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
QQ在线