细分参数对细分曲线形状的控制作用分析

Study on control effect of subdivision parameters

多数有关细分法的文献侧重于研究细分法的构造、收敛性光滑性分析及其在光滑曲线曲面造型中的应用,少有文献致力于细分参数对细分曲线形状影响的理论分析。首先引入仿射坐标的观点,从几何直观的角度对三点ternary插值细分法中细分参数的几何意义进行研究。接着通过对细分法的C0和C1参数域及新顶点域的等价描述,从理论化的角度对细分参数对细分曲线形状的局部和整体控制作用进行分析,描述它们对细分曲线行为的影响。在给定初始数据的条件下,可通过对形状参数的适当选择来有的放矢地实现对三点ternary插值细分曲线曲面的形状调整和控制。该结果可用于工业领域中产品的外形设计及形状控制。

Hassan and Dodgson proposed a 3-point ternary interpolatory subdivision scheme which contains two parameters.h has good locality and can be C^1-continuous when the two parameters are kept within a proper range.But the clear meanings of the two parameters and how the parameters affect the shape of the subdivision curve are not known,which limits the application of the subdivision scheme in a way.In this paper we focus on the theoretical analysis of the roles of the parameters.The control effect of the parameters on the shape of the 3-point ternary interpolatory subdivision curve is analyzed and described.The obtained results suggest a clear direction to control the shape of the 3-point ternary interpolatory subdivision curve or surface.