approximation of the distance function

TL;DR

This paper introduces a novel numerical method for computing the cut locus of a compact submanifold in 3D space using a convex variational approach, with applications to Voronoi cell approximation on surfaces.

Contribution

It presents a new variational technique with proven convergence for the numerical computation of the cut locus, extending its application to Voronoi diagram approximation on embedded surfaces.

Findings

Method successfully computes cut loci with convergence guarantees.

Approach effectively approximates Voronoi cells on surfaces.

Demonstrates versatility on various embedded surfaces.

Abstract

We propose a new method for the numerical computation of the cut locus of a compact submanifold of $\mathbb{R}^3$ without boundary. This method is based on a convex variational problem with conic constraints, with proven convergence. We illustrate the versatility of our approach by the approximation of Voronoi cells on embedded surfaces of $\mathbb{R}^3$.