Delaunay Hodge Star

TL;DR

This paper introduces signed dual volumes for circumcentric dual meshes in Delaunay triangulations, enabling the correct definition of the discrete Hodge star operator in Discrete Exterior Calculus, thus broadening its applicability.

Contribution

It defines signed dual volumes for all dimensions in circumcentric dual meshes and proves their positivity in pairwise Delaunay triangulations, facilitating DEC on more general meshes.

Findings

Signed dual volumes are positive in pairwise Delaunay triangulations.

The discrete Hodge star operator can be correctly defined using these volumes.

DEC can now be applied to a wider class of meshes.

Abstract

We define signed dual volumes at all dimensions for circumcentric dual meshes. We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive. This allows the use of such Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes. This operator is crucial for DEC and is a diagonal matrix with the ratio of primal and dual volumes along the diagonal. A correct definition requires that all entries be positive. DEC is a framework for numerically solving differential equations on meshes and for geometry processing tasks and has had considerable impact in computer graphics and scientific computing. Our result allows the use of DEC with a much larger class of meshes than was previously considered possible.