Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

RULED SURFACES GENERATED BY SALKOWSKI CURVE AND ITS FRENET VECTORS IN EUCLIDEAN 3-SPACE

  • Ebru Cakil (Department of Mathematical Engineering, Gumushane University) ;
  • Sumeyye Gur Mazlum (Department of Computer Technology, Gumushane University)
  • Received : 2024.03.22
  • Accepted : 2024.05.24
  • Published : 2024.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

In present study, we introduce ruled surfaces whose base curve is the Salkowski curve in Euclidean 3-space and whose generating lines consist of the Frenet vectors of this curve (tangent, principal normal and binormal vectors). Then, we produce regular surfaces from a vector with real coefficients, which is a linear combination of these vectors, and we examine some special cases for these surfaces. Moreover, we present some geometric properties and graphics of all these surfaces.

Keywords

References

  1. P. Alegre, K. Arslan, A. Carriazo, C. Murathan, G. Ozturk, Some special types developable ruled surface, Hacettepe J. Math. Stat. 39 (3) (2010), 319-325.
  2. A. T. Ali, Position vectors of slant helices in Euclidean 3-space, J. Egyptian Math. Soc. 20 (1) (2012), 1-6. https://doi.org/10.1016/j.joems.2011.12.005
  3. A. T. Ali, H. S. Abdel Aziz, A. H. Sorour, Ruled surfaces generated by some special curves in Euclidean 3-Space, J. Egyptian Math. Soc. 21 (2013), 285-294. https://doi.org/10.1016/j.joems.2013.02.004
  4. M. Altin, A. Kazan, H. B. Karadag, Ruled Surfaces in 3 with Density, Honam Math. J. 41(4) (2019), 683-695. https://doi.org/10.5831/HMJ.2019.41.4.683
  5. M. Altin, A. Kazan, H. B. Karadag, Ruled surfaces constructed by planar curves in Euclidean 3-space with density, Celal Bayar Univ. J. Sci. 16 (1) (2020), 81-88.
  6. A. Gray, E. Abbena, S. Salamon, Modern differential geometry of curves and surfaces with Mathematica, CRC Press. New York (2006).
  7. S. Gur, S. Senyurt, Frenet vectors and geodesic curvatures of spheric indicators of Salkowski curves in E3, Hadronic J. 33 (5) (2010), 485-512.
  8. S. Gur Mazlum, S. Senyurt, M. Bektas, Salkowski curves and their modified orthogonal frames in E3, J. New Theory. 40 (2022), 1226. https://doi.org/10.53570/jnt.1140546
  9. S. Gur Mazlum, M. Bektas, (k, m)- type slant helices for the null cartan curve with the Bishop frame in E41, Honam Math. J. 45 (4) (2023), 610-618. https://doi.org/10.5831/HMJ.2023.45.4.610
  10. S. Gur Mazlum, Bishop frames of Salkowski curves in E3, Bitlis Eren Univ. J. Sci. 13(1) (2024), 79-91. https://doi.org/10.17798/bitlisfen.1345438
  11. S. Gur Mazlum, On the Gaussian curvature of timelike surfaces in Lorentz-Minkowski 3-space, Filomat, 37(28) (2023), 9641-9656. https://doi.org/10.2298/FIL2328641G
  12. S. Izumiya, N. Takeuchi, Special curves and ruled surfaces, Beitr Algebra Geom. 44(1) (2003), 203-212.
  13. S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turkish J. Math. 28(2) (2004), 153-163.
  14. Y. H. Kim, D. W. Yoon, On the Gauss Map of Ruled Surfaces in Minkowski Space, Rocky Mountain J. Math. 35(5) (2005), 1555-1581. https://doi.org/10.1216/rmjm/1181069651
  15. L. Kula, N. Ekmek,ci, Y. Yayli, K. Ilarslan, Characterizations of slant helices in Euclidean 3-space, Turkish J. Math. 33 (2009), 1-13. https://doi.org/10.3906/mat-0809-17
  16. Y. Li, X. Jiang, Z. Wang, Singularity properties of Lorentzian Darboux surfaces in Lorentz-Minkowski spacetime, Res. Math. Sci. 11 (2024), 7. https://doi.org/10.1007/s40687-023-00420-z
  17. Y. Li, Z. Chen, S. H. Nazra, R. A. Abdel-Baky, Singularities for Timelike Developable Surfaces in Minkowski 3-Space, Symmetry. 15 (2023), 277. https://doi.org/10.3390/sym15020277
  18. Y. Li, Z. Wang, T. Zhao, Geometric algebra of singular ruled surfaces, Adv. Appl. Clifford Algebr. 31 (2021), 1-19. https://doi.org/10.1007/s00006-020-01097-1
  19. Y. Li, Z. Wang, T. Zhao, Slant helix of order n and sequence of Darboux developables of principal-directional curves, Math. Methods Appl. Sci. 43(17) (2020), 9888-9903. https://doi.org/10.1002/mma.6663
  20. J. Monterde, Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, Comput. Aided Geom. Des. 26(3) (2009), 271-278. https://doi.org/10.1016/j.cagd.2008.10.002
  21. J. Monterde, The Bertrand curve associated to a Salkowski curve, J. Geom. 111(2) (2020), 21. https://doi.org/10.1007/s00022-020-00533-8
  22. S. Ouarab, A. O. Chahdi, Special family of ruled surfaces in Euclidean 3-space, Int. J. Sci.Eng. Res. 10(5) (2019), 320-327.
  23. M. Onder, O. Kaya, Characterizations of slant ruled surfaces in the Euclidean 3-space, Caspian J. Math. Sci. 6(1) (2017), 31-46. https://doi.org/10.22080/CJMS.2017.1637
  24. E. Salkowski, Zur transformation von raumkurven, Math. Ann. 66(4) (1909), 517-557. https://doi.org/10.1007/BF01450047
  25. A. Sarioglugil, A. Tutar, On ruled surfaces in Euclidean space E3, Int. J. Contemp. Math. Sci. 2(1) (2007), 1-11. https://doi.org/10.12988/ijcms.2007.07001
  26. S. S,enyurt, S. Gur Mazlum, L. Grilli, Gaussian curvatures of parallel ruled surfaces, Appl. Math. Sci. 149(4) (2020), 171-183. https://doi.org/10.12988/ams.2020.912175
  27. Y. Tun,cer, N. Ekmek,ci, A study on ruled surface in Euclidean 3-space, J. Dyn. Sys. Geom. Theor. 8(1) (2013), 49-57. https://doi.org/10.1080/1726037X.2010.10698577
  28. O. G. Yildiz, M. Akyigit, M. Tosun, On the trajectory ruled surfaces of framed base curves in the Euclidean space, Math. Meth. Appl. Sci. 44(9) (2021), 7463-7470. https://doi.org/10.1002/mma.6267
  29. Y. Yu, H. Liu, S. D. Jung, Structure and characterization of ruled surfaces in Euclidean 3-space, Appl. Math. Comput. 233 (2014), 252-259. https://doi.org/10.1016/j.amc.2014.02.006