The Duality of Geodesic Voronoi/Delaunay Diagrams For An Intrinsic Discrete Laplace-Beltrami Operator on Simplicial Surfaces

An intrinsic discrete Laplace-Beltrami operator on simplicial surfaces S proposed in [2] was established via an intrinsic Delaunay tessellation on S. Up to now, this intrinsic Delaunay tessellations can only be computed by an edge flipping algorithm without any provable complexity analysis. In the paper, we show that the intrinsic Delaunay triangulation can be obtained from a duality of geodesic Voronoi diagram on S with a proof that this duality exists under two practical assumptions. Then the fast and stable computation of geodesic Voronoi diagrams provides a new way to compute the intrinsic discrete Laplace-Beltrami operator on S. Given the duality, the time and space complexities of the intrinsic Delaunay triangulation are the same as that of geodesic Voronoi diagram, which are O(m logm) and O(m), respectively, where m is the number of vertices in the intrinsic Delaunay triangulation.