Convex Combination Spherical Parameterization Using Spherical Coordinates

Parameterization is the key step in digital geometry processing. And spherical parameterization is an important parameterization approach with broad application. The solution cost of convex combination spherical parameterization using Cartesian coordinates is very expensive because it needs to solve a high nonlinear equation group. In this paper, a new spherical parameterization method for closed and genus zero mesh is presented. By importing several spherical coordinates cover, convex combination under spherical coordinates is used to calculate spherical parameterization, which only needs to solve a quasi linear equation group. Compared to other convex combination spherical parameterization method using Cartesian coordinates, this approach lowers the nonlinear extent, and the solution time decreases greatly. Moreover, several degenerate cases under Cartesian coordinates are avoided, such as point lies in the opposite side of the sphere, since there is no quadratic term using spherical coordinates. The shortcoming brought by this method is that the result is a little un uniform along longitude. The mesh near the equation is apt to be denser than the mesh near the poles. At the end of this paper, several problems in convex combination spherical parameterization are discussed and experiment results are given. This spherical parameterization method can be used in the consistent mesh construction.