中文摘要：

渐进迭代逼近（PIA）方法在CAD领域有很好的自适应性和收敛稳定性,在曲线或曲面的逼近和拟合问题上具有很好的应用前景.文中将该方法应用于二维自由曲线的等距曲线（也称offset曲线）的逼近,提出基于PIA的等距曲线逼近算法.首先在等距曲线上采样数据点,采用Floater的方法对数据点进行参数化,并以这些采样点作为初始控制顶点,由这些初始控制顶点产生初始逼近曲线;然后考察相同参数值处采样点和逼近点的误差,并运用PIA方法逐步逼近等距曲线.该算法分别考虑了等距曲线的多项式逼近和有理逼近.数值实例结果表明,综合控制顶点数和算法误差这2项因素,文中算法具备较好的优势.

英文摘要：

In CAD, progressive iterative approximation （PIA） method has a wide range of applications for solving curve and surface approximation and fitting problems due to its good adaptability and stable convergence. In this paper, an efficient approximation method of the plane freeform offset curves based on PIA is proposed. Firstly, original data points are sampled on the offset curve. Secondly, parameterization is carried out on these points based on the Floater＇s method. By taking these sample points as original control points, an initial approximating curve is obtained. Then, by studying the errors between sample points and approximating points which have same parameter values, the PIA method is applied to minimize the corresponding errors. In our approach, the polynomial approximation method and rational approximation method are both considered in the offset approximation. Numerical examples show that our method has distinct advantages.