A Shape Preserving Class of Two-Frequency Trigonometric B-Spline Curves

IF 2.2 3区 综合性期刊 Q2 MULTIDISCIPLINARY SCIENCES Symmetry-Basel Pub Date : 2023-11-10 DOI:10.3390/sym15112041
Gudrun Albrecht, Esmeralda Mainar, Juan Manuel Peña, Beatriz Rubio
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Abstract

This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space U4(Iα)=span{1,cost,sint,cos2t,sin2t} defined on compact intervals Iα=[0,α], where α is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space U4(Iα) to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as α→0. As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of C(3 Hermite interpolation problems.
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一类双频三角b样条曲线的保形性
本文提出了一种定义具有有趣保形特性的两频三角样条曲线的新方法。这种构造需要空间U4(Iα)=span{1,cost,sint,cos2t,sin2t}的归一化b基,定义在紧区间Iα=[0,α]上,其中α是一个全局形状参数。将证明归一化的b基在三角空间U4(Iα)中可视为伯恩斯坦多项式基的等价,并具有其众所周知的对称性。事实上,当α→0时,归一化b基函数收敛于Bernstein多项式。因此,所得到的分段三角曲线收敛于一致的四次b样条曲线也将得到证明。所提出的三角样条曲线可用于凸轮设计、轨迹生成、球面上的数据拟合,甚至可以定义新的代数-三角毕达哥拉斯-霍图曲线及其分段对应曲线,从而解决C(3) Hermite插值问题。
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来源期刊
Symmetry-Basel
Symmetry-Basel MULTIDISCIPLINARY SCIENCES-
CiteScore
5.40
自引率
11.10%
发文量
2276
审稿时长
14.88 days
期刊介绍: Symmetry (ISSN 2073-8994), an international and interdisciplinary scientific journal, publishes reviews, regular research papers and short notes. Our aim is to encourage scientists to publish their experimental and theoretical research in as much detail as possible. There is no restriction on the length of the papers. Full experimental and/or methodical details must be provided, so that results can be reproduced.
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