Bertrand Offsets of Spacelike Ruled Surfaces With Blaschke Approach

Main Article Content

Awatif Al-Jedani

Abstract

Dual parametrizaions of the Bertrand offset- spacelike ruled surfaces are assigned and sundry modern outcomes are acquired in view of their integral invariants. A modern characterization of the Bertrand offsets of spacelike developable surfaces is specified. Further, many connections among the striction curves of Bertrand offsets of spacelike ruled surfaces and their integral invariants are gained.

Article Details

References

  1. G.R. Veldkamp, On the Use of Dual Numbers, Vectors and Matrices in Instantaneous, spatial kinematics, Mech. Mach. Theory. 11 (1976), 141–156. https://doi.org/10.1016/0094-114x(76)90006-9.
  2. O. Bottema, B. B. Roth, Theoretical Kinematics, North-Holland Press, New York, (1979).
  3. A. Karger, J. Novak, Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, New York, (1985).
  4. H. Pottman, J. Wallner, Computational Line Geometry, Springer-Verlag, Berlin, Heidelberg, (2001).
  5. B. Ravani, T.S. Ku, Bertrand Offsets of Ruled and Developable Surfaces, Computer-Aided Design. 23 (1991), 145–152. https://doi.org/10.1016/0010-4485(91)90005-h.
  6. A. Küçük, O. Gürsoy, On the Invariants of Bertrand Trajectory Surface Offsets, Appl. Math. Comput. 151 (2004), 763–773. https://doi.org/10.1016/s0096-3003(03)00534-4.
  7. E. Kasap, N. Kuruoglu, Integral Invariants of the Pairs of the Bertrand Ruled Surface, Bull. Pure Appl. Sci. Sect. E Math. 21 (2002), 37–44.
  8. E. Kasap, N. Kuruoglu, The Bertrand Offsets of Ruled Surfaces in R3 1 , Acta Math. Vietnam. 31 (2006), 39–48.
  9. E. Kasap, S. Yuce, N. Kuruoglu, The Involute-Evolute Offsets of Ruled Surfaces, Iran. J. Sci. Tech. Trans. A. 33 (2009), 195–201.
  10. K. Orbay, E. Kasap, ˙I. Aydemir, Mannheim Offsets of Ruled Surfaces, Mathematical Problems in Engineering. 2009 (2009) 160917. https://doi.org/10.1155/2009/160917.
  11. M. Önder, H.H. U ˘gurlu, Frenet Frames and Invariants of Timelike Ruled Surfaces, Ain Shams Eng. J. 4 (2013), 507–513. https://doi.org/10.1016/j.asej.2012.10.003.
  12. M.T. Aldossary, R.A. Abdel-Baky, On the Bertrand Offsets for Ruled and Developable Surfaces, Boll. Unione Mat. Ital. 8 (2015), 53–64. https://doi.org/10.1007/s40574-015-0025-1.
  13. G.Y. Senturk, S. Yuce, Properties of Integral Invariants of the Involute-Evolute Offsets of Ruled Surfaces, Int. J. Pure Appl. Math. 102 (2015), 757–768. https://doi.org/10.12732/ijpam.v102i4.13.
  14. R.A. Abdel-Baky, M.F. Naghi, A Study on a Line Congruence as Surface in the Space of Lines, AIMS Math. 6 (2021), 11109–11123. https://doi.org/10.3934/math.2021645.
  15. M.A. Gungor, S. Ersoy, M. Tosun, Dual Lorentzian Spherical Motions and Dual Euler-savary Formula, Eur. J. Mech. - A/Solids. 28 (2009), 820–826. https://doi.org/10.1016/j.euromechsol.2009.03.007.
  16. N. Alluhaibi, R.A. Abdel-Baky, On the One-Parameter Lorentzian Spatial Motions, Int. J. Geom. Meth. Mod. Phys. 16 (2019), 1950197. https://doi.org/10.1142/s0219887819501974.
  17. N. Alluhaibi, R.A. Abdel-Baky, On the Kinematic-Geometry of One-Parameter Lorentzian Spatial Movement, Int. J. Adv. Manuf. Technol. 121 (2022), 7721–7731. https://doi.org/10.1007/s00170-022-09812-x.
  18. Y. Li, N. Alluhaibi, R.A. Abdel-Baky, One-Parameter Lorentzian Dual Spherical Movements and Invariants of the Axodes, Symmetry. 14 (2022), 1930. https://doi.org/10.3390/sym14091930.
  19. S.H. Nazra, R.A. Abdel-Baky, Bertrand Offsets of Ruled Surfaces with Blaschke Frame in Euclidean 3-Space, Axioms. 12 (2023), 649. https://doi.org/10.3390/axioms12070649.
  20. J.W. Bruce, P.J. Giblin, Curves and Singularities, 2nd ed. Cambridge University Press, Cambridge, (1992).
  21. Y. Li, Y. Zhu, Q.Y. Sun, Singularities and Dualities of Pedal Curves in Pseudo-Hyperbolic and De Sitter Space, Int. J. Geom. Meth. Mod. Phys. 18 (2020), 2150008. https://doi.org/10.1142/s0219887821500080.
  22. Y. Li, S.H. Nazra, R.A. Abdel-Baky, Singularity Properties of Timelike Sweeping Surface in Minkowski 3-Space, Symmetry. 14 (2022), 1996. https://doi.org/10.3390/sym14101996.
  23. 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.