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Received：10 November 2017，
Revised：2017-12-26，
Published：16 June 2018
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目的
2
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
2
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
2
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
2
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。
Objective
2
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
2
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
2
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
2
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.
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