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带参数的同次Ball曲线

李军成,李兵,易叶青(湖南人文科技学院数学与金融学院, 娄底 417000;四川航天技术研究院总体部, 成都 610100;湖南人文科技学院信息学院, 娄底 417000)

摘 要
目的 虽然Ball曲线具有很好的几何特性,但当控制顶点保持不变时,曲线的形状却无法进行调整,这无疑限制了其在几何造型中的应用。为了使得任意次Ball曲线在控制顶点保持不变的情形下具有形状可调性,提出了一种构造带参数的同次Ball曲线的简单方法。方法 首先通过将传统三次Ball基的定义区间由[0,1]扩展为[0,α],构造了一种带参数α的三次Ball基,并称之为三次α-Ball基;然后基于三次α-Ball基定义了相应的三次α-Ball曲线,并讨论了三次α-Ball曲线的拼接、参数对曲线的影响以及参数的3种选取方案;最后借助传统高次Ball基的递推性构造了任意次α-Ball基及其对应的α-Ball曲线,并给出了任意次α-Ball基与α-Ball曲线的性质。结果 实例表明,所构造的α-Ball曲线是传统Ball曲线的同次扩展,不仅保留了传统Ball曲线的性质,而且还由于带有参数α使得曲线具有更好的表现能力。利用所给出的3种参数选取方案可构造出满足相应要求的α-Ball曲线。结论 所提出的α-Ball曲线克服了传统Ball曲线在形状调整方面的不足,是一种构造形状可调的任意次Ball曲线的有效方法。
关键词
Ball curve of the same degree with a parameter

Li Juncheng,Li Bing,Yi Yeqing(College of Mathematics and Finances, Hunan University of Humanities, Science and Technology, Loudi 417000, China;The General Design Department, Sichuan Academy of Aerospace Technology, Chengdu 610100, China;College of Information, Hunan University of Humanities, Science and Technology, Loudi 417000, China)

Abstract
Objective The Ball curve has excellent geometric properties. However, its shape cannot be adjusted when the control points remain unchanged. This condition undoubtedly limits its application in geometrical modeling. A simple method for constructing the Ball curve of the same degree with a parameter is presented to enable the Ball curve with arbitrary degree to obtain shape adjustment capability under fixed control points. Method The cubic Ball basis, referred to as cubic α-Ball basis, is constructed by extending the definition interval of the traditional cubic Ball basis from [0, 1] to [0, α]. Then, the corresponding cubic α-Ball curve is defined base on the cubic α-Ball basis. The splicing of the curves, the influence of the parameter on the curve, and the three selection schemes for the parameter are discussed. Finally, the α-Ball basis and α-Ball curve with arbitrary degree are established by the recursion of the transitional high-degree Ball basis, and the properties of the α-Ball basis and α-Ball curve with arbitrary degree are provided. Result Examples show that the proposed α-Ball curve is an extension of the same degree to the traditional Ball curve. The curve not only preserves the properties of the traditional Ball curve, such as convex hull, symmetry, geometric invariance, variance reduction, and convexity, but also has better performance because of the parameter α. The α-Ball curve can be constructed to satisfy the requirements by using the three selection schemes for the parameter, including the scheme for the curve with the shortest arc length, the curve with minimum energy, and the curve with the shortest arc length and minimum energy. Conclusion The α-Ball curve overcomes the disadvantage of the traditional Ball curve in shape adjustment, which is an effective method for constructing the shape-adjustable Ball curve with arbitrary degree.
Keywords
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