圆弧曲线段和球面曲面片的多项式逼近

The Polynomial Approximants for Circular Segments and Spherical Patches

鉴于现有的CAD/CAM造型系统不能处理圆和球面的隐式方程以及用三角函数所表示的参数方程,因此为了使现有的CAD/CAM造型系统能够处理圆弧、圆以及球面曲面片、球面,人们只能采用参数多项式和参数有理多项式来逼近它们。为了能更好地对圆弧曲线段和球面曲面片进行逼近,提出了一种基于最小二乘范数的参数Bézier多项式逼近方法。该方法根据在最小二乘范数L2下所定义的距离函数取最小值,首先得到了一个圆弧曲线段和球面曲面片的参数Bézier多项式逼近式,并把该逼近多项式表示成两个行列式的商的形式。如果所取圆弧曲线段或球面曲面片为圆或球面时,则可得到圆或球面的参数Bézier多项式逼近式。另外,用该方法也可得到椭圆弧曲线段和椭球面曲面片的参数Bézier多项式逼近式。最后给出了一些数值实例,数值实验结果表明,该方法是有效的。

Modem CAD/CAM systems do not dispose the circle and the sphere represented by the implicit equation and the parameter equation with trigonometric function. Ones approximate the circular arc and the circle as well as the spherical surface and the sphere by using the parameter polynomial and parameter rational polynomial such that they can be disposed by the modem CAD/CAM systems. In order to effectively approximate the circular arc and the circle as well as the spherical surface and the sphere, the parameter B6zier form polynomial approximants of circular segments and spherical patches are obtained by minimizing the defined distance function with respect to the best least squares norms L2. Meanwhile these approximants are expressed as the quotient of two determinants. If the circular segment or the spherical patch to be approximated is a full circle or a sphere, The parameter B6zier form polynomial approximants of a full circle or a sphere can be given by the same process. Furthermore, by using this paper＇s method, The parameter B6zier form polynomial approximants of elliptic arcs and ellipsoid patches can be presented. Finally we show some graphical examples in order to prove the validity of our method.