计算机科学 ›› 2018, Vol. 45 ›› Issue (3): 76-82.doi: 10.11896/j.issn.1002-137X.2018.03.013

• 第十届全国几何设计与计算学术会议 • 上一篇    下一篇

二次三角Hermite插值样条控制点的选取

刘成志,韩旭里,李军成   

  1. 中南大学数学与统计学院 长沙410083;湖南人文科技学院数学与金融学院 湖南 娄底417000,中南大学数学与统计学院 长沙410083,湖南人文科技学院数学与金融学院 湖南 娄底417000
  • 出版日期:2018-03-15 发布日期:2018-11-13
  • 基金资助:
    本文受国家自然科学基金(11272376),湖南省自然科学基金资助

Selection of Control Points of Quadratic-trigonometric Hermite Interpolation Splines

LIU Cheng-zhi, HAN Xu-li and LI Jun-cheng   

  • Online:2018-03-15 Published:2018-11-13

摘要: 文中对C1连续的二次三角Hermite插值样条曲线的自由控制点进行了进一步研究。首先讨论了给定中点条件时自由控制点的选取问题。为了获得光顺及弧长最短的二次三角Hermite插值样条曲线,基于能量优化法建立了一个求解最优自由控制点取值的优化模型,求解得到的最优控制点使得曲线的能量值达到最小;然后建立了一个优化模型来求解出最优控制点,使得插值曲线的近似弧长最短。数值实例表明,通过优化模型求出的控制点能使得二次三角Hermite插值样条曲线具有较好的光顺性及近似最短弧长。

关键词: Hermite插值样条,二次三角样条曲线,中点条件,能量优化,弧长最短

Abstract: This paper studied the selection of the free control points of the C1 continuous quadratic-trigonometric Hermite interpolation curves.Firstly,this paper discussed the selection of the free control points when the conditions of midpoint were given.In order to obtain the most smooth or the shortest arc length interpolation curves,an optimization model for solving the optimal control points was established based on the energy optimization method.By solving the optimization model,the optimal control points were obtained to minimize energy value of the curve.Then,an optimization model was also established for solving the shortest arc length.Numerical examples show that the optimal control points can make the curves smooth or have the shortest arc length.

Key words: Hermite interpolation spline,Quadratic-trigonometric curve,Conditions of midpoint,Energy optimization,Shortest arc length

[1] LORENTZ R A.Multivariate Hermite interpolation by algebraic polynomials:a survey [J].Journal of Computational and Applied Mathematics,2000,122(2):167-201.
[2] GFRERRER A,ROSCHEL O.Blended Hermite interpolations [J].Computer Aided Geometric Design,2001,18(9):865-873.
[3] YONG J H,CHENG F H.Geometric Hermite curves with minimum strain energy [J].Computer Aided Geometric Design,2004,21(3):281-301.
[4] HAll C A,MEYER W W.Optimal error bounds for cubic splineinterpolation [J].Journal of Approximation Theory,1976,16(2):105-122.
[5] DUAN Q,DJIDJELI K,PRICE W G,et al.Rational cubic spline based on function values [J].Computer and Graphics,1998,22(4):479-486.
[6] DUAN Q,DJIDJELI K,PRICE W G,et al.The approximation properties of some rational cubic splines [J].International Journal of Computer Mathematics,1999,72(2):155-166.
[7] SARFRAZ M.Cubic spline curves with shape control [J].Computer and Graphics,1994,18(5):707-713.
[8] DUAN Q,LIU A K,CHENG F H.Constrained interpolationusing rational cubic spline with linear denominators [J].Korean Journal of Computational and Applied Mathematics,1999,6(1):203-215.
[9] XIE J,TAN J Q,LI S F.Rational cubic Hermite interpolating spline and its approximation properties[J].Chinese Journal of Engineering Mathematics,2011,28(3):385-392.(in Chinese) 谢进,檀结庆,李声锋.有理三次Hermite插值样条及其逼近性质[J].工程数学学报,2011,28(3):385-392.
[10] LI J C,LIU C Y,YANG L.Quartic Herimite interpolatingsplines with parameters[J].Journal of Computer Applications,2012,32(7):1868-1870.(in Chinese) 李军成,刘纯英,杨炼.带参数的四次Hermite插值样条[J].计算机应用,2012,32(7):1868-1870.
[11] LI J C,ZHONG Y E,XIE C.Cubic trigonometric Hermite interpolating splines curves with shape parameters[J].Computer Engineering and Applications,2014,50(17):182-185.(in Chinese) 李军成,钟月娥,谢淳.带形状参数的三次三角Hermite插值样条曲线[J].计算机工程与应用,2014,50(17):182-185.
[12] HAN X.Piecewise trigonometric Hermite interpolation[J].Applied Mathematics & Computation,2015,268(C):616-627.
[13] KARCIAUSKAS K,PETERS J.Biquintic G2 surfaces via functionals[J].Computer Aided Geometric Design,2015,33:17-29.
[14] YAN L L,LI S P.Parameter selection of shape-adjustable interpolation curve and surface[J].Journal of Image and Graphics,2016,21(12):1685-1695.(in Chinese) 严兰兰,李水平.形状可调插值曲线曲面的参数选择[J].中国图象图形学报,2016,21(12):1685-1695.

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