有理Bézier曲线的多项式逼近新方法

针对有理曲线多项式Hybrid逼近未必收敛及计算较繁的局限性,给出了以原有理Bézier曲线之升阶曲线的控制顶点为顶点的多项式Bézier曲线,来逼近原有理曲线的一类简单逼近方法.与此同时,为追求较高逼近速度,导出了有理Bézier曲线多项式逼近的一个矛盾方程组,并进一步基于广义逆矩阵理论,给出了其用矩阵表示的最小二乘解.最后借助以原有理曲线权因子为Bézier纵标的多项式的升阶,使得多项式逼近的曲线次数保持不变的同时大幅度提高了逼近精度．

New way of approximating rational Bézier curve with polynomial curve

In order to resolve the problem that hybrid polynomial  approximation cannot guarantee the property of convergence, a simple  approximation method was given which used the polynomial Bézier curve whose points are the control points of the degree-elevated curve to approximate the original rational curve. Meanwhile,   the contradictory equations of precise approximating rational curve by polynomial curve were deduced to achieve higher approximation efficiency. Then based on the theory of generalized inverse matrix, the least square solution in matrix form was obtained. Combined with the degree elevation of the function which took the weights of the original rational curve as Bézier lengths, the new way  got better approximating result with less error with the same approximating degree．