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

TL;DR

This paper introduces a new variational principle based on hyperbolic volume for improving 2D triangle meshes, leading to more regular and well-shaped meshes through an efficient optimization algorithm.

Contribution

It proposes a convex energy functional related to hyperbolic volume for mesh improvement, with a unique maximizer of equilateral triangles, and develops an efficient algorithm for constrained optimization.

Findings

The algorithm produces meshes with more uniform angles.

Meshes have tighter aspect ratio distributions.

Performance surpasses that of CVT in experiments.

Abstract

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.