曲线插值的一种具有还圆性的细分方法

A circle-restoring subdivision scheme for curve interpolation

传统的线性四点插值细分方法不能表示圆等非多项式曲线,为了解决这种问题,基于几何特性提出了一种带有一个参数的四点插值型曲线细分方法。细分过程中,过相邻三插值点作圆,过相邻二插值点的圆弧有两个中点,将其加权平均得到新插值点,文中给出了插值公式和算法描述。所给方法具有还圆性,可以实现保凸性。实例分析对比了本方法与多种细分方法的差异,说明本方法是有效的,当参数取值较小时,曲线靠近控制多边形。

A geometric 4-points interpolatory subdivision scheme with a parameter is proposed to overcome the deficiency of traditional 4-points interpolatory subdivision scheme that it can not generate non-polynomial curve,for example,circle.As three adjacent points confirm a circle,there are two arcs between every two adjacent points.The new generating point is determined by weighted average of two midpoints on the arcs.Interpolation formula and algorithm are described.This subdivision scheme can be convexity-preserving and restore a circle if all initial knots are on the same circle.Examples show the difference between this scheme and some traditional schemes.As the parameter becomes smaller,limit curve gets closer to initial controlling polygon.