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h-Said-Ball基与h-Said-Ball曲线

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h-B´ezier曲线是B´ezier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-B´ezier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B´ezier曲线的转化关系,结合h-B´ezier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-B´ezier曲线的一半.借助从h-Said-Ball曲线到h-B´ezier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.
h-Said-Ball bases and h-Said-Ball curves
The h-B´ezier curve is a generalized model of B´ezier 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-B´ezier 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 B´ezier curve,combining the recursive valuation algorithm of h-B´ezier 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-B´ezier curve.With the help of the corner cutting algorithms from the h-Said-Ball curve to the h-B´ezier 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 and flexibility of the h-Said-Ball curve over the Said-Ball curve.

h-B´ezier curveSaid-Ball curveh-Said-Ball basesh-Said-Ball curvetotally positive basisrecursive evaluation algorithm

刘婉柔、解滨、韩力文

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河北师范大学数学科学学院,河北石家庄 050024

河北师范大学计算机与网络空间安全学院,河北石家庄 050024

河北省计算数学与应用重点实验室,河北石家庄 050024

河北省数学与交叉科学国际联合研究中心,河北石家庄 050024

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h-B´ezier曲线 Said-Ball曲线 h-Said-Ball基函数 h-Said-Ball曲线 全正基 递归求值算法

国家自然科学基金河北省中央引导地方科技发展资金项目河北省研究生创新项目基金

62076088236Z0104Gxycxzz2023005

2024

高校应用数学学报
浙江大学 中国工业与应用数学学会

高校应用数学学报

CSTPCD北大核心
影响因子：0.396
ISSN：1000-4424
年,卷(期)：2024.39(3)