Blending Tipi Bézier Yüzeylerde Alan Yaklaşımı

Abstract

Çalışmamızın odak noktası, Bézier yüzeylerinin geometrik analizidir. Blending Tipi Bézier yüzeylerinin Birinci Temel Formu ve yüzey alan elemanı EG-F^2 (8)’i inceliyoruz. Kinematik geometride temel bir yapı şeklindeki, regle yüzey bir referans modeli olarak alınmıştır. Bu özel örneğe dayanarak, yüzey alanı hesaplamalarındaki EG-F^2 integrandı açısından, Blending Tipi Bézier yüzey yaklaşımlarının Klasik Bézier yüzey yaklaşımlarından daha doğru sonuçlar sağladığı gözlemlenmiştir.

Keywords

• Tensör Çarpımı Bézier Yüzeyleri
• Blending Tipi Bézier Yüzeyleri
• I. Esas Form
• Alan Elemanı

Area Approximation on Blending Type Bézier Surfaces

Abstract

The focus of our study is the geometric analysis of Bézier surfaces. We investigate the First Fundamental Form and the surface area element EG-F^2 (8) of Blending Type Bézier surfaces. As a fundamental structure in kinematic geometry, the ruled surface has been taken as a reference model. Based on this specific example, it has been observed that the Blending Type Bézier surface approximations provide more accurate results than Classical Bézier surface approaches, in terms of the integrand EG-F^2 used in surface area computations.

Keywords

• Tensor Product Bézier Surfaces
• Blending Type Bézier Surfaces
• First Fundamental Form
• Area Element

References

1. Farin G. Curves and Surfaces for CAGD. 5th ed. Waltham: Academic Press; 2002. doi:10.1016/B978-1-55860-737-8.X5000-5.
2. Cheng HH, Zeng XM. Bézier Curves with Shape Parameter. Journal of Xiamen University 2006;45:320-322. doi:10.1007/BF02841755.
3. Kaur J, Goyal M. On α-Bézier Curves and Surfaces. Bollettino dell’unione Matematica Italiana 2022;16:459-470. doi:10.1007/s40574-022-00341-9.
4. Ye Z, Long X, Zeng XM. Adjustment Algorithms for Bézier Curve and Surface. In: The 5th Int. Conference on Computer Science & Education, Hefei, China; 2010, p. 1712-1716. doi:10.1109/ICCSE.2010.5593563.
5. Zhou G, Cai QB. Approximation Properties of (λ,μ)-Bernstein-DurrMeyer Operators. Mathematical Methods in the Applied Sciences 2024;48:5946-5953. doi:10.1002/mma.10647.
6. Cai QB, Aslan R, Özger F, Srivastava HM. Approximation by a New Stancu Variant of Generalized (λ,μ)- Bernstein Operators. Alexandra Engineering Journal 2024;107:205-214. doi:10.1016/j.aej.2024.07.015.
7. Turhan N, Özger F, Mursaleen M. Kantorich-Stancu type (α,λ,s)- Bernstein Operators and Their Approximation Properties. Mathematical and Computer Modelling of Dynamical Systems 2024;30(1):228-265. doi:10.1080/13873954.2024.2335382.
8. Karakılıç İ, Karakılıç S, Budakçı G, Özger F. Bézier Curves and Surfaces with the Blending (α,λ,s)- Bernstein Basis. Symmetry 2025;17(2):219. doi:10.3390/sym17020219.

9. Kahn K, Lobiyal DK, Kilicman A. Bézier Curves and Surfaces Based on Modified Bernstein Polynomials. Azerbaijan Journal of Mathematics 2019;9(1):3-21.
10. Cai QB, Lian BY, Zhou G. Approximation Properties of λ-Bernstein Operators. J Inequal Appl 2018;61. doi:10.1186/s13660-018-1653-7.
11. Gezer H, Aktuğlu H, Baytunç E, Atamert MS. Generalized Blending Type Bernstein Operators Based on the Shape Parameter λ. J Inequal Appl 2022;96. doi:10.1186/s13660-022-02832-x.
12. Presley A. Elementary Differential Geometry. Springer Undergraduate Mathematics Series; 2010. doi:10.1007/978-1-84882-891-9.