A Variational Principle for Improving 2D Triangle Meshes based on Hyperbolic Volume

摘要

In this paper, we consider the problem of improving 2D triangle meshestessellating planar regions. We propose a new variational principle forimproving 2D triangle meshes where the energy functional is a convex functionover the angle structures whose maximizer is unique and consists only ofequilateral triangles. This energy functional is related to hyperbolic volumeof ideal 3-simplex. Even with extra constraints on the angles for embedding themesh into the plane and preserving the boundary, the energy functional remainswell-behaved. We devise an efficient algorithm for maximizing the energyfunctional over these extra constraints. We apply our algorithm to variousdatasets and compare its performance with that of CVT. The experimental resultsshow that our algorithm produces the meshes with both the angles and the aspectratios of triangles lying in tighter intervals.