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

Xiaopeng Zheng, Yiming Zhu, Na Lei, Zhongxuan Luo, Xianfeng Gu
2020 arXiv   pre-print
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
more » ... dition 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.
arXiv:2007.07334v1 fatcat:raoc2fxgrfgnxhrh7lj7lq2fsi