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Differential Equations of the Space-Like Loxodromes on the Helicoidal Surfaces in Minkowski 3-Space

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Abstract

In the present paper, we investigate the differential equations of the space-like loxodromes on the helicoidal surfaces having space-like meridians and time-like meridians, respectively in Minkowski 3-space. Also we illustrate our main results by using Mathematica.

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References

  1. Babaarslan, M., Munteanu, M.I.: Time-like loxodromes on rotational surfaces in Minkowski 3-space. An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 61(2), 471–484 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Babaarslan, M., Yayli, Y.: Space-like loxodromes on rotational surfaces in Minkowski 3-space. J. Math. Anal. Appl. 409(1), 288–298 (2014)

    Article  MathSciNet  Google