POSITION VECTOR OF SPACELIKE SLANT HELICES IN MINKOWSKI 3-SPACE

• Ali, Ahmad T. ;
• Mahmoud, S.R.
• Received : 2014.01.03
• Accepted : 2014.06.05
• Published : 2014.06.25
• 71 7

Abstract

In this paper, position vector of a spacelike slant helix with respect to standard frame are deduced in Minkowski space $E^3_1$. Some new characterizations of a spacelike slant helices are presented. Also, a vector differential equation of third order is constructed to determine position vector of an arbitrary spacelike curve. In terms of solution, we determine the parametric representation of the spacelike slant helices from the intrinsic equations. Thereafter, we apply this method to find the parametric representation of some special spacelike slant helices such as: Salkowski and anti-Salkowski curves.

Keywords

Minkowski 3-space;slant helix;intrinsic equations

References

1. A.T. Ali and M. Turgut, Position vectors of a timelike general helices in Minkowski 3-space, Glob. J. Adv. Res. Class. Mod. Geom. 2(1) (2013), 1-10.
2. A.T. Ali, Spacelike Salkowski and anti-Salkowski curves with a spacelike principal normal in Minkowski 3-space, Int. J. Open Problems Comp. Math. 2 (2009), 451- 460.
3. A.T. Ali, Timelike Salkowski curves in Minkowski space $E^3_1$, J. Adv. Res. Dyn. Cont. Syst. 2 (2010), 17-26.
4. A.T. Ali, Position vectors of spacelike general helices in Minkowski 3-space, Nonl. Anal. Theory Meth. Appl. 73 (2010), 1118-1126. https://doi.org/10.1016/j.na.2010.04.051
5. A.T. Ali and M. Turgut, Position vector of a timelike slant helix in Minkowski 3-space, J. Math. Anal. Appl. 365 (2010), 559-569. https://doi.org/10.1016/j.jmaa.2009.11.026
6. S. Yilmaz, Determination of spacelike curves by Vector Differential Equations in Minkowski space $E^3_1$, J. Adv. Res. Pure Math., 1 (2009), 10-14.
7. B O'Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983.
8. J.G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, 2006.
9. L.P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Ginn and Co., 1909.
10. X. Yang, High accuracy approximation of helices by quintic curve, Comput. Aided Geomet. Design, 20 (2003), 303-317. https://doi.org/10.1016/S0167-8396(03)00074-8
11. J.D. Watson and F.H. Crick, Molecular structures of nucleic acids, Nature, 171 (1953), 737-738. https://doi.org/10.1038/171737a0
12. N. Chouaieb, A. Goriely and J.H. Maddocks, Helices, PNAS, 103 (2006), 398-403.
13. T.A. Cook, The curves of life, Constable, London 1914; Reprinted (Dover, London, 1979).
14. K. Ilarslan and O. Boyacioglu, Position vectors of a spacelike W-curve in Minkowski Space $E^3_1$, Bull. Korean Math. Soc., 44 (2007), 429-438. https://doi.org/10.4134/BKMS.2007.44.3.429
15. K. Ilarslan, and O. Boyacioglu, Position vectors of a timelike and a null helix in Minkowski 3-space, Chaos, Solitons and Fractals, 38 (2008), 1383-1389. https://doi.org/10.1016/j.chaos.2008.04.003
16. M. Barros, General helices and a theorem of Lancret, Proc. Amer. Math. Soc., 125 (1997), 1503-1509. https://doi.org/10.1090/S0002-9939-97-03692-7
17. M. Barros, A. Ferrandez, P. Lucas and M.A. Merono, General helices in the three dimensional Lorentzian space forms, Rocky Mountain J. Math., 31 (2001), 373-388. https://doi.org/10.1216/rmjm/1020171565
18. A. Ferrandez, A. Gimenez and P. Lucas, Null helices in Lorentzian space forms, Int. J. Mod. Phys. A, 16 (2001), 4845-4863. https://doi.org/10.1142/S0217751X01005821
19. J. Walrave, Curves and Surfaces in Minkowski Space, Doctoral thesis, K.U. Leuven, Faculty of Science, Leuven, 1995.
20. M.S. El Naschie, Notes on superstings and the infinite sums of Fibonacci and Lucas numbers, Chaos, Solitons and Fractals, 12 (2001), 1937-1940. https://doi.org/10.1016/S0960-0779(00)00139-9
21. M.S. El Naschie, Experimental and theoretial arguments for the number and mass of the Higgs particles, Chaos, Solitons and Fractals, 23 (2005), 1901-1908. https://doi.org/10.1016/j.chaos.2004.07.033
22. S. Falcon and A. Plaza, On the 3-dimensional k-Fibonacci spirals, Chaos, Solitons and Fractals, 38 (2008), 993-1003. https://doi.org/10.1016/j.chaos.2007.02.009
23. A.T. Ali and R. Lopez, Slant helices in Minkowski space $E^3_1$, J. Korean Math. Soc. 48 (2011), 159-167. https://doi.org/10.4134/JKMS.2011.48.1.159
24. J. Monterde, Salkowski curves revisted: A family of curves with constant curvature and non-constant torsion, Comput. Aided Geomet. Design, 26 (2009), 271-278. https://doi.org/10.1016/j.cagd.2008.10.002
25. E. Salkowski, Zur transformation von raumkurven, Mathematische Annalen, 66 (1909), 517-537. https://doi.org/10.1007/BF01450047
26. A. Jain, G. Wang and K.M. Vasquez, DNA triple helices: biological consequences and therapeutic potential, Biochemie, 90 (2008), 1117-1130. https://doi.org/10.1016/j.biochi.2008.02.011
27. Y. Yin, T. Zhang, F. Yang and X. Qui, Geometric conditions for fractal super carbon nanotubes with strict self-similarities, Chaos, Solitons and Fractals, 37 (2008), 1257-1266. https://doi.org/10.1016/j.chaos.2008.01.005