Quadrilateral Mesh Generation III: Optimizing Singularity Configuration Based on Abel-Jacobi Theory

TL;DR

This paper introduces a rigorous algorithm for quad-mesh generation based on Abel-Jacobi theory, involving complex algebraic geometry computations to optimize singularity configurations for improved mesh quality.

Contribution

The work presents a novel, practical algorithm leveraging Abel-Jacobi theory for generating meromorphic quartic differentials to optimize singularity placement in quad-meshes.

Findings

Algorithm effectively generates high-quality quad-meshes.

Experimental results demonstrate efficiency and accuracy.

Method enables direct construction of T-Splines from T-meshes.

Abstract

This work proposes a rigorous and practical algorithm for generating meromorphic quartic differentials for the purpose of quad-mesh generation. The work is based on the Abel-Jacobi theory of algebraic curve. The algorithm pipeline can be summarized as follows: calculate the homology group; compute the holomorphic differential group; construct the period matrix of the surface and Jacobi variety; calculate the Abel-Jacobi map for a given divisor; optimize the divisor to satisfy the Abel-Jacobi condition by an integer programming; compute the flat Riemannian metric with cone singularities at the divisor by Ricci flow; isometric immerse the surface punctured at the divisor onto the complex plane and pull back the canonical holomorphic differential to the surface to obtain the meromorphic quartic differential; construct the motor-graph to generate the resulting T-Mesh. The proposed method is rigorous and practical. The T-mesh results can be applied for constructing T-Spline directly. The efficiency and efficacy of the proposed algorithm are demonstrated by experimental results.