h-Said-Ball基与h-Said-Ball曲线

h-Said-Ball bases and h-Said-Ball curves

h-Bezier曲线是Bezier曲线基于h-微积分意义下的推广模型.为增强Said-Ball曲线的造型能力,提高h-Bezier曲线递归求值速度,该文提出任意次的h-Said-Ball基函数,构造了h-Said-Ball曲线.通过分析Said-Ball曲线递归求值算法与Bezier曲线的转化关系,结合h-Bezier曲线的递归求值算法和h-Bernstein基函数的构造方式,得到任意次h-Said-Ball基函数的表达式.h-Said-Ball基具有非负,单位分解,端点插值等优良性质,和h-Bernstein基之间存在显式转换矩阵.进一步,定义h-Said-Ball曲线并分析其基本性质,推导递归求值算法和包络表示,h-Said-Ball曲线的求值计算量是h-Bezier曲线的一半.借助从h-Said-Ball曲线到h-Bezier曲线的割角算法,证明了h-Said-Ball基是全正基,从而h-Said-Ball曲线具有变差缩减性和保凸性.数值实例显示了h-Said-Ball曲线相比Said-Ball曲线的造型优势和灵活性.

The h-Bezier curve is a generalized model of Bezier curve based on the sense of h-calculus.In order to enhance the modeling ability of Said-Ball curve and improve the speed of recursive valuation of h-Bezier curve,this paper proposes the h-Said-Ball basis function of arbitrary order and constructs the h-Said-Ball curve.By analyzing the transformation relationship between the recursive valuation algorithm of Said-Ball curve and Bezier curve,combining the recursive valuation algorithm of h-Bezier curve and the construction method of h-Bernstein basis function,the expressions of arbitrary times of h-Said-Ball basis function are obtained.The h-Said-Ball basis has excellent properties such as non-negativity,unit decomposition,and endpoint interpolation,and there is an explicit transformation matrix between it and the h-Bernstein basis.Further,the h-Said-Ball curve is defined and its basic properties are analyzed,and the recursive valuation algorithm and envelope representation are derived.h-Said-Ball curve is half the computational effort of the h-Bezier curve.With the help of the corner cutting algorithms from the h-Said-Ball curve to the h-Bezier curve,it is shown that the h-Said-Ball basis is a fully positive basis,and thus the h-Said-Ball curve has variational reduction and convexity preservation.Numerical examples show the modeling advantages andflexibility of the h-Said-Ball curve over the Said-Ball curve.