  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.
