基于最小二乘法的张量积Said-Ball曲面降多阶逼近

New Method of Approximate Multi-degree Reduction of Tensor Product Bézier Surfaces

给出张量积Said-Ball曲面降多阶逼近的一种方法.该方法根据原张量积Said-Ball曲面Pn,m(u,v)与降多阶张量积Said-Ball曲面Qn1,m1(u,v)(n1≤n-1,m1≤m-1)在最小二乘范数下的距离函数在单位正方形[0,1]×[0,1]上取最小值,从而得到了用矩阵表示的降多阶张量积Said-Ball曲面Qn1,m1(u,v)的控制顶点{qij}in1=,0,m1j=0的显示表示式.在降多阶过程中,分别考虑了带角点高阶插值条件和不带角点插值条件的情形.文末附有数值例子,并将本文方法与参考文献(9)的方法做了比较.

This paper offers a new method for approximate multi-degree reduction of tensor product Said-Ball surfaces, which is based on the minimizing the distance function between Pnm （u, v） and Qn1, m1 （u, v） （n1≤n1 - 1, m1≤m - 1） over the unit square [0,1]×[0,1] in the sense of least squares normal （L2）, which gives the explicit representation of control points {qij}i^n1=0,^.m}=0 of the reduced multi-degree tensor product Said-Ball surface Qn1, m1 （u, v）. During the multi-degree reduction process, we also consider two cases, one has the constraint of high-order interpolations over corners and another has on any constraint. The examples show that the multi-degree reduction surfaces obtained by our method have better approximation than that of the current methods.