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C^2 de Rasyonel Cebirsel Eğrilerin İzometri ve Simetrilerinin Hesaplanması

Year 2024, Volume: 29 Issue: 1, 131 - 143, 30.04.2024
https://doi.org/10.53433/yyufbed.1323369

Abstract

Bu çalışmada, iki boyutlu karmaşık uzayda rasyonel cebirsel eğrilerin izometrilerinin ve simetrilerinin hesaplanması için yeni ve etkili bir algoritma sunulmaktadır. Metot, problemin, karmaşık rasyonel cebirsel eğrilerin parametrizasyonlarına indirgenmesine dayanmaktadır. İki eğri arasındaki karmaşık izometriler, bir üniter matris ve iki boyutlu karmaşık vektörden meydana gelmektedir. Karmaşık izometrilerden etkilenmeyen invaryantlar sayesinde oluşturulan polinom denklemlerinin çözümü çarpanlara ayırma ve en büyük ortak çarpan bulma işlemleri ile bulunacaktır. Bu sayede, doğrusal olmayan büyük denklem sistemlerinin çözümünden sakınılacaktır. Girdi eğrilerinin özdeş olması durumunda metot, karmaşık rasyonel cebirsel eğrilerin tüm üniter simetrilerini tespit etmektedir. Sunulan algoritma, Maple bilgisayar cebir sistemi kullanılarak bilgisayar ortamına uyarlanmış ve bu uyarlama kullanılarak geniş çaplı testler yürütülmüştür.

Thanks

Yazarlar Prof. Juan Gerardo Alcazár’a, sonsuz desteği, yardımını hiçbir zaman esirgemediği ve tüm süreçlerde yanlarında olduğu için saygı ve teşekkürlerini sunar.

References

  • Alcázar, J. G., Hermoso, C., & Muntingh, G. (2015). Symmetry detection of rational space curves from their curvature and torsion. Computer Aided Geometric Design, 33, 51-65. doi:10.1016/j.cagd.2015.01.003
  • Alcázar, J. G., Díaz Toca, G. M., & Hermoso, C. (2019a). The problem of detecting when two implicit plane algebraic curves are similar. International Journal of Algebra and Computation, 29(5), 775-793. doi:10.1142/S0218196719500279
  • Alcázar, J. G., Lávička, M., & Vršek, J. (2019b). Symmetries and similarities of planar algebraic curves using harmonic polynomials. Journal of Computational and Applied Mathematics, 357, 302-318. doi:10.1016/j.cam.2019.02.036
  • Alcázar, J. G., & Quintero, E. (2020a). Affine equivalences of trigonometric curves. Acta Applicandae Mathematicae, 170, 691-708. doi:10.1007/s10440-020-00354-6
  • Alcázar, J. G., & Quintero, E. (2020b). Affine equivalences, isometries and symmetries of ruled rational surfaces. Journal of Computational and Applied Mathematics, 364, 112339. doi:10.1016/j.cam.2019.07.004
  • Alcázar, J. G., Gözütok, U., Çoban, H. A., & Hermoso, C. (2022). Detecting affine equivalences between implicit planar algebraic curves. Acta Applicandae Mathematicae, 182, 2. doi:10.1007/s10440-022-00539-1
  • Alcázar, J. G., Lávička, M., & Vršek, J. (2023). Computing symmetries of implicit algebraic surfaces. Computer Aided Geometric Design, 104, 102221. doi:10.1016/j.cagd.2023.102221
  • Bizzarri, M., Lávička, M., & Vršek, J. (2020). Computing projective equivalences of special algebraic varieties. Journal of Computational and Applied Mathematics, 367, 112438. doi:10.1016/j.cam.2019.112438
  • Gözütok, U. (2023). Testler, örnekler ve kaynak kodları. https://www.ugurgozutok.com/ Erişim Tarihi: 05.07.2023.
  • 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. doi:10.1016/j.cam.2022.114782
  • Hauer, M., Jüttler, B., & Schicho, J. (2018). Projective and affine symmetries and equivalences of rational and polynomial surfaces. Journal of Computational and Applied Mathematics, 349, 424-437. doi:10.1016/j.cam.2018.06.026
  • Jüttler, B., Lubbes, N., & Schicho, J. (2022). Projective isomorphisms between rational surfaces. Journal of Algebra, 594, 571-596. doi:10.1016/j.jalgebra.2021.11.045
  • Maple, (2022). Maplesoft, a division of Waterloo Maple Inc. Waterloo, Ontario.
  • Sendra, J. R., Winkler, F., & Perez-Diaz, S. (2007). Rational Algebraic Curves: A Computer Algebra Approach, 2008 ed., Algorithms and Computation in Mathematics, V. 22, Springer.

