导数振荡与应变能极小的平面分段三次参数Hermite插值

由于分段三次参数Hermite插值的切矢往往被作为变量，故可对其进行优化以使得构造的插值曲线满足特定的要求.为了构造兼具保形性与光顺性的平面分段三次参数Hermite插值曲线，给出了一种通过同时极小化导数振荡和应变能来确定切矢的方法.首先以导数振荡函数和应变能函数为双目标建立了切矢满足的方程系统；然后证明了方程系统存在唯一解，并给出了解的具体表达式；最后给出了误差分析，并通过数值算例表明方法的有效性.结果表明，相对于导数振荡极小化方法和应变能极小化方法，所提出的导数振荡和应变能极小化方法同时兼顾了平面分段三次参数Hermite插值曲线的保形性和光顺性.

PLANAR PIECEWISE CUBIC PARAMETRIC HERMITE INTERPOLATION WITH MINIMAL DERIVATIVE OSCILLATION AND STRAIN ENERGY

Since the tangent vectors of the piecewise cubic parametrie Hermite interpolation are often used as variables,they could be optimized to make the interpolation curve meet some specific requirements.In order to construet the planar piecewise cubic parametrie Hermite interpolation curve with both shape preservation and smoothness,a method for determining the tangent vectors by minimizing both derivative oscillation and strain energy simultaneously is presented.Firstly,the derivative oscillation function and the strain energy function are taken as a bi-objective to establish the equation system for solving the tangent vectors.Then,the equation system has a unique solution is proved and the concrete expression of the solution is given.Finally,the error analysis is proposed and the effectiveness of the method is demonstrated by some numerical examples.Compared with the derivative oscillation minimization method and the strain energy minimization method,the derivative oscillation and strain energy minimization method proposed in this paper takes into account the shape preservation and smoothness of the planar piecewise cubic parametric Hermite interpolation curve.