Metric Based Quadrilateral Mesh Generation

TL;DR

This paper introduces a rigorous, automatic metric-based algorithm for quadrilateral mesh generation using discrete Ricci flow and conformal deformation, ensuring specific geometric conditions for high-quality mesh creation.

Contribution

It presents a novel method that guarantees the generation of quad-meshes satisfying complex geometric and topological conditions through metric deformation techniques.

Findings

The algorithm is efficient and effective in generating quad-meshes.

It ensures the induced metric satisfies Gauss-Bonnet and holonomy conditions.

Experimental results validate the method's robustness and practicality.

Abstract

This work proposes a novel metric based algorithm for quadrilateral mesh generating. Each quad-mesh induces a Riemannian metric satisfying special conditions: the metric is a flat metric with cone signualrites conformal to the original metric, the total curvature satisfies the Gauss-Bonnet condition, the holonomy group is a subgroup of the rotation group $\{e^{ik\pi/2}\}$, furthermore there is cross field obtained by parallel translation which is aligned with the boundaries, and its streamlines are finite geodesics. Inversely, such kind of metric induces a quad-mesh. Based on discrete Ricci flow and conformal structure deformation, one can obtain a metric satisfying all the conditions and obtain the desired quad-mesh. This method is rigorous, simple and automatic. Our experimental results demonstrate the efficiency and efficacy of the algorithm.