导　　师： 江平

学科专业： G0102

授予学位： 硕士

作　　者： ;

机构地区： 合肥工业大学

摘　　要： 计算机辅助几何设计(CAGD)和计算机辅助机器制造(CAM)中，圆弧和球面是非常重要和基础的几何研究对象．现有的CAD/CAM造型系统不能处理圆和球面的参数方程和隐式方程，因此为了使现有的CAD/CAM造型系统可以处理圆弧、圆、以及球面曲面片和球面，人们只有采用参数多项式和参数有理多项式来逼近它们．因而圆弧和球面的逼近表示问题一直是研究的热点问题之一．本文从基于函数的LEGENDRE展开为基础，利用LEGENDRE多项式去逼近圆弧和球面，得到了圆弧和球面的LEGENDRE多项式逼近函数．随后用LEGENDRE基函数与BERNSTEIN基函数的基转化矩阵得到了圆弧和球面的任意次数的BéZIER多项式逼近，同时在此基础上得到了椭圆和椭球面的BéZIER多项式逼近．最后，通过实例验证了这种逼近方法的效果，并且与已知的其他方法做了比较． Circular arcs and sphere are very important and basic object of geometric research in CAGD and CAM.The Modern CAD//CAM systems can not deal with the parameter equation and the implicit equation of circle and the sphere.Therefore,people can only use the parametric polynomial and parametric rational polynomial to approximate the circular arc,the circle,the spherical surface and the sphere enabling the Modern CAD//CAM systems to deal with them.Approximation and representation of circular arcs and sphere have been hot question.In the thesis,based on the Legendre expansion of function, using Legendre polynomial to approximate circular arcs and sphere,approximation function of the circular arcs and sphere are obtained in Legendre polynomial.Then,the arbitrary order Bézier polynomial approximation of circular arcs and sphere are obtained by transformation between Legendre and Bernstein polynomial,and we also use this method to obtain Bézier polynomial approximation of ellipse and ellipsoid.Finally,some examples are given to show the effectiveness of these methods,and the results are also compared with other methods.

关 键 词： 造型系统 圆弧 球面 多项式逼近函数 参数多项式

分 类 号： [O241.5]

领　　域： [理学] [理学]