Journals / Uşak Üniversitesi Fen ve Doğa Bilimleri Dergisi / 2026 / Cilt: 10 Sayı: 1

Geometric Modeling of Queen Alexandra Birdwing with the Help of Bezier Curves

Kraliçe Alexandra Kuş Kanatlısının Bezier Eğrileri Yardımıyla Geometrik Modellemesi

Betül Bulca Sokur(Bursa Uludağ Üniversitesi)0000-0001-5861-0184
Nisanur Aktaş(Bursa Uludağ Üniversitesi)0009-0000-0294-314X
Pages
—
Publication Date
2026-10-02
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AI Summary & Key Takeaways

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

During this modeling process, models were generated using both polynomial and trigonometric Bézier curves, and a comparison between them is provided.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this study, Bézier curves, which play an important role in Computer-Aided Geometric Design (CAGD), are discussed.
📊
Key Findings & Data:Geometric modeling using these curves is significant in many fields, and within this study, the geometric modeling of both the male and female forms of the butterfly species Queen Alexandra's birdwing (Ornithoptera alexandrae) is presented.
💡
Conclusion & Impact:During this modeling process, models were generated using both polynomial and trigonometric Bézier curves, and a comparison between them is provided.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Bezier eğrisi · Trigonometrik Bezier eğrisi · Kraliçe Alexandra kuş kanatlısı

Abstract

In this study, Bézier curves, which play an important role in Computer-Aided Geometric Design (CAGD), are discussed. Polynomial Bézier curves and Bézier-like curves defined using trigonometric functions are examined. Geometric modeling using these curves is significant in many fields, and within this study, the geometric modeling of both the male and female forms of the butterfly species Queen Alexandra's birdwing (Ornithoptera alexandrae) is presented. During this modeling process, models were generated using both polynomial and trigonometric Bézier curves, and a comparison between them is provided.

Özet

Bu çalışmada, Bilgisayar Destekli Geometrik Tasarım (CAGD) için önemli rol oynayan Bezier eğrileri ele alınmıştır. Bu çalışmada, polinom Bezier eğrileri ve trigonometrik fonksiyonlar yardımıyla tanımlanan Bezier benzeri eğriler ele alınmıştır. Bu eğriler yardımıyla yapılan geometrik modellemeler birçok alanda önemli olup bu çalışma içinde bir kelebek türü olan Kraliçe Alexandra (Ornithoptera Alexandra) kuş kanatlısının dişi ve erkek türleri için geometrik modellemesi verilmiştir. Bu modelleme yapılırken polinom ve trigonometrik Bezier eğrileri yardımıyla modeller elde edilmiş ve karşılaştırmaları verilmiştir.

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Cite This Article

Sokur, B. B., Aktaş, N. (2026). Geometric Modeling of Queen Alexandra Birdwing with the Help of Bezier Curves. Uşak Üniversitesi Fen ve Doğa Bilimleri Dergisi, Cilt: 10 Sayı: 1. https://doi.org/10.47137/usufedbid.1816664

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References (41)

