  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.
