球域Bézier曲面的精确边界及其多项式逼近  

Exact boundary of ball Bézier surface and its approximation by polynomial form

作　　者：胡倩倩[1,2] 王国瑾[1] 

机构地区：[1]浙江大学计算机图象图形研究所,浙江杭州310027 [2]浙江工商大学统计与数学学院,浙江杭州310018

出　　处：《浙江大学学报（工学版）》2008年第11期1906-1909,共4页Journal of Zhejiang University：Engineering Science

基　　金：国家自然科学基金资助项目(60373033,60333010);国家“973”重点基础研究发展规划资助项目(2002CB312101)

摘　　要：为全面控制产品表面与理论曲面之间的偏差,引入球域Bézier曲面的定义,作为圆域Bézier曲线在三维空间的推广形式.根据经典微分几何中双参数曲面族的包络原理,运用球面参数坐标和Cramer法则,给出了球域Bézier曲面边界的精确数学显式表达式.依据函数逼近论中Legendre多项式的正交性,得到了采用多项式形式表示的球域Bézier曲面的精确边界的最佳平方逼近.进一步利用Legendre基与Bernstein基的转换公式,采用计算机辅助设计(CAD)系统中常用的Bézier形式表示球域Bézier曲面的近似边界.该算法表示简单,易于实现.通过具体实例对逼近效果进行演示与分析,结果表明该算法的逼近误差小,效果好.Ball Bézier surface was introduced as a generalization of disk Bézier curve in 3D space in order to totally control the deviation between the exterior of a product and the theoretical surface. An explicit mathematical representation of the exact boundary of a ball Bézier surface was given by recurring to the envelope theory of the family of bivariate surfaces in the classic differential geometry, as well as the spherical coordinates and the Cramer＇s rule. According to the orthonormality of Legendre polynomials in the function approximation theory, the exact boundary of the hall Bézier surface was optimally squarely approximated in polynomial form. Furthermore, using the transformation formulae between Legendre basis and Bernstein basis, the approximate boundary of the surface was expressed in Bézier form, which is usually used in computer aided design （CAD） systems. Finally numerical examples were presented and analyzed. This algorithm is simple and easy to realize. Examples showed that the approximation error is small and the approach has good results.

关 键 词：外形检测 球域Bézier曲面 包络 边界 LEGENDRE多项式 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术]