三角域上双变量Jacobi-Bernstein的基转换及应用  

Transformation between Bivariate Jacobi and Bernstein Basis on Triangular Domain and its Application

作　　者：蔡华辉[1,2,3] 王国瑾[1,2] 

机构地区：[1]浙江大学数学系计算机图象图形研究所,杭州310027 [2]浙江大学CAD&CG国家重点实验室,杭州310058 [3]景德镇陶瓷学院信息工程学院,景德镇333001

出　　处：《计算机辅助设计与图形学学报》2009年第10期1394-1400,共7页Journal of Computer-Aided Design & Computer Graphics

基　　金：国家自然科学基金(60873111);国家"九七三"重点基础研究发展计划项目(2004CB719400)

摘　　要：为了在CAGD中有效地求解三角域上Bézier曲面的最小平方逼近问题,给出了三角域上双变量Jacobi基和Bernstein基的相互转换矩阵.首先利用Bernstein基构造了三角域上的Jacobi多项式;然后利用单变量Jacobi基和Bernstein基的转换关系,给出了三角域上双变量Bernstein基与Jacobi基的相互转换矩阵.进一步,利用该矩阵得到了在加权L2范数下基于正交基的Bézier曲面最佳降多阶逼近算法,给出了具体的最佳降多阶矩阵以及该降阶逼近的可预报的误差公式.For solving least squares approximation problem simply and effectively on triangular domains in CAGD, this paper derives the matrices of transformation of the bivariate Bernstein basis form into the Jacobi basis of the same degree and vice versa. A method for constructing bivariate Jacobi-weighted orthogonal polynomials in the Bernstein form on triangular domains is formulated firstly. And then, by using connection coefficients between the univariate Bernstein and Jacobi basis, the transformation matrices between bivariate Jacobi and Bernstein basis are presented. Finally, by using the matrices, an explicit form of the multi-degree reduction matrix for Bezier surface on triangular domains with respect to Jacobi weighted L2 norm is proposed, and the error of the degree reduction is given.

关 键 词：三角域 BERNSTEIN基 Jacobi基 转换矩阵 BÉZIER曲面 降阶 矩阵表示 

分 类 号：TP391.41[自动化与计算机技术—计算机应用技术]