Evolutes of Hyperbolic Dual Spherical Curve in Dual Lorentzian 3-Space

Main Article Content

Rashad A. Abdel-Baky


Based on the E. Study's map, we study a timelike ruled surface as a curve on the hyperbolic dual unit sphere in dual Lorentzian 3-space $\mathbb{D}_{1}^{3}$. Then, as applications of the singularity theory of smooth functions, we define the notation of evolutes for timelike ruled surfaces and establish the relationships between their geometric invariants. Finally, an example of application is introduced and explained in detail.

Article Details


  1. O. Bottema, and B. Roth. Theoretical Kinematics, North-Holland Press, New York 1979.
  2. A. Karger, and J. Novak. Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, New York 1985.
  3. J M. MC Carthy J M. An introduction to theoretical kinematics, London: The MIT Press 1990.
  4. H. Pottman, and J. Wallner. Computational Line Geometry, Springer-Verlag, Berlin, Heidelberg 2001.
  5. R.A. Abdel-Baky, and F.R. Al-Solamy. A new geometrical approach to one-parameter spatial motion, J. Eng. Math. 60 (2008). 149-172.
  6. R.A. Abdel-Baky, and R.A Al-Ghefari. On the one-parameter dual spherical motions, Comput. Aided Geom. Des. 28 (2011), 23-37.
  7. R.A Al-Ghefari, and R.A. Abdel-Baky. Kinematic geometry of a line trajectory in spatial motion, J. Mech. Sci. Tech. 29 (9) (2015), 3597-3608.
  8. B. O'Neil. Semi-Riemannian Geometry geometry, with applications to relativity, Academic Press, New York, 1983.
  9. W. Sodsiri. Ruled linear Weingarten surfaces in Minkowski 3-space, Soochow J. Math., 29(4) (2003), 435-443.
  10. W. Kuhnel. Differential Geometry (2nd Edition), Amer. Math. Soc., 2006.
  11. R. Lopez. Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space, arxiv.org/abs/0810.3351v1 2008.
  12. Y. Yayli, A. Caliskan, and H. H. U ¬łgurlu, The E. Study Maps of Circles on Dual Hyperbolic and Lorentzian Unit Spheres H 2 0 and S 2 0 , Math. Proc. R. Ir. Acad. 102A (2002), no. 1, 37-47.
  13. M. Onder, and HH Ugurlu. Frenet frames and invariants of timelike ruled surfaces, Ain Shams Eng. J. 4 (2013), 507-513.
  14. Bruce, J. W.; Giblin, P. J.: Curves and Singularities. 2nd. ed. Cambridge Univ. Press, Cambridge 1992.
  15. IR. Porteous. Geometric differentiation for the intelligence of Curves and Surfaces, Second edition, Cambridge University Press, Cambridge, 2001.
  16. S. Izumiya and N. Takeuchi, Geometry of ruled surfaces, Applicable Math., in the golden age, (2003), 305-338.
  17. S. Izumiya and N. Takeuchi. Special Curves and Ruled Surfaces, Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry, 44 (2003), No. 1, 203-212.
  18. S. Izumiya and A. Takiyama, A time-like surface in Minkowski 3-space which contains pseudocircles, Proc. Edinburgh Math. Soc. (2) 40, 1 (1997), 27-136.
  19. S. Izumiya, D-H. Pe. and T. Sano. The lightcone Gauss map and lightcone developable of a spacelike curve in Minkowski 3-space, J. Glasgow Math. J. 42 (2000), 75-89.
  20. S. Izumiya, D. Pei and T. Sano, Singularities of hyperbolic Gauss maps, Proc. London Math. Soc., 86 (2003), 485-512.
  21. S. Izumiya and M. C. Romero Fuster, The horospherical Gauss-Bonnet type theorem in hyperbolic space, J. Math. Soc. Japan 58 (2006), 965-984.
  22. S. Izumiya, K. Saji and N. Takeuchi, Circular surfaces, Adv. Geom. 7 (2007), 295-313.
  23. S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, Mosc. Math. J. 9 (2009), 325-357.