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**Planar piecewise cubic parametric Hermite interpolation with minimal derivative oscillation and strain energy.**
*(Chinese.
English summary)*
Zbl 1513.65028

Summary: Since the tangent vectors of the piecewise cubic parametric Hermite interpolation are often used as variables, they could be optimized to make the interpolation curve meet some specific requirements. In order to construct the planar piecewise cubic parametric 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.

### MSC:

65D17 | Computer-aided design (modeling of curves and surfaces) |

41A05 | Interpolation in approximation theory |

65D07 | Numerical computation using splines |

### Keywords:

cubic parametric Hermite interpolation; shape-preserving; smoothness; derivative oscillation; strain energy; minimization### Software:

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\textit{J. Li} et al., Math. Numer. Sin. 44, No. 1, 97--106 (2022; Zbl 1513.65028)

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[25] | PLANAR PIECEWISE CUBIC PARAMETRIC HERMITE INTERPOLATION WITH MINIMAL DERIVATIVE OSCILLATION AND STRAIN ENERGY Li Juncheng¡ Liu Chengzhi (College of Mathematics and Finance, Hunan University of Humanities, Science and Technology, Loudi 417000, China) Guo Xiao (College of Mathematics Science, Changsha Normal University, Changsha 410100, China) |

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