Quadrilateral mesh generation III : Optimizing singularity configuration based on Abel–Jacobi theory

This work proposes a rigorous and practical algorithm for quad-mesh generation based the Abel–Jacobi theory of algebraic curves. We prove sufficient and necessary conditions for a flat metric with cone singularities to be compatible with a quad-mesh, in terms of the deck-transformation, then develop an algorithm based on the theorem. The algorithm has two stages: first, a meromorphic quartic differential is generated to induce a T-mesh; second, the edge lengths of the T-mesh are adjusted by solving a linear system to satisfy the deck transformation condition, which produces a quad-mesh.