Research ArticleNo Access

Magnetic spherical indicatrix of a ruled surface

Alperen Yıldırım

https://orcid.org/0009-0005-9326-0534

Mathematics Department, Science Faculty, Ondokuz Mayıs University, Samsun, Türkiye

Search for more papers by this author

and

Emin Kasap

https://orcid.org/0000-0002-5017-5170

Mathematics Department, Science Faculty, Ondokuz Mayıs University, Samsun, Türkiye

E-mail Address: kasape@omu.edu.tr

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0219887824500683Cited by:1 (Source: Crossref)

Abstract

The spherical indicatrix of a ruled surface is the image curve of its rulings on the sphere $S^{2}$ $S^{2}$ . In this paper, we present Lorentz forces and magnetic curves produced by the Frenet frame ${T_{e}, N_{e}, B_{e}}$ ${T_{e}, N_{e}, B_{e}}$ of the spherical indicatrix of a ruled surface in a magnetic field. We calculate magnetic vector fields of magnetic curves for $T_{e}, N_{e}, B_{e}$ $T_{e}, N_{e}, B_{e}$ . Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatrix. We obtain developability conditions for these surfaces. Finally, we give some examples to show magnetic flux surfaces.

Keywords:

References

1. M. Barros and A. Romero, Magnetic vortices, Europhys. Lett. 77(3) (2007) 34002. Crossref, Google Scholar
2. M. Barros Díaz, J. L. Cabrerizo Jaraíz, M. Fernández Andrés and A. Romero Sarabia, Magnetic vortex filament flows, J. Math. Phys. 48(8) (2007) 082904. Crossref, Web of Science, Google Scholar
3. M. Barros, General helices and a theorem of Lancret, Proc. Amer. Math. Soc. 125(5) (1997) 1503–1509. Crossref, Web of Science, Google Scholar
4. M. Barros, A. Romero, J. L. Cabrerizo and M. Fernández, The Gauss–Landau–Hall problem on Riemannian surfaces, J. Math. Phys. 46(11) (2005) 112905. Crossref, Web of Science, Google Scholar
5. M. Barros, A. Ferrandez, P. Lucas and M. A. Merono, General helices in the three-dimensional Lorentzian space forms, Rocky Mountain J. Math. 31 (2001) 373–388. Crossref, Web of Science, Google Scholar
6. T. Turhan, Magnetic trajectories in three-dimensional Lie groups, Math. Methods Appl. Sci. 43(5) (2020) 2747–2758. Crossref, Web of Science, Google Scholar
7. Z. Özdemir, A geometrical and physical interpretation of quaternionic generalized magnetic flux tubes, Chaos Solitons Fractals 143 (2021) 110541. Crossref, Web of Science, Google Scholar
8. Z. Özdemir, Magnetic trajectories on lightlike surfaces, Differential Equations Dynam. Systems 31(2) (2023) 269–281. Crossref, Web of Science, Google Scholar
9. M. I. Munteanu and A. I. Nistor, Magnetic curves in the generalized Heisenberg group, Nonlinear Anal. 214 (2022) 112571. Crossref, Web of Science, Google Scholar
10. J. I. Inoguchi and M. I. Munteanu, Magnetic Jacobi fields in 3-dimensional Sasakian space forms, J. Geom. Anal. 32(3) (2022) 96. Crossref, Web of Science, Google Scholar
11. M. I. Munteanu and A. I. Nistor, Magnetic Jacobi fields in 3-dimensional cosymplectic manifolds, Mathematics 9(24) (2021) 3220. Crossref, Web of Science, Google Scholar
12. E. Bayram, F. Güler and E. Kasap, Magnetic flux ruled surfaces, Math. Methods Appl. Sci. 46(5) (2023) 5989–6001. Crossref, Web of Science, Google Scholar
13. F. Güler, E. Bayram and E. Kasap, Magnetic spherical indicatricies in Minkowski 3-space, Int. J. Geom. Methods Mod. Phys. 20 (2023) 2350179. Link, Web of Science, Google Scholar
14. F. Guler, E. Kasap and N. Altinisik, A new approach to find the magnetic flux surfaces, Int. J. Geom. Methods Mod. Phys. (2023). Web of Science, Google Scholar
15. S. Aslan, M. Bekar and Y. Yaylı, Ruled surfaces constructed by quaternions, J. Geom. Phys. 161 (2021) 104048. Crossref, Web of Science, Google Scholar
16. L. G. Nartova, R. R. Anamova and S. A. Leonova, Modeling ideas for ruled surfaces and their implementation in applied geometry, J. Phys. Conf. Ser. 1889(5) (2021) 052017. Crossref, Google Scholar
17. F. Taş and K. İlarslan, A new approach to design the ruled surface, Int. J. Geom. Methods Mod. Phys. 16(06) (2019) 1950093. Link, Web of Science, Google Scholar
18. Y. Li, Z. Wang and T. Zhao, Geometric algebra of singular ruled surfaces, Adv. Appl. Clifford Algebras 31 (2021) 1–19. Crossref, Web of Science, Google Scholar
19. M. Incesu, The new characterization of ruled surfaces corresponding dual Bézier curves, Math. Methods Appl. Sci. 45(18) (2022) 12030–12045. Crossref, Web of Science, Google Scholar
20. Ö. G. Yıldız, M. Akyiğit and M. Tosun, On the trajectory ruled surfaces of framed base curves in the Euclidean space, Math. Methods Appl. Sci. 44(9) (2021) 7463–7470. Crossref, Web of Science, Google Scholar
21. B. Ravani and T. S. Ku, Bertrand offsets of ruled and developable surfaces, Comput.-Aided Des. 23(2) (1991) 145–152. Crossref, Web of Science, Google Scholar
22. D. Somasundaram, Differential Geometry (Nasora Publishing House, New Delhi, India, 2005). Google Scholar
23. B. O’neill, Elementary Differential Geometry (Elsevier, 2006). Google Scholar