非光滑曲线等长样条逼近

The Spline Curve Approximation for Non-smooth Curve with the Fixed Length

在工程计算中,常常需要对非光滑曲线进行光滑逼近.本文讨论了非光滑曲线在局部小区间上,可用保长度不变的光滑样条曲线来近似的问题.首先利用介值定理,从理论上证明:对定义在有限区间上的任意连续函数,只要曲线长度有限,则在该区间上可用光滑样条曲线来等长近似原曲线.在此基础上,通过选择恰当的插值节点,可以构造出一条唯一的光滑曲线,使其具有和原曲线形同的长度.同时,本文还给出具体构造光滑曲线的方法及详细计算过程.最后的数值结果表明这种方法既简单又行之有效,从而完整地解决了工程计算中曲线近似这一问题.

It is a necessary condition that the curve is continuous and sufficient smooth when we solve the differential equation defined on the curve in engineering calculation. This paper discusses the smooth curve approximation for non-smooth curves with the same length on local interval. We prove that the following theoretical results based on the intermediate value theorem: if the measurable length of non-smooth points in a continuous curve has limit length,on any local sub-intervals, there always existences at least one smooth spline curve can be used to approximate the curve with same length as long as the length can be measured on this fixed sub-intervals. Furthermore, if a reference point is given, the unique smooth spline curve can be derived easily. Based on these theoretic results, we also propose the algorithm and present some numerical results to show the simplicity and efficiency of this method, which leads to the optimal solution for the non-smooth curve approximation.