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 necessary conditions flat metric with cone singularities to be compatible quad-mesh, in terms deck-transformation, then develop an on theorem. The has two stages: first, meromorphic quartic differential is generated induce T-mesh; second, edge lengths T-mesh are adjusted by solving linear system satisfy deck transformation condition, which produces quad-mesh. In first stage, pipeline can summarized as follows: calculate homology group; compute holomorphic construct period matrix surface Jacobi variety; map given divisor; optimize divisor condition integer programming; Riemannian at Ricci flow; isometrically immerse punctured onto complex plane pull back canonical obtain differential; motorcycle graph generate T-Mesh. second constraints formulated equation T-mesh. solution provides integral transformations, leads final proposed method practical. results applied constructing Splines directly. efficiency efficacy demonstrated experimental surfaces complicated topologies geometries.