Optimal Surface Quadrilateral Mesh Generation

Structured mesh generation has fundamental importance, but controlling the qualities of structured meshes remains a challenge. This work proposes a rigorous and practical algorithm to generate quadrilateral meshes on topological poly-annuli with the least number of singularities. It is shown that each quad-mesh with 4k vertex degree induces a holomorphic differential. All the holomorphic differentials form a finite-dimensional linear space. A geometric distortion energy is proposed to measure the local area distortion. One can achieve quad-meshes as uniformly as possible by optimizing the distortion energy in the linear space. Experimental results show the proposed algorithm is able to improve the uniformity of the quad-meshes, and the method has the potential to be generalized to handle quad-meshes with other constraints.