  1. Levent A. Bezier yüzeyleri ve kıvrımlı yüzeyler, Yüksek Lisans Tezi, İnönü Üniversitesi Fen Bilimleri Enstitüsü, Malatya, Türkiye, 2013.
  2. Farin G. Curves and surfaces for CAGD: A Practical Guide, Elsevier Science; 2002.
  3. Laurent PJ and Sablonniere. P. Pierre Bezier: An engineer and a mathematician, Computer Aided Geometric Design, 2001;18:609–617.
  4. Rabut, C. On Pierre Bezier’s life and motivations, Computer Aided Design, 2002;34:493–510.
  5. Rogers, DF. Pierre Etienne Bezier (1910–1999), in memoriam, Computer Aided Design, 2002;34:489-491.
  6. Böhm M. Ornithoptera alexandrae, The IUCN Red List of Threatened Species, 2018.
  7. Marsh D. Applied geometry for computer graphics and CAD. 2nd edition. Springer-Verlag, London; 2005.
  8. Forrest AR. Curves and surfaces for computer aided design, PhD Thesis, University of Cambridge, Department of Mathematics, Cambridge, England, 1968.
  9. Bezier P. Example of an existing system in the motor industry: The Unisurf system, Proceedings of the Royal Society of London, 1971;321(1545):207–218.
  10. Kaplan CS and Mann S. Introduction to computer graphics, Lecture notes, University of Waterloo, 2006.
  11. David S. Curves and surfaces for computer graphics, Springer Science+Business Media. Inc., USA; 2006.
  12. Ameer M, Abbas M, Shafig M, Nazir T and Birhanu A. Generalized Bezier-like model and its applications to curve and surface modeling, Plos One, 2024;19(6):1–38.
  13. Ameer M, Abbas M, Abdeljawad T and Nazir T. A novel generalization of Bezier-like curves and surfaces with shape parameters, Mathematics, 2022;10(3):1–19.
  14. Ameer M, Abbas M, Miura K, Majeed A and Nazir T. Curve and surface geometric modeling via generalized Bezier-like model, Mathematics, 2022;10(7):1–14.
  15. Hu G, Wu J and Qin X. A novel extension of the Bezier model and its applications to surface modeling, Advances in Engineering Software, 2018;125:27–54.
  16. Li J. A novel Bezier curve with a shape parameter of the same degree, Results in Mathematics, 2018;73(4):1–11.
  17. Qin X, Hu G, Zhang N, Shen X and Yang Y. A novel extension to the polynomial basis functions describing Bezier curves and surfaces of degree n with multiple shape parameters, Applied Mathematics and Computation, 2013;223:1–16.
  18. Yan L. Adjustable Bezier curves with simple geometric continuity conditions, Mathematical and Computational Applications, 2016;21(4):1–11.
  19. Yan L and Liang J. An extension of the Bezier model, Applied Mathematics and Computation, 2011;18(6):2863–2879.
  20. Yang L and Zeng XM. Bézier curves and surfaces with shape parameters, International Journal of Computer Mathematics, 2009;86(7):1253–1263.
  21. Zhu Y, Han X and Liu S. Curve construction based on four αβ-Bernstein-like basis functions, Journal of Computational and Applied Mathematics, 2015;273:160–181.
  22. Nouri B and Saeidian J. On a class of Bezier-like model for shape- preserving approximation, Iranian Journal of Numerical Analysis and Optimization, 2022;12(2):449–466.
  23. Han XA, Ma YC and Huang XL. The cubic trigonometric Bezier curve with two shape parameters, Applied Mathematics Letters, 2009;22:226–231.
  24. Maqsood S, Abbas M, Miura KT, Majeed A and Iqbal A. Geometric modeling and applications of generalized blended trigonometric Bezier curves with shape parameters, Advances in Difference Equations, 2020;550:1–18.
  25. Maqsood S, Abbas M, Hu G, Ramli A and Miura KT. A novel generalization of trigonometric Bezier curve and surface with shape parameters and its applications, Mathematical Problems in Engineering, 2020;4036434:1–25.
  26. Ammad M, Misro MY and Ramli A. A novel generalized trigonometric Bezier curve: properties, continuity conditions and applications to the curve modelling, Mathematics and Computers in Simulation, 2022;194:744–763.
  27. Saeidian J, Nouri B and Azizi A. A Bernstein-like trigonometric basis: properties, curve design, and operator construction, Journal of Applied Mathematics, 2025;5676548:1-12.
  28. Misro MY, Ramli A and Ali JM. Quintic trigonometric Bezier curve with two shape parameters, Sains Malaysiana, 2017; 46(5):825–831.
  29. Kaur H and Goyal MR. New generalized blended trigonometric Bezier curves with one shape parameter, Filomat, 2024; 38(2):705–725.
  30. Meek AS. Letter to Karl Jordan dated 3rd Feb 1906 from “Above “Biagi” (5000 ft) Head of Mambare River”, Papua New Guinea, Meek letter 155, BMNH Archives, 1906.
  31. Tennent J. The man who shot butterflies. (Albert Stewart Meek 1871–1943, Naturalist and Explorer), Storm Entomological Publications, Oxford; 2021.
  32. Sourakov A and Sourakov AAT. The little book of butterflies, Princeton University Press, Princeton; 2024.
  33. Parsons MJ. The world’s largest butterfly endangered: the ecology, status and conservation of Ornithoptera alexandrae (Lepidoptera: Papilionidae), Tropical Lepidoptera Research, 1992;3(1):33–60.
  34. Mitchell DK, Dewhurst CF, Tennent WJ and Page WW. Queen Alexandra’s birdwing butterfly Ornithoptera alexandrae (Rothschild, 1907): A review and conservation proposals, Kuala Lumpur, Southdene Sdn Bhd; 2016.
  35. Reboud, E.L., Nabholz, B. Chevalier, E. Tilak, M., Bito, D. & Condamine, F.L. Genomics, population divergence, and historical demography of the world’s largest and endangered butterfly, the Queen Alexandra’s birdwing, Genome Biology and Evolution, 2023;15(4):1–19.
  36. Anand V.B. Computer graphics and geometric modeling for engineers. 1st edition. John Wiley and Sons. Inc., New York; 1992.
  37. De Casteljau P. Mathematiques et CAO Tome 2: Formes a poles, Hermes; 1986.
  38. Majeed A, Naureen M, Abbas M and Miura K. Construction of cubic trigonometric curves with an application of curve modelling, Mathematics, 2022; 10(7):1087.
  39. Sharma R. Cubic trigonometric Bezier curve with shape parameter, International Journal of Innovative Research in Computer and Communication Engineering, 2016;4(4):7718–7723.
  40. Adnan SBZ, Ariffin AAM and Misro MY. Curve fitting using quintic trigonometric Bezier curve, AIP Conference Proceedings, 2020;2266(1):040009.
  41. Misro MY, Ramli A, Ali JM and Hamid NNA. Pythagorean hodograph quintic trigonometric Bezier transtion curve, 14th International Conference on Computer Graphics, Imaging and Visualization; 2017. p.