中文摘要：

将数值计算中的函数插值和外形设计中的参数曲线插值相结合,提出构造具有指定多项式重构精度的函数插值和具有指定连续阶的参数曲线插值的一般方法.该方法以Hermite插值的基本形式为桥梁,首先以用于函数插值时达到指定的精度为目标来推导基本形式中的导向量表达式,通过解方程获取导向量中的系数;然后将导向量代入Hermite插值的基本形式,并将其按照插值数据点进行整理,得出插值基函数表达式;最后给出以插值数据点和插值基函数的线性组合形式表达的插值曲线.数值实验结果表明,曲线形状可以固定也可以做局部调整,所给2n+1次Hermite插值多项式的重构精度一般会超过n次.

英文摘要：

This paper aims at combining the function interpolation in numerical calculation and the parametriccurve interpolation in shape design,and provide the general method of constructing the function interpolationwith given polynomial reconstruction precision and the parametric curve interpolation with givencontinuity order.The method takes the cardinal form of Hermite interpolation as a bridge.Firstly,we deducethe expression of the derivative vectors in the cardinal form with the goal of making the function interpolationreaches the given precision.And the coefficients in the derivative vectors are obtained by solving linearequations.Secondly,we substitute the derivative vectors into the cardinal form.Rearranging it in accordancewith the interpolation data points,we can obtain the expression of the interpolation basis functions.Lastly,we provide the interpolation curve with the form of the linear combination of the interpolation points and theinterpolation basis.The interpolation curve construction method given here does not require one to reversethe control points.Numerical experiment results show that the shape of the resulting curve can be fixed andcan do partial adjustment.The reconstruction precision of the degree2n+1Hermite interpolation polynomialis generally more than n.