球面坐标下的凸组合球面参数化

球面参数化是一种应用价值很广的几何参数化方法.对于封闭且亏格为零的三角形网格,该文提出了一种新的球面参数化方法.通过引入多个球面坐标覆盖,在球面坐标系下,用凸组合方法,得到了接近线性的球面参数化求解方法.与已有的直角坐标系下的凸组合参数化方法相比,该文所提出的方法大大降低了求解方程组的非线性程度,因此求解时间大幅度降低.此外,还避免了直角坐标系下求解的多种退化情况.最后,给出了实验结果,并对凸组合球面参数化中存在的几个问题进行了讨论.

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.