带参数的Catmull-Clark细分曲面

Catmull-Clark Subdivision Surfaces with Parameters

在均匀B样条曲线的Lane-Riesenfeld细分算法中,每一步细分可看成是对原控制多边形的"切角"操作.文章通过引入一个参数来控制切角的程度,提出加权的Lane-Riesenfeld算法,并从均匀三次B样条曲线出发,得到光滑性为C^1的单参数曲线细分格式.进一步将该算法推广到任意拓扑的四边形网格上,得到除奇异点外处处C^1的细分曲面(称之为带参数的Catmull-Clark(C-C)细分曲面).格式中的参数在一定范围内调整时,可以使细分曲线/曲面不同程度地逼近控制多边形/控制网格,具有较好的灵活性.

In Lane-Riesenfeld algorithm for uniform B-spline curves, each subdivision step can be regarded geometrically as the corner-cutting process of the control polygon. In this paper, we control the corner cutting＇s degree by introducing a parameter to derive a weighted Lane-Riesenfeld algorithm, and from which a C^1-continuous subdivision scheme is obtained. We further generalize the subdivision scheme to arbitrary topological quad-meshes which produces a family of C^1 continuous surfaces except at the extraordinary points（i.e., Catmull-Clark subdivision surfaces with parameters）. By adjusting the values of the parameter, we obtain different subdivision surfaces which approximate the control meshes with different degree.