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浙江大学学报（理学版）
浙江大学学报（理学版）  2021, Vol. 48 Issue (2): 159-166    DOI: 10.3785/j.issn.1008-9497.2021.02.004
图形计算     
优化端点条件的平面二次均匀B样条插值曲线
李军成, 刘成志, 赵文才
湖南人文科技学院 数学与金融学院，湖南 娄底 417000
Planar quadratic uniform B-spline interpolation curve with optimized endpoint condition
LI Juncheng, LIU Chengzhi, ZHAO Wencai
College of Mathematics and Finance, Hunan University of Humanities ,Science and Technology, Loudi 417000, Hunan Province, China
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摘要： 在利用反求法构造B样条插值曲线时，往往需要选取端点条件。 因此，可对端点条件进行优化选取，使得构造的B样条插值曲线满足特定要求。提出了一种利用曲线内能极小选取平面二次均匀B样条插值曲线端点条件的算法。首先给出了二次均匀B样条插值曲线分控制顶点与首个控制顶点（即端点条件）的递推关系式；然后给出了利用曲线内能极小优化选取首个控制顶点的算法，证明了利用该算法构造的C1连续二次均匀B样条插值曲线为保形插值，并通过数值算例证明了算法的有效性；最后，为便于实际应用，基于MATLAB平台设计了算法所对应的图形用户界面，用户通过简单的操作即可获得光顺的C1连续二次均匀B样条保形插值曲线。
关键词： 内能极小;  二次均匀B样条;  插值曲线;  保形插值    
Abstract: In the construction of B-spline interpolation curve by reverse method,it is often necessary to set the endpoint conditions. The endpoint conditions can be optimized so that the constructed B-spline interpolation curve can meet some specific requirements.An algorithm for selecting the endpoint conditions of plane quadratic uniform B-spline interpolation curve by minimizing the internal energy is proposed.Firstly,the recurrence relation between other control points and the first control point (viz.the endpoint condition) is given.Then,an algorithm for choosing the first control point by minimizing the internal energy of the curve is proposed. It is proved that the quadratic uniform B-spline interpolation curve with C1 continuity constructed by the proposed algorithm is shape-preserving.Some numerical examples are presented to show the effectiveness of the proposed algorithm.Finally,to facilitate the practical application,the corresponding graphical user interface of the proposed algorithm is designed based on MATLAB platform. Users can generate the shape-preserving smooth C1 quadratic uniform B-spline interpolation curve by simple operation.
Key words: internal energy minimization    quadratic uniform B-spline    shape-preserving interpolation    interpolation curve
收稿日期: 2020-03-19 出版日期: 2021-03-18
CLC:  TP  
基金资助: 湖南省教育厅资助科研项目(18A415;18C877);湖南省自然科学基金资助项目(2017JJ3124).
作者简介: 李军成(1982—)，ORCID:http://orcid.org/0000-0002-1904-4068，男，博士，教授，主要从事计算机辅助几何设计及其应用研究，E-mail：lijuncheng82@126.co；
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引用本文:

李军成, 刘成志, 赵文才. 优化端点条件的平面二次均匀B样条插值曲线[J]. 浙江大学学报（理学版）, 2021, 48(2): 159-166.

LI Juncheng, LIU Chengzhi, ZHAO Wencai. Planar quadratic uniform B-spline interpolation curve with optimized endpoint condition. Journal of Zhejiang University (Science Edition), 2021, 48(2): 159-166.

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https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2021.02.004    或    https://www.zjujournals.com/sci/CN/Y2021/V48/I2/159

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