A Generalized Quasi Cubic Trigonometric Bernstein Basis Functions and Its B-Spline Form

Yunyi Fu and Yuanpeng Zhu
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Yunyi Fu: School of Mathematics, South China University of Technology, Guangzhou 510640, China
Yuanpeng Zhu: School of Mathematics, South China University of Technology, Guangzhou 510640, China

Mathematics, 2021, vol. 9, issue 10, 1-25

Abstract: In this paper, under the framework of Extended Chebyshev space, four new generalized quasi cubic trigonometric Bernstein basis functions with two shape functions ? ( t ) and ? ( t ) are constructed in a generalized quasi cubic trigonometric space span { 1 , sin 2 t , ( 1 ? sin t ) 2 ? ( t ) , ( 1 ? cos t ) 2 ? ( t ) } , which includes lots of previous work as special cases. Sufficient conditions concerning the two shape functions to guarantee the new construction of Bernstein basis functions are given, and three specific examples of the shape functions and the related applications are shown. The corresponding generalized quasi cubic trigonometric Bézier curves and the corner cutting algorithm are also given. Based on the new constructed generalized quasi cubic trigonometric Bernstein basis functions, a kind of new generalized quasi cubic trigonometric B-spline basis functions with two local shape functions ? i ( t ) and ? i ( t ) is also constructed in detail. Some important properties of the new generalized quasi cubic trigonometric B-spline basis functions are proven, including partition of unity, nonnegativity, linear independence, total positivity and C 2 continuity. The shape of the parametric curves generated by the new proposed B-spline basis functions can be adjusted flexibly.

Keywords: Extended Chebyshev (EC) space; trigonometric Bernstein basis functions; trigonometric B-spline basis functions; total positivity; corner cutting algorithm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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