Bézier曲线细分收敛定理的推广

摘要

在CAGD中,基于de Casteljau算法对Bézier曲线进行迭代细分时收敛定理成立,即假设每一次在相同的位置参数r(0＜r＜1)处对曲线进行细分,那么迭代得到的控制多边形收敛到初始控制多边形定义的Bézier曲线.文中对这一定理进行推广,给出了允许在每一次细分时采用不同的位置参数,得到了细分后产生的控制多边形收敛到初始控制多边形所定义的Bézier曲线的充要条件,并讨论了收敛速度.%In computer aided geometric design, based on the de Casteljau algorithm, the theory of subdivision convergence is established by subdividing Bezier curve iteratively. The control polygon converges to the original Bezier curve after the iterative subdivision at the same local parameter r, 0< r<1. In this paper, the theory above is extended and generalized. Different parameters associated with different steps of subdivision iteration are permitted and the necessary and sufficient condition that the control polygon converges to the original Bezier curve is obtained. Furthermore, the speed of convergence is discussed.