李军成.利用近似能量极小构造平面C1三次Hermite插值曲线[J].数学研究及应用,2019,39(4):433~440
利用近似能量极小构造平面C1三次Hermite插值曲线
Constructing Planar $C^1$ Cubic Hermite Interpolation Curves Via Approximate Energy Minimization
投稿时间:2018-09-04  修订日期:2018-10-26
DOI:10.3770/j.issn:2095-2651.2019.04.009
中文关键词:  Hermite插值  应变能  曲率变化  极小
英文关键词:Hermite interpolation  strain energy  curvature variation  minimization
基金项目:湖南省自然科学基金(Grant No.2017JJ3124); 湖南省教育厅资助科研项目(Grant No.18A415).
作者单位
李军成 湖南人文科技学院数学与金融学院, 湖南 娄底 417000 
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中文摘要:
      研究了利用近似能量极小构造平面$C^1$三次Hermite插值曲线的方法.该方法的主要的目是求出$C^1$三次Hermite插值曲线的最佳切矢.通过将应变能、曲率变化能和组合能的近似函数极小化,得到了求解最佳切矢的线性方程组.通过求解发现,近似曲率变化能极小不存在唯一解, 而近似应变能极小和近似组合能极小由于方程系统的系数矩阵为严格对角占优故都存在唯一解.最后, 通过实例表明了本文方法构造平面$C^1$三次Hermite插值曲线的有效性.
英文摘要:
      The methods for constructing planar $C^1$ cubic Hermite interpolation curves via approximate energy minimization are studied. The main purpose of the proposed methods are to obtain the optimal tangent vectors of the $C^1$ cubic Hermite interpolation curves. By minimizing the appropriate approximate functions of the strain energy, the curvature variation energy and the combined energy, the linear equation systems for solving the optimal tangent vectors are obtained. It is found that there is no unique solution for the minimization of approximate curvature variation energy minimization, while there is unique solution for the minimization of approximate strain energy and the minimization of approximate combination energy because the coefficient matrix of the equation system is strictly diagonally dominant. Some examples are provided to illustrate the effectiveness of the proposed method in constructing planar $C^1$ cubic Hermite interpolation curves.
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