Angle-Preserving Quadrilateral Mesh Parameterization

In response to their growing use in the real world, this article presents two algorithms for direct parameterization of quadrilateral meshes. The proposed algorithms are angle-preserving mappings, with one mapping a topological disk surface onto a Euclidean plane and one mapping a topological sphere surface onto a unit sphere. Specifically, for topological disk surfaces, the authors devise a discrete conformal energy function to flatten the quadrilateral meshes with a length-preserving boundary condition. For topological sphere surfaces, a derived Tuette energy function is applied to the initialization of parameterization for a mesh, and then the final spherical parametrization result is obtained by minimizing a devised harmonic energy function. Experimental results demonstrate the efficiency of the proposed methods.