张量积Bézier曲面降多阶逼近

Multi-degree reduction of tensor product Bézier surfaces

给出了一种基于最小二乘范数下的Bézier曲面降多阶逼近误差的矩阵计算公式。根据带角点高阶插值条件下原张量积Bézier曲面与降多阶张量积Bézier曲面的误差函数在[0,1]×[0,1]上取极小值,得到降多阶张量积Bézier曲面的控制顶点的矩阵表达式。通过数值例子显示采用该方法所得的降多阶曲面对原曲面有较好的逼近效果。将Bézier曲线降阶逼近的迭代方法推广到曲面,得到曲面降阶逼近的迭代方法,并给出了相应的数值实例。

A matrix formula of the multi-degree reduction of tensor product Bézier surface approximation error is presented based on least squares normal （L2）. It gives the explicit representation of control points of the reduced multi-degree tensor prod- uct Bézier surface, through minimizing the distance function between the original B6zier surface and the reduced multi-degree tensor product Bézier surface over unit square [0, 1] x[0, 1]. During the multi-degree reduction process, it is considered that the constraint of high-order interpolations over corners. Examples show that the proposed approach has better approximation of the reduced surfaces than that of current methods. An iterative aleorithm for degree reduction of B6zier surfaces is given.