A New and Efficient Algorithm for Detecting Affine Equivalences and Symmetries of Ruled Rational Surfaces

Yıl 2024, Cilt: 24 Sayı: 3, 574 - 582, 27.06.2024

Hüsnü Anil Çoban , Uğur Gözütok

https://doi.org/10.35414/akufemubid.1374733

Öz

A novel and efficient method for computing all affine equivalences between two ruled surfaces using rational parameterizations is presented in this paper. The technique essentially uses affine differential invariants to find equivalences in contrast to other methods, our method does not require polynomial system solutions, which complicate computations and increase computation time. The performance of the method was thoroughly tested using the Maple^TM (2021) computer algebra system after it had been transformed into an algorithm.

Anahtar Kelimeler

Ruled surfaces, Affine equivalence, Affine symmetry, Algorithm

Rasyonel Regle Yüzeylerin Afin Denklikleri ve Simetrilerinin Tespiti Üzerine Yeni ve Etkili Bir Algoritma

Öz

Bu çalışmada, rasyonel parametrizasyonlar yardımıyla verilen iki regle yüzey arasındaki tüm afin denklikleri hesaplayan yeni ve etkili bir yöntem sunulmaktadır. Yöntem, temelde, afin diferansiyel invaryantları kullanılarak denklikleri tespit etmektedir. Bu sayede, benzer problemi çözen metotlardan farklı olarak hesaplamayı zorlaştıran ve hesaplama süresinin uzamasına sebep olan polinom sistemi çözümlerine ihtiyaç duymamaktadır. Bu yöntem bir algoritma haline getirilmiş ve yöntemin performansı Maple^TM (2021) bilgisayar cebir sistemi kullanılarak, geniş kapsamlı testlerle incelenmiştir.

Anahtar Kelimeler

Regle yüzeyler, Afin denklik, Afin simetri, Algoritma

Kaynakça

• Alcázar, J.G., Hermoso, C. and Muntingh, G., 2015. Symmetry detection of rational space curves from their curvature and torsion. Comput. Aided. Geom. Design, 33, 51-65. https://doi.org/10.1016/j.cagd.2015.01.003
• Alcázar, J.G. and Hermoso, C., 2016. Involutions of polynomially parametrized surfaces. J. Comput. Appl. Math., 294, 23-38. https://doi.org/10.1016/j.cam.2015.08.002
• Alcázar, J.G. and Goldman, R., 2017. Detecting when an implicit equation or a rational parametrization defines a conical or cylindrical surface, or a surface of revolution. IEEE Trans. Vis. Comput. Graphics, 12 (23), 2550-2559. https://doi.ieeecomputersociety.org/10.1109/tvcg.2016.2625786
• Alcázar, J.G. and Quintero, E., 2020. Affine equivalences, isometries and symmetries of ruled rational surfaces. Journal of Computational and Applied Mathematics, 364, 112339. https://doi.org/10.1016/j.cam.2019.07.004
• Alcázar, J.G., Gözütok, U., Çoban, H.A. and Hermoso, C., 2022. Detecting affine equivalences between implicit planar algebraic curves. Acta Applicandae Mathematicae, 2 (182), 23-38. https://doi.org/10.1007/s10440-022-00539-1
• Bizzarri, M., Làvi˘cka, M., Vr˘sek, J., 2020. Computing projective equivalences of special algebraic varieties. J. Comput. Appl. Math, 367, 112438. https://doi.org/10.1016/j.cam.2019.112438
• Chen, F., Zheng J. and Sederberg, T.W., 2001. The μ-basis of a rational ruled surface. Comput. Aided. Geom. Design, 18, 61-72. https://doi.org/10.1016/S0167-8396(01)00012-7
• Chen, F., Wang W., 2003. Revisiting the μ-basis of a rational ruled surface. J. Symbolic Comput, 36, 699-716. https://doi.org/10.1016/S0747-7171(03)00064-6
• Gözütok, U., Çoban, H.A., Sağıroğlu, Y., Alcázar, J.G., 2023. A new method to detect projective equivalences and symmetries of rational 3D curves. Journal of Computational and Applied Mathematics, 419, 114782. https://doi.org/10.1016/j.cam.2022.114782
• Hauer, M., Jüttler, B., 2018. Projective and affine symmetries and equivalences of rational curves in arbitrary dimension. J. Symbolic Comput., 87, 68-86. https://doi.org/10.1016/j.jsc.2017.05.009
• Hauer, M., Jüttler, B. and Schicho, J., 2018. Projective and affine symmetries and equivalences of rational and polynomial surfaces. J. Comput. Appl. Math., 349, 424-437. https://doi.org/10.1016/j.cam.2018.06.026
• O’ Neill, B., 2006. Elementary Differential Geometry. Revised Second Eddition. NY, USA, 145-146.
• Perez-Diaz, S., Shen, L-Y., 2014. Characterization pf rational ruled surfaces. J. Symbolic. Comput., 63, 21-45. https://doi.org/10.1016/j.jsc.2013.11.003
• https://avesis.ktu.edu.tr/hacoban/dokumanlar, (10.10.2023)