圆弧的四次Bézier曲线逼近

Approximation of Circular Arcs by Quartic Bézier Curves

针对Bézier曲线不能精确表示圆弧,导致在基于Bézier曲线曲面造型的CAD系统中存在圆弧的Bézier曲线逼近问题,提出一种用四次Bézier曲线逼近圆弧的方法.根据圆弧与Bézier曲线都具有的对称性确定带待定参数的Bézier曲线的控制顶点;再由误差函数的零点分布情况确定待定参数,给出控制顶点的计算公式、误差的解析表达式和逼近阶.与采用已有方法得到的最好结果相比较,文中方法的逼近阶虽然也是8,但系数不到已有方法的一半,因而具有更好的逼近精度.

To address the problem that Bézier curves can not accurately represent circular arcs, a new approximation method for circular arcs by quartic Bézier curves is presented. Firstly, based on the symmetry of circular arcs and Bézier curves, the control points with unknown parameters are determined. Then according to the distribution of roots of the error function, the parameters of control points are further determined. The analytic expression of error function and the approximation order are given in this paper. Compared to the previously known best results, the approximation order of the proposed method is also eight, but the coefficient is less than half of the previously best results and thus our method has better approximation accuracy.