Weighted triangulations for geometry processing
Résumé
In this paper, we investigate weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. We show that the addition of weights to an arbitrary triangle mesh provides a geometric construction of an orthogonal dual structure obtained by displacing the primal mesh's circumcenters along a gradient vector field. From this flexible, yet principled definition of a dual we derive a discrete Laplace-Beltrami operator that preserves core properties of its continuous counterpart. We also propose several metric representations of these primal-dual structures for numerical convenience. In the process, we relate our work to the circle and sphere packing literature, and uncover closed-form expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate that weighted triangulations offer a valuable extension to pairwise, intrinsic, and weighted Delaunay triangulations for the design of efficient and accurate computational tools useful in a variety of geometry processing tasks.