  In this paper, we consider the problem of improving 2D triangle meshes
tessellating planar regions. We propose a new variational principle for
improving 2D triangle meshes where the energy functional is a convex function
over the angle structures whose maximizer is unique and consists only of
equilateral triangles. This energy functional is related to hyperbolic volume
of ideal 3-simplex. Even with extra constraints on the angles for embedding the
mesh into the plane and preserving the boundary, the energy functional remains
well-behaved. We devise an efficient algorithm for maximizing the energy
functional over these extra constraints. We apply our algorithm to various
datasets and compare its performance with that of CVT. The experimental results
show that our algorithm produces the meshes with both the angles and the aspect
ratios of triangles lying in tighter intervals.