Computing Isometries and Symmetries of Rational Algebraic Curves in C^2

Year 2024, Volume: 29 Issue: 1, 131 - 143, 30.04.2024
https://doi.org/10.53433/yyufbed.1323369

Abstract

In this study, we present a new and efficient algorithm for computing isometries and symmetries of rational algebraic curves in the two dimensional complex space. The method is based on reducing the problem to parameterizations of complex rational algebraic curves. Complex isometries between two curves consist of a unitary matrix and a two dimensional complex vector. The solution of polynomial equations formed by invariants that are not affected by complex isometries will be found by factoring and finding the greatest common factor. In this way, we avoid solving large nonlinear systems. If the input curves are identical, the method detects all unitary symmetries of complex rational algebraic curves. The presented algorithm was implemented using the computer algebra system Maple, and using the implementation, we provide an extensive experimentation.

References

  • Alcázar, J. G., Hermoso, C., & Muntingh, G. (2015). Symmetry detection of rational space curves from their curvature and torsion. Computer Aided Geometric Design, 33, 51-65. doi:10.1016/j.cagd.2015.01.003
  • Alcázar, J. G., Díaz Toca, G. M., & Hermoso, C. (2019a). The problem of detecting when two implicit plane algebraic curves are similar. International Journal of Algebra and Computation, 29(5), 775-793. doi:10.1142/S0218196719500279
  • Alcázar, J. G., Lávička, M., & Vršek, J. (2019b). Symmetries and similarities of planar algebraic curves using harmonic polynomials. Journal of Computational and Applied Mathematics, 357, 302-318. doi:10.1016/j.cam.2019.02.036
  • Alcázar, J. G., & Quintero, E. (2020a). Affine equivalences of trigonometric curves. Acta Applicandae Mathematicae, 170, 691-708. doi:10.1007/s10440-020-00354-6
  • Alcázar, J. G., & Quintero, E. (2020b). Affine equivalences, isometries and symmetries of ruled rational surfaces. Journal of Computational and Applied Mathematics, 364, 112339. doi:10.1016/j.cam.2019.07.004
  • Alcázar, J. G., Gözütok, U., Çoban, H. A., & Hermoso, C. (2022). Detecting affine equivalences between implicit planar algebraic curves. Acta Applicandae Mathematicae, 182, 2. doi:10.1007/s10440-022-00539-1
  • Alcázar, J. G., Lávička, M., & Vršek, J. (2023). Computing symmetries of implicit algebraic surfaces. Computer Aided Geometric Design, 104, 102221. doi:10.1016/j.cagd.2023.102221
  • Bizzarri, M., Lávička, M., & Vršek, J. (2020). Computing projective equivalences of special algebraic varieties. Journal of Computational and Applied Mathematics, 367, 112438. doi:10.1016/j.cam.2019.112438
  • Gözütok, U. (2023). Testler, örnekler ve kaynak kodları. https://www.ugurgozutok.com/ Erişim Tarihi: 05.07.2023.
  • 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. doi:10.1016/j.cam.2022.114782
  • Hauer, M., Jüttler, B., & Schicho, J. (2018). Projective and affine symmetries and equivalences of rational and polynomial surfaces. Journal of Computational and Applied Mathematics, 349, 424-437. doi:10.1016/j.cam.2018.06.026
  • Jüttler, B., Lubbes, N., & Schicho, J. (2022). Projective isomorphisms between rational surfaces. Journal of Algebra, 594, 571-596. doi:10.1016/j.jalgebra.2021.11.045
  • Maple, (2022). Maplesoft, a division of Waterloo Maple Inc. Waterloo, Ontario.
  • Sendra, J. R., Winkler, F., & Perez-Diaz, S. (2007). Rational Algebraic Curves: A Computer Algebra Approach, 2008 ed., Algorithms and Computation in Mathematics, V. 22, Springer.
There are 14 citations in total.

Details

Primary Language Turkish
Subjects Algorithms and Calculation Theory, Algebraic and Differential Geometry
Journal Section Natural Sciences and Mathematics / Fen Bilimleri ve Matematik
Authors

Uğur Gözütok 0000-0002-6072-3134

Hüsnü Anil Çoban 0000-0001-8175-4960

Publication Date April 30, 2024
Submission Date July 5, 2023
Published in Issue Year 2024 Volume: 29 Issue: 1

Cite

APA Gözütok, U., & Çoban, H. A. (2024). C^2 de Rasyonel Cebirsel Eğrilerin İzometri ve Simetrilerinin Hesaplanması. Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 29(1), 131-143. https://doi.org/10.53433/yyufbed.1323369