Bézier曲线降多阶逼近的一种方法

文献1,2]讨论了Bezier曲线一次降多阶逼近问题,得到了很好的结果.文献1]利用广义逆矩阵得到不保端点插值的降多阶逼近曲线的控制顶点的表达式.但却没有得到带端点任意阶插值条件的降多阶逼近曲线的控制顶点的表达式.文献2]得到了带端点任意阶插值的降多阶逼近曲线的控制顶点的解析表达式.本文首先给出两Bezier曲线间距离的定义;然后根据降阶曲线与原曲线间的距离最小,分别得到了用矩阵表示的不保端点插值和保端点任意阶插值的降多阶逼近曲线的控制顶点的显示表达式.所给数值例子显示,用本文方法得到的降多阶逼近曲线对原曲线有很好的逼近效果.

A Method for Multidegree Reduction Approximation of Bézier Curves

In paper 1,2], multidegree reduction approximation problem of Bezier curves is disscussed, nice results have been obtained. In paper 1], matrix express for control points of multidegree reduction approximate curve without interpolation contitions of two endpoits has been obtained, but matrix express of control points of multidegree reduction approximate courve with arbitrary degree interpolation conditions of two endpoints sin't given. In paper 2], analytic express of control points of multidegree reduction approximate curve with arbitrary degree interpolation conditions is given. In this paper, the definition of distance between two Bezier curves is given. Matrix expresses of control points of multidegree reduction approximate curves without interpolation contidionts of two endpoints and with arbitrary degree interpolation conditions of two endpoints are obtained re-proesently by letting the distance between the orginal curve and multidegree reduction approximate curve be minimum. The examples of this paper show that multidegree reduction approximate curve obtained by method of this paper have good approximate effects to the orginal curves.