Honam Mathematical Journal (호남수학학술지)

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A STUDY OF THE TUBULAR SURFACES ACCORDING TO MODIFIED ORTHOGONAL FRAME WITH TORSION

  • Gulnur SAFFAK ATALAY (Ondokuz Mayis University, Faculty of Education, Department of Mathematical Sciences)
  • Received : 2023.11.02
  • Accepted : 2023.12.16
  • Published : 2024.06.25
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Abstract

In this study, tubular surfaces were introduced according to the modified orthogonal frame defined at the points where the torsion is different from zero in the 3-dimensional Euclidean space. First, the relations between the Frenet frame and the modified orthogonal frame with torsion are given. Then, the singularity, Gaussian curvature, mean curvature and basic forms of the tubular surface given according to the modified orthogonal frame with torsion were calculated. In addition, the conditions for the parameter curves of the tubular surface to be geodesic, asymptotic and line of curvature were examined. Finally, tubular surface examples based on both the Frenet frame and the modified orthogonal frame with torsion were given to support the study.

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References

  1. M. Akyigit, K. Eren, and H. H. Kosal, Tubular surfaces with modified orthogonal frame in Euclidean 3-space, Honam Math. J. 3 (2021), no. 43, 453-463.
  2. A. Z. Azak, Involute-evolute curves according to modified orthogonal frame, Journal of Science and Arts. 2 (2021), no. 55, 385-394.
  3. S. Bas and T. Korpinar, Modified roller coaster surface in space, Mathematics 7 (2019), no. 2, 195-205. https://doi.org/10.3390/math7020195
  4. R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly 82 (1975), no. 3, 246-251. https://doi.org/10.1080/00029890.1975.11993807
  5. P. A. Blaga, On tubular surfaces in computer graphics, Stud. Univ. Babes-Bolyai Inform. 50 (2005), no. 2, 81-90.
  6. B. Bukcu and M. K. Karacan, On the modified orthogonal frame with curvature and torsion in 3-space, Math. Sci. Appl. E-Notes 4 (2016), no. 1, 184-188. https://doi.org/10.36753/mathenot.421429
  7. F. Dogan and Y. Yayli, On the curvatures of tubular surface with Bishop frame, Communications Fac. Sci. Univ. Ank. Ser. A1 60 (2011), no. 1, 59-69.
  8. F. Dogan and Y. Yayli, Tubes with Darboux frame, Int. J. Contemp. Math. Sci. 7 (2012), no. 16, 751-758.
  9. M. K. Karacan and Y. Tuncer, Tubular surfaces of Weingarten types in Galilean and Pseudo-Galilean, Bull. Math. Anal. Appl. 5 (2013), no. 2, 87-100.
  10. M. K. Karacan and Y. Yayli, On the geodesics of tubular surfaces in Minkowski 3- space, Bull. Malays. Math. Sci. Soc. 31 (2008), no. 1, 1-10.
  11. M. K. Karacan, D. W. Yoon, and Y. Tuncer, Weingarten and linear Weingarten type tubular surfaces in E3, Math. Probl. Eng. 2011 (2011), no. 3, 1-11.
  12. M. K. Karacan, D. W. Yoon, and Y. Tuncer, Tubular surfaces of Weingarten types in Minkowski 3-space, Gen. Math. Notes 22 (2014), no. 1, 44-56.
  13. S. Kiziltug, S. Kaya, and O. Tarakci, Tube surfaces with type-2 Bishop frame of Weingarten types in E3, Int. J. Math. Anal. 7 (2013), no. 2, 9-18. https://doi.org/10.12988/ijma.2013.13002
  14. T. Korpinar and E. Turhan, Tubular surfaces around timelike biharmonic curves in Lorentzian Heisenberg group Heis3, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat. 20 (2012), no. 1, 431-446.
  15. M. S. Lone, E. S. Hasan, M. K. Karacan, and B. Bukcu, On some curves with modified orthogonal frame in Euclidean 3-space, Iran. J. Sci. Technol. Trans. A Sci. 43 (2019), no. 4, 1905-1916. https://doi.org/10.1007/s40995-018-0661-2
  16. M. S. Lone, E. S. Hasan, M. K. Karacan, and B. Bukcu, Mannheim curves with modified orthogonal frame in Euclidean 3-space, Turkish J. Math. 43 (2019), no. 2, 648-663. https://doi.org/10.3906/mat-1807-177
  17. T. Sasai, The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations, Tohoku Math. J. 36 (1984), 17-24. https://doi.org/10.2748/tmj/1178228899