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.


65D17 Computer-aided design (modeling of curves and surfaces)
41A05 Interpolation in approximation theory
65D07 Numerical computation using splines


Full Text: DOI


[1] Cripps R J, Hussain M Z. C 1 monotone cubic Hermite interpolant[J]. · Zbl 1248.41008
[2] Applied Mathematics Letter, 2012, 25(8): 1161-1165. · Zbl 1248.41008
[3] Daniel C V, Mario M H, Ibai L M, et al. Reconstruction of 2D river beds by appropriate inter-polation of 1D cross-sectional information for flood simulation[J]. Environmental Modelling and Software, 2014, 61: 206-228.
[4] Hagstrom T, Appelö D. Solving PDEs with Hermite interpolation[C]. In: Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, 2015, Springer, Cham. · Zbl 1352.65452
[5] Bica A. Fitting data using optimal Hermite type cubic interpolating splines[J]. · Zbl 1252.65033
[6] Applied Mathe-matics Letter, 2012, 25(12): 2047-2051. · Zbl 1252.65033
[7] Han X L, Guo X. Cubic Hermite interpolation with minimal derivative oscillation[J]. Journal of Computational and Applied Mathematics, 2018, 331: 82-87. · Zbl 1377.65015
[8] Vassilev T I. Fair interpolation and approximation of B-splines by energy minimization and points insertion[J]. Computer-Aided Design, 1996, 28(9): 753-760.
[9] Zhang C M, Zhang P F, Cheng F H. Fairing spline curves and surfaces by minimizing energy[J]. · Zbl 1206.65040
[10] Computer-Aided Design, 2001, 33(13): 913-923. · Zbl 1206.65040
[11] Hofer M, Pottmann H. Energy-minimizing splines in manifolds[J].
[12] ACM Transaction on Graphics, 2004, 23(3): 284-93.
[13] Yong J H, Cheng F. Geometric Hermite curves with minimum strain energy[J]. · Zbl 1069.65541
[14] Computer Aided Geometric Design, 2004, 21(3): 281-301. · Zbl 1069.65541
[15] Jaklič G,Žagar E. Curvature variation minimizing cubic Hermite interpolants[J]. · Zbl 1243.41001
[16] Applied Math-ematics and Computation, 2011, 218(7): 3918-3924. · Zbl 1243.41001
[17] Xu G, Wang G Z, Chen W Y. Geometric construction of energy-minimizing Bézier curves[J].
[18] Science China (Information Sciences), 2011, 54(7): 1395-1406. · Zbl 1267.65021
[19] Xu G, Zhu Y F, Deng L S, et al. Efficient construction of B-Spline curves with minimal internal energy[J].
[20] Computers, Materials and Continua, 2019, 58(3): 879-892. · Zbl 1480.47035
[21] Kobza J. Cubic splines with minimal norm[J]. Applications of Mathematics, 2002, 47(3): 285-295. · Zbl 1090.65012
[22] Wang F S, Zhang K C. The theoretical analysis and algorithm on a class of optimal curve fitting problems[J]. Applied Mathematics and Computation, 2006, 183(1): 622-633. · Zbl 1116.65012
[23] Li J C. Constructing planar C 1 cubic Hermite interpolation curves via approximate energy mini-mization[J]. Journal of Mathematical Research with Applications, 2019, 39(4): 433-440. · Zbl 1449.65012
[24] Õ , ͹ . ℄ Á Ø [M].´¾: ¼©µ, 1992.
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.