New Trigonometric Basis Possessing Exponential Shape Parameters

New Trigonometric Basis Possessing Exponential Shape Parameters

Year:    2015

Author:    Yuanpeng Zhu, Xuli Han

Journal of Computational Mathematics, Vol. 33 (2015), Iss. 6 : pp. 642–684

Abstract

Four new trigonometric Bernstein-like basis functions with two exponential shape parameters are constructed, based on which a class of trigonometric Bézier-like curves, analogous to the cubic Bézier curves, is proposed. The corner cutting algorithm for computing the trigonometric Bézier-like curves is given. Any arc of an ellipse or a parabola can be represented exactly by using the trigonometric Bézier-like curves. The corresponding trigonometric Bernstein-like operator is presented and the spectral analysis shows that the trigonometric Bézier-like curves are closer to the given control polygon than the cubic Bézier curves. Based on the new proposed trigonometric Bernstein-like basis, a new class of trigonometric B-spline-like basis functions with two local exponential shape parameters is constructed. The totally positive property of the trigonometric B-spline-like basis is proved. For different values of the shape parameters, the associated trigonometric B-spline-like curves can be $C^2$ ∩ $FC^3$ continuous for a non-uniform knot vector, and $C^3$ or $C^5$ continuous for a uniform knot vector. A new class of trigonometric Bézier-like basis functions over triangular domain is also constructed. A de Casteljau-type algorithm for computing the associated trigonometric Bézier-like patch is developed. The conditions for $G^1$ continuous joining two trigonometric Bézier-like patches over triangular domain are deduced.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1509-m4414

Journal of Computational Mathematics, Vol. 33 (2015), Iss. 6 : pp. 642–684

Published online:    2015-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    43

Keywords:    Trigonometric Bernstein-like basis Trigonometric B-spline-like basis Corner cutting algorithm Totally positive property Shape parameter Triangular domain.

Author Details

Yuanpeng Zhu

Xuli Han