优化张力参数与边界条件的平面三次Cardinal样条

doi:10.3785/j.issn.1008-9497.2019.02.004

浙江大学学报（理学版）

2019, Vol. 46

Issue (2): 164-171 DOI: 10.3785/j.issn.1008-9497.2019.02.004

Chinagraph 2018 会议专栏

优化张力参数与边界条件的平面三次Cardinal样条

李军成¹, 刘成志¹, 易叶青²

1.湖南人文科技学院数学与金融学院，湖南娄底 417000
2.湖南人文科技学院信息学院，湖南娄底 417000

Planar cubic Cardinal spline with tension parameter and boundary condition optimization

Juncheng ¹, Chengzhi LIU¹, Yeqing YI²

1.College of Mathematics and Finance, Hunan University of Humanities, Science and Technology, Loudi 417000, Hunan Province, China
2.College of Information, Hunan University of Humanities, Science and Technology, Loudi 417000, Hunan Province, China

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摘要： 当数据点给定时,三次Cardinal样条的张力参数和边界条件均为自由变量，因此可对这些自由变量进行优化，以得到满足某种特定要求的最佳三次Cardinal样条。讨论了如何通过优化张力参数与边界条件使得构造的平面三次Cardinal样条尽可能光顺。首先,分析了三次Cardinal参数样条曲线形状的影响因素；然后，利用曲率变化能极小对三次Cardinal参数样条曲线的张力参数与边界条件进行优化，获得张力参数与边界条件的唯一解；最后，给出了对应三次Cardinal样条函数的张力参数与边界条件的优化方法。实例表明，经曲率变化极小优化张力参数与边界条件后的三次Cardinal样条比三次Catmull-Rom样条更为光顺，插值效果更好。

关键词： 三次Cardinal样条; 张力参数; 边界条件; 曲率变化极小; 三次Catmull-Rom样条

Abstract: The tension parameter and boundary condition of the cubic Cardinal spline can be optimized to obtain the best cubic Cardinal spline satisfying some specific requirements, because they are free variables when the data points are fixed. How to construct the smooth planar cubic Cardinal spline by optimizing the tension parameter and boundary condition is discussed in this paper. Firstly, the influence factors on the shape of the parametric cubic Cardinal spline curve are analyzed. Then, the tension parameter and boundary condition of the parametric cubic Cardinal parameter spline curve are optimized by the curvature variation minimization, and the unique solution of the tension parameter and the boundary condition is obtained. Finally, the corresponding method of the cubic Cardinal spline function is given. Some examples show that the cubic Cardinal spline obtained by optimizing the tension parameter and boundary condition is smoother, and the interpolation effect is better than that of the cubic Catmull-Rom spline.

Key words: cubic Cardinal spline tension parameter boundary condition curvature variation minimization cubic Catmull-Rom spline

收稿日期: 2018-08-14 出版日期: 2019-03-25

CLC:

TP391

基金资助: 国家自然科学基金资助项目(61472135); 湖南省自然科学基金资助项目(2017JJ3124)；湖南省教育厅资助科研项目（18A415）.

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引用本文:

李军成, 刘成志, 易叶青. 优化张力参数与边界条件的平面三次Cardinal样条[J]. 浙江大学学报（理学版）, 2019, 46(2): 164-171.

Juncheng LI, Chengzhi LIU, Yeqing YI. Planar cubic Cardinal spline with tension parameter and boundary condition optimization. Journal of Zhejiang University (Science Edition), 2019, 46(2): 164-171.

链接本文:

https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2019.02.004 或 https://www.zjujournals.com/sci/CN/Y2019/V46/I2/164

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