端点约束加权正交基与Bernstein基的转换及应用  

Transformation between weighted orthogonal basis satisfying end point constraints and Bernstein basis and its application

作　　者：蔡华辉[1,2] 王国瑾[1] 

机构地区：[1]浙江大学CAD&CG国家重点实验室,浙江杭州310027 [2]景德镇陶瓷学院信息工程学院,江西景德镇333403

出　　处：《浙江大学学报（工学版）》2011年第4期602-606,613,共6页Journal of Zhejiang University：Engineering Science

基　　金：国家自然科学基金资助项目(60933007;61070065);江西省教育厅基金资助项目(GJJ11200);景德镇陶瓷学院博士启动基金资助项目

摘　　要：为了在计算机辅助几何设计(CAGD)中,有效地求解在Jacobi加权L2范数下Bézier曲线约束最佳降多阶逼近问题,推导具有端点约束特征的加权正交基与Bernstein基之间的转换矩阵.利用Bernstein基构造端点约束加权正交基,给出约束加权正交基与Bernstein基的相互转换矩阵,利用该矩阵给出具体的端点约束最佳降多阶矩阵和该降阶逼近的可预报的误差公式,提出在L2、L1、L∞范数下适合于最佳降阶逼近的相应Jacobi基的权函数的选取方案.通过具体实例对逼近算法进行演示与分析.结果表明,该算法表示简单,易于实现.The transformation matrices between the weighted orthogonal basis which possesses end point constraints characteristic and Bernstein basis were derived in order to effectively obtain the optimal algorithm for constrained multi-degree reducing Bezier curve based on Jacobi weighted L2 norm in computer aided geometric design （CAGD）. A method for constructing Jacobi-weighted orthogonal polynomials satisfying end point constraints in the Bernstein form was formulated, and the transformation matrices between Jacobi-weighted orthogonal basis and Bernstein basis were presented. Then the matrix representation for constrained multi-degree reducing Bezier curve was presented by the matrices, and the degree reduction error that can be forecasted was given. The Jacobi weighted function adapting to optimal degree reduction was selected with respect to L2. L1. L∞ norm, respectively. Numerical examples were presented and analyzed. The method is simple and easy to realize.

关 键 词：BERNSTEIN基 加权正交基 端点约束 转换矩阵 降多阶 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]