Delaunay Hodge star

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.

► DEC is a framework for differential equations and geometry processing in graphics.
► Discrete Hodge star is a diagonal matrix in DEC and contains all metric information.
► Hodge star is needed to define discrete Laplacians and for mass matrices in FEM.
► Well-centered meshes are sufficient for DEC but hard to generate in 3D.
► This paper shows that Delaunay meshes with one-sided boundary simplices are enough.