基于C-B样条的Catmull-Clark细分曲面

Catmull-Clark Subdivision Surfaces Modelling with C-B splines

为了解决 Catum ull- Clark细分曲面在工程上难以推广的问题 ,给出了一种基于 C- B样条的 Catumull-Clark细分曲面的算法 .C- B样条曲线是 B样条曲线的拓广 ,但它们的形状依赖于参数 α.由于新的曲面细分方法充分利用 C- B样条能够精确表示圆、椭圆等规则形体的特性 ,因而使通过此方法生成的细分曲面 ,除了在奇异点处能保持二阶导数连续外 ,还能够像 C- B样条曲线、曲面一样 ,精确地表示圆柱等常规曲面、统一工程曲面等的造型 ;同时它仍然保持细分曲面的造型特点 ,即能够解决 NU RBS曲面难以处理的任意拓扑结构的造型问题 ,另外 ,还可依赖控制参数 α的调节作用来增加造型的自由度 ,而且当 α→ 0时 ,它们就退化成 Catm ul- Clark细分曲面 .在工程图形上的应用实例表明 ,这种算法简单、有效 .

In order to overcome the difficulty in applying the Catmull-Clark subdivision surfaces in engineering, a new algorithm about Catmull-Clark subdivision surfaces based on C-B splines is presented. C-B splines curves are extension of B splines, they depend on a parameter α . By use of characteristics of C-B splines, for example, they can provide exact reproduction of circles and cylinders and they can be generated by subdivision scheme while keeping C 2 , a new surface scheme is generated. The limit surfaces generated by this surface scheme are C 2 except at extraordinary points. In conclusion, this method not only solves the problem of the precise representation of standard analytic shapes such as circle encountered by Catmull-Clark subdivision surfaces, but also overcomes the difficulty of generating surfaces on arbitrary topological meshes faced by NURBS. Meanwhile, the shaapes of the subdivision surfaces can be adjusted using controlling parameter α and the particular case ( α →0) of this scheme is Catmull-Clark subdivision scheme. An application in engineering garphics demonstrates freedom and efficiency of this algorithm.