Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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ON ADJOINT CURVES OF FRAMED CURVES AND SOME RULED SURFACES

  • Bahar Dogan Yazici (Faculty of Science, Department of Mathematics, Bilecik Seyh Edebali University) ;
  • Siddika Ozkaldi Karakus (Faculty of Science, Department of Mathematics, Bilecik Seyh Edebali University) ;
  • Murat Tosun (Faculty of Science, Department of Mathematics, Sakarya University)
  • Received : 2021.09.12
  • Accepted : 2023.03.21
  • Published : 2023.09.14
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Abstract

In this study, we introduce the adjoint curves of framed curves. We examine some characterizations of adjoint curve of a framed curve. In addition, we give the conditions for framed curves and adjoint curves to be Bertrand and Mannheim curves. Then, we introduce adjoint curves of Frenet-type framed curves and give ruled surfaces related to adjoint curves. Finally, we create normal and binormal surfaces of the framed adjoint curves and obtain some characterizations of these surfaces and we support by the results with figures.

Keywords

Acknowledgement

The results presented in this paper were part of the PhD thesis of Bahar Dogan Yazici [17].

References

  1. C. Boyer, A History of Mathematics, New York, Wiley, 1968.
  2. J. H. Choi and Y. H. Kim, Associated curves of a Frenet curve and their applications, Appl. Math. Comput. 218 (2012), no. 18, 9116-9124.
  3. T. Fukunaga and M. Takahashi, Existence conditions of framed curves for smooth curves, J. Geom. 108 (2017), 763-774. https://doi.org/10.1007/s00022-017-0371-5
  4. T. Fukunaga and M. Takahashi, Framed surfaces in the Euclidean space, Bull. Braz. Math. Soc, New Series, 50 (2019), 37-65.
  5. H. H. Hacisalihoglu, Diferensiyel Geometri, Ankara Universitesi Fen Fakultesi, 2000.
  6. S. Honda and M. Takahashi, Framed curves in the Euclidean space, Adv. Geom. 16 (2016), no. 3, 265-276.
  7. S. Honda, Rectifying developable surfaces of framed base curves and framed helices, Adv. Pure Appl. Math. 78 (2018), 273-292.
  8. S. Honda, S. Izumiya, and M. Takahashi, Developable surfaces along frontal curves on embedded surfaces, J. Geom. 110 (2019), 27.
  9. S. Honda and M. Takahashi, Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space, Turk. J. Math. 44 (2020), 883-899. https://doi.org/10.3906/mat-1905-63
  10. J. Huang and D. Pei, Singularities of non-developable surfaces in three-dimensional Euclidean space, Mathematics, 7 (2019), 1106.
  11. S. Izumiya and N. Takeuchi, Generic properties of helices and Bertrand curves, J. Geom. 74 (2002), 97-109. https://doi.org/10.1007/PL00012543
  12. S. Izumiya and N. Takeuchi, Geometry of Ruled Surfaces, In Applicable Mathematics in the Golden Age, 305-338, Misra, J. C., Ed., Narosa Publishing House, New Delhi, India, 2003.
  13. J. M. Lee, Introduction to Smooth Manifolds-Integral Curves and Flows, Graduate Texts in Math., 205-248, New York, 2003.
  14. Y. Li, K. Eren, K. H. Ayvaci, and S. Ersoy, The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space, AIMS Math. 8 (2023), 2226-2239.
  15. H. Liu and F. Wang, Mannheim partner curves in 3-space, J. Geom. 88 (2008), 120-126. https://doi.org/10.1007/s00022-007-1949-0
  16. S. K. Nurkan, I. A. Guven, and M. K. Karacan, Characterizations of adjoint curves in Euclidean 3-space, Proc. Nat. Acad. Sci. India Sect. A. 89 (2019), no. 1, 155-161.
  17. B. Dogan Yazici, Ozel Singuler Egrilerin Geometrisi Uzerine, PhD Thesis, Sakarya University, 2022.
  18. B. Dogan Yazici, O. Z. Okuyucu, and M. Tosun, Framed curves in three-dimensional Lie groups and a Berry phase model, J. Geom. Phys. 182 (2022), 104682.
  19. B. Dogan Yazici, S. Ozkaldi Karakus, and M. Tosun, On the classification of framed rectifying curves in Euclidean space, Math. Meth. App. Sci. 45 (2022), no. 18, 12089-12098.
  20. B. Dogan Yazici, S. Ozkaldi Karakus, and M. Tosun, Framed Normal Curves in Euclidean Space, TBILISI-MATHEMATICS, 27-37, Sciendo, 2019.
  21. B. Dogan Yazici, S. Ozkaldi Karakus, and M. Tosun, Framed curves and their applications based on a new differential equation, Int. Electron. J. Geom. 15 (2022), no. 1, 47-56.
  22. O. G. Yildiz, M. Akyigit, and M. Tosun, On the trajectory ruled surfaces of framed base curves in the Euclidean space, Math. Meth. App. Sci. 44 (2021), no. 9, 7463-7470.
  23. Y. Wang, D. Pei, and R. Gao, Generic properties of framed rectifying curves, Mathematics, 7 (2019), 37.