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

Author

Listed:
  • 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)

Abstract

In this paper, under the framework of Extended Chebyshev space, four new generalized quasi cubic trigonometric Bernstein basis functions with two shape functions α ( t ) and β ( t ) are constructed in a generalized quasi cubic trigonometric space span { 1 , sin 2 t , ( 1 − sin t ) 2 α ( t ) , ( 1 − cos t ) 2 β ( 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 α i ( t ) and β 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.

Suggested Citation

  • Yunyi Fu & Yuanpeng Zhu, 2021. "A Generalized Quasi Cubic Trigonometric Bernstein Basis Functions and Its B-Spline Form," Mathematics, MDPI, vol. 9(10), pages 1-25, May.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:10:p:1154-:d:558500
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    References listed on IDEAS

    as
    1. Yuanpeng Zhu & Zhuo Liu, 2019. "A Class of Trigonometric Bernstein-Type Basis Functions with Four Shape Parameters," Mathematical Problems in Engineering, Hindawi, vol. 2019, pages 1-16, April.
    2. Kai Wang & Guicang Zhang, 2018. "New Trigonometric Basis Possessing Denominator Shape Parameters," Mathematical Problems in Engineering, Hindawi, vol. 2018, pages 1-25, October.
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