SavedAdd your journal

Journals / Turkish Journal of Mathematics and Computer Science / 2026 / Volume: 18 Issue: 2

Design of B´ezier Curves of Some Surfaces with Matlab Applications

Ceyda Yılmaz Luzum(VAN YUZUNCU YIL UNIVERSITY)0009-0003-9441-9020
Şenay Baydaş(VAN YUZUNCU YIL UNIVERSITY)0000-0002-1026-9471
Bülent Karakaş(BARTIN UNIVERSITY)0000-0002-3915-6526
Pages
—
Publication Date
2026-06-30
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

In addition, the theory is exemplified using the Matlab program.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:B´ezier curves are special types of curves defined by control points.
📊
Key Findings & Data:When the control points of a B´ezier curve are the points of a surfacein $\mathbb{R}^3$, the B´ezier curve will be obtained depending on the surface as well as the control points.
💡
Conclusion & Impact:In addition, the theory is exemplified using the Matlab program.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Bernstein polynomials · B´ezier curves · De Casteljau algorithms

Abstract

B´ezier curves are special types of curves defined by control points. The large number of control points naturally affects the form of the B´ezier curve. When the control points of a B´ezier curve are the points of a surfacein $\mathbb{R}^3$, the B´ezier curve will be obtained depending on the surface as well as the control points. As the number of points on the surface increases, the obtained B´ezier curve will approach the limit of positioning on the surface. The study examines this approach. In addition, the theory is exemplified using the Matlab program.

🔖

Cite This Article

Luzum, C. Y., Baydaş, Ş., Karakaş, B. (2026). Design of B´ezier Curves of Some Surfaces with Matlab Applications. Turkish Journal of Mathematics and Computer Science, Volume: 18 Issue: 2. https://doi.org/10.47000/tjmcs.1584879

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (30)

  1. Ameer, M., Abbas, M., Shafiq, M., Nazir, T., Birhanu, A., Generalized B´ezier-like model and its applications to curve and surface modeling, PLoS ONE, 19(6)(2024), e0303340.
  2. Bulut, V., Design of optimal quasi-developable surfaces using B´ezier surface representations, Ann. Math. Artif. Intell., 93(2024), 611–631.
  3. B´ezier, P., Example of an existing system in the motor industry: the Unisurf system, Proc. R. Soc. Lond. A., 321(1971), 207-218.
  4. B´ezier, P., Sioussiou, S., Semi-automatic system for defining free-form curves and surfaces, Comput. Aided Des., 15(1983), 65-72.
  5. Borden, M.J., Scott, M.A., Evans, J.A., Hughes, T.J.R., Isogeometric finite element data structures based on B´ezier extraction, International Journal for Numerical Methods in Engineering, 87(1–5)(2016), 15–47.
  6. Baydaş, Ş., Karakaş, B., Defining a curve as a B´ezier curve, J. Taibah Univ. Sci., 13(2019), 522-528.
  7. Chen, X.,Wang, G., Shape adjustable B´ezier curves and surfaces and their applications, Applied Mathematical Modelling, 77(2020), 491–505.
  8. De Casteljau, P., POLynomials, POLar forms, and interPOLation, In: Lyche, T., Schumaker, L.L. (Eds.), Mathematical Methods in Computer Aided Geometric Design II, Academic Press, (1992), 57–68.
  9. Deng, J., Chen, F., Matrix representations and efficient computation of B´ezier and spline surfaces, Graphical Models, 90(2017), 1–12.
  10. Du, X., Zhao, G., Wang, W., Guo, M., Zhang, R., Yang, J., Triangular and quadrilateral B´ezier discretizations of trimmed CAD surfaces and its application to the isogeometric analysis, Comput. Aided Des., 18(4)(2020), 738–759.
  11. Du, X. et al., NLIGA: A MATLAB framework for nonlinear isogeometric analysis, Computer Aided Geometric Design, 80(2020), p. 101869.
  12. Farin, G., Algorithms for rational B´ezier curves, Comput. Aided Des., 15(1983), 73–77.
  13. Farin, G., Curves and Surfaces for CAGD a Practical Guide, Fifth edition, Morgan Kaufmann, U.S.A., 2002.
  14. Hu, G., Cao, H., A class of generalized B´ezier curves with multiple shape parameters, Journal of Computational and Applied Mathematics, 356(2019), 1–14.
  15. Joy, K. I., On-line Geometric Modeling Notes, Bernstein Polynomials, CAGDNotes, 2000.
  16. Karakılıç, I., Karakılıç, S., Budakçı, G., Ozger, F., B´ezier curves and Surfaces with the blending (α, λ, s)-Bernstein basis, Symmetry, 17(2)(2025), 219.
  17. Kaplan, C. S., Mann, S., Introduction to Computer Graphics. Lecture Notes CS 488/688 of University of Waterloo, 2006.
  18. Kılıçoğlu, S¸., Matrix representation of bicubic B´ezier surfaces in E3, DergiPark Journal, 2026.
  19. Laurent, P.J., Sablonniere, P., Pierre B´ezier: An engineer and a mathematician, Comput. Aided Des., 18(2001), 609–617.
  20. Lee, C., Kim, D., Park, S., MATLAB implementation of B´ezier and NURBS-based geometric modeling for engineering analysis, Mathematics, 9(18)(2021), Article 2241.
  21. Li, Y., Zhang, J., Fairing of B´ezier surfaces using curvature-based energy minimization, Computer-Aided Design, 102(2018), 1–12.
  22. Marsh, D., Applied Geometry for Computer Graphics and CAD., Springer-Verlag, London, 2005.
  23. Nguyen, V.P., Anitescu, C., Bordas, S.P.A., Rabczuk, T., An open-source MATLAB framework for isogeometric analysis of plates and shells, Engineering with Computers, 34(3)(2018), 541–560.
  24. Rabut, C., On Pierre B´ezier’s life and motivations, Comput. Aided Des., 34(2002), 493–510.
  25. Rogers, D.F., Pierre Etienne B´ezier (1910-1999), In Memoriam, Comput. Aided Des., 34(2002), 489–491.
  26. Steffens, K.G., The History of Approximation Theory, Birkhauser, Boston, U.S.A., 2006.
  27. Sun, Y., Wang, W., B´ezier surface parameterization for complex geometric modeling, Advances in Engineering Software, 168(2022), Article 103123.
  28. V´azquez, R. et al., A MATLAB/Octave framework for isogeometric analysis, Computers & Mathematics with Applications, 72(3)(2016), 523–554.
  29. Zhang, X. et al., B-ITO: A MATLAB toolbox for isogeometric topology optimization with B´ezier extraction of NURBS, Advances in Engineering Software, (2024).
  30. Kılıçoğlu, S¸., Matrix representation of bicubic B´ezier surfaces in E3, DergiPark Journal, 2026.