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A polynomial basis with a shape parameter for curve and surface modeling

Author

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  • Nouri, Bahareh
  • Juhász, Imre
  • Saeidian, Jamshid

Abstract

Based on Bernstein polynomials, a system of functions with a free parameter is proposed in the space of polynomials of degree at most n. The system inherits several properties of Bernstein polynomials, such as linear independence, non-negativity, partition of unity and symmetry. This new family of functions are employed to construct control point based parametric curves. The free parameter serves as a shape adjustment parameter, by means of which a one-parameter family of polynomial curves is obtained. The new family of curves is in common with Bézier curves in most of the geometric properties, providing a smooth transition between the Bézier curve and the straight line segment joining the first and last control points. Shape preserving properties, such as monotonicity preservation, as well as length, hodograph and variation diminishing are studied. The proposed basis can also be used to create tensor product surfaces. The extent to which the suggested basis generation method can be applied to other (non-polynomial) function spaces is also being investigated.

Suggested Citation

  • Nouri, Bahareh & Juhász, Imre & Saeidian, Jamshid, 2025. "A polynomial basis with a shape parameter for curve and surface modeling," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 229(C), pages 690-705.
    • Handle: RePEc:eee:matcom:v:229:y:2025:i:c:p:690-705
    DOI: 10.1016/j.matcom.2024.10.029
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    References listed on IDEAS

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    1. Sidra Maqsood & Muhammad Abbas & Gang Hu & Ahmad Lutfi Amri Ramli & Kenjiro T. Miura, 2020. "A Novel Generalization of Trigonometric Bézier Curve and Surface with Shape Parameters and Its Applications," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-25, May.
    2. Salma Naseer & Muhammad Abbas & Homan Emadifar & Samia Bi Bi & Tahir Nazir & Zaheer Hussain Shah & Muhammad Aslam, 2021. "A Class of Sextic Trigonometric Bézier Curve with Two Shape Parameters," Journal of Mathematics, Hindawi, vol. 2021, pages 1-16, June.
    3. Ammad, Muhammad & Misro, Md Yushalify & Ramli, Ahmad, 2022. "A novel generalized trigonometric Bézier curve: Properties, continuity conditions and applications to the curve modeling," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 744-763.
    4. Muhammad Ammad & Md Yushalify Misro & Muhammad Abbas & Abdul Majeed, 2021. "Generalized Developable Cubic Trigonometric Bézier Surfaces," Mathematics, MDPI, vol. 9(3), pages 1-17, January.
    5. Moavia Ameer & Muhammad Abbas & Thabet Abdeljawad & Tahir Nazir, 2022. "A Novel Generalization of Bézier-like Curves and Surfaces with Shape Parameters," Mathematics, MDPI, vol. 10(3), pages 1-19, January.
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