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A NEW CONSTRUCTION OF TIMELIKE RULED SURFACES WITH CONSTANT DISTELI-AXIS

  • Abdel-Baky, Rashad A. (Department of Mathematics, Sciences Faculty for Girls, University of Jeddah) ;
  • Unluturk, YasIn (Department of Mathematics, Kirklareli University)
  • Received : 2019.12.12
  • Accepted : 2020.02.15
  • Published : 2020.09.25

Abstract

In this study, we construct timelike ruled surfaces whose Disteli-axis is constant in Minkowski 3-space 𝔼31. Then we attain a general system characterizing these surfaces, and also give necessary and sufficient conditions for a timelike ruled surface to get a constant Disteli-axis.

Keywords

References

  1. R.A. Abdel-Baky, and R.A Al-Ghefari, On the kinematic geometry of relative screw motions, J. of Mech. Sci. Tech. 26 (8) (2012), 2497-2503. https://doi.org/10.1007/s12206-012-0624-z
  2. R.A. Abdel-Baky, Evolutes of hyperbolic dual spherical curve in dual Lorentzian 3-space, Int. J. Anal. Appl. 15 (2) (2017), 114-124.
  3. M. Bilici, On the invariants of ruled surfaces generated by the dual involute Frenet trihedron, Commun. Fac. Sci. Univ. Ank. Series A1 66 (2) (2017), 62-70.
  4. E. Bayram, and M. Bilici, Surface family with a common involute asymptotic curve, Int. J. Geom. Meth. Mod. Phys. 13 (5) (2016), 1650062. https://doi.org/10.1142/S0219887816500626
  5. O. Bottema, and B. Roth, Theoretical Kinematics, North-Holland Press New York (1979).
  6. M. Disteli, Uber instantane schraubengeschwindigkeiten un die verzahnung der hyperboloidrader, Z. Math. Phys. 51 (1904), 51-88.
  7. M. Disteli, Uber die verzahnung der hyperbololoidrader mit geradlinigem eingriff, Z. Math. Phys. 59 (1911), 244-298.
  8. M. Disteli, Uber das Analon der Savaryschen Formel und Konstruktion in der kinematischen Geometrie des Raumes. Z. Math. Phys. 62 (1914), 261-309.
  9. Z. Ekinci, H.H. Ugurlu, Disteli diagram of dual hyperbolic spherical motion in dual Lorentzian space, Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 84 (3) (2014), 397-407. https://doi.org/10.1007/s40010-014-0149-1
  10. N. Hassan, R.A. Abdel-Baky, and F.M. Hamdoon, Ruled surfaces with timelike rulings, Appl. Math. Comput. 147 (1) (2004), 241-253. https://doi.org/10.1016/S0096-3003(02)00664-1
  11. A. Karger, and J. Novak, Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, New York (1985).
  12. Y.H. Kim, and D.W. Yoon, Classification of ruled surfaces in Minkowski 3-spaces, J. Geom. Phys. 49 (1) (2004), 89-100. https://doi.org/10.1016/S0393-0440(03)00084-6
  13. A. Kucuk, On the developable timelike trajectory ruled surfaces in Lorentz 3-space $R^3_1$, Appl. Math. Comput. 157 (2004), 483-489. https://doi.org/10.1016/j.amc.2003.09.001
  14. R. Lopez, Differential geometry of curves and surfaces in Lorentz-Minkowski space, Int. Electronic J. Geom. 7 (2014), 44-107. https://doi.org/10.36890/iejg.594497
  15. J.M. McCarthy, An Introduction to Theoretical Kinematics, London: The MIT Press (1990).
  16. M. Onder and H.H. Ugurlu, Frenet frames and invariants of timelike ruled surfaces, Ain Shams Engineering Journal 4 (2013), 507-513. https://doi.org/10.1016/j.asej.2012.10.003
  17. B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York (1983).
  18. H. Pottman, and J. Wallner, Computational Line Geometry, Springer-Verlag Berlin Heidelberg (2001).
  19. J.G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer Science+Business Media LLC New York (2006).
  20. A. Turgut, and H.H. Hacisalihoglu, Timelike ruled surfaces in the Minkowski 3-space, Far East J. Math. Sci. 5 (1) (1997), 83-90.
  21. Z. Yapar, and Y. Sagiroglu, On the geometry of closed timelike ruled surfaces in dual Lorentzian space, Int. J. Appl. Math. 29 (1) (2016), 7-18.
  22. Y. Yayli, A. Caliskan, and H. H. Ugurlu, The E. Study maps of circles on dual hyperbolic and Lorentzian unit spheres $H^2_0$ and $S^2_0$, Math. Proc. Roy. Ir. Acad. 102A (1) (2002), 37-47.
  23. Z.K. Yuzbasi, and D.W. Yoon, On constructions of surfaces using a geodesic in Lie group, J. Geom. 110 (29) (2019).