# Quadrilateral Mesh Generation II : Meromorphic Quartic Differentials and Abel-Jacobi Condition

## TL;DR

This paper establishes a theoretical and computational framework linking quadrilateral mesh generation to meromorphic quartic differentials and the Abel-Jacobi condition, enabling algebraic geometric methods for mesh construction.

## Contribution

It introduces a novel equivalence between quad-meshes and meromorphic quartic differentials, along with algorithms for verifying Abel-Jacobi conditions and constructing differentials on zero surfaces.

## Key findings

- Algorithm for Abel-Jacobi condition verification demonstrated efficiency.
- Constructive method for meromorphic quartic differentials on zero surfaces proposed.
- Theoretical link between quad-mesh singularities and meromorphic differentials established.

## Abstract

This work discovers the equivalence relation between quadrilateral meshes and meromorphic quartic. Each quad-mesh induces a conformal structure of the surface, and a meromorphic differential, where the configuration of singular vertices correspond to the configurations the poles and zeros (divisor) of the meroromorphic differential. Due to Riemann surface theory, the configuration of singularities of a quad-mesh satisfies the Abel-Jacobi condition. Inversely, if a satisfies the Abel-Jacobi condition, then there exists a meromorphic quartic differential whose equals to the given one. Furthermore, if the meromorphic quadric differential is with finite, then it also induces a a quad-mesh, the poles and zeros of the meromorphic differential to the singular vertices of the quad-mesh. Besides the theoretic proofs, the computational algorithm for verification of Abel-Jacobi condition is explained in details. Furthermore, constructive algorithm of meromorphic quartic differential on zero surfaces is proposed, which is based on the global algebraic representation of meromorphic. Our experimental results demonstrate the efficiency and efficacy of the algorithm. This opens up a direction for quad-mesh generation using algebraic geometric approach.