Shape-Preserving Parametrization of Genus 0 Surfaces

The parametrization of 3-d meshes can be used in many fields of computer graphics. Mesh-texturing, mesh-retriangulation or 3-d morphing are only few applications for which a mesh parametrization is needed. Because, many polygonal surfaces are manifolds of genus 0 (topological equivalent to a sphere), we can apply a mapping, in which 2-d polar coordinates of a sphere can be directly transformed onto the 3-d coordinates of a polygonal object. In this paper we present a hierarchical mapping algorithm, that preserves the local surface properties. Our method consists of three main-steps. First, the mesh is simplified to a tetrahedron. Next, the tetrahedron will be transformed to a spherical surface in which the previous simplification process will be reversed on the surface of the sphere. Hereby, in every refinement step the new vertices are inserted and the resulting parametrization mesh is optimized to be barycentric. Finally, the resulting barycentric mesh is used as the basis for a shape-preserving optimization process. The efficiency of these method will be shown by using our parametrization algorithm on different 3-d objects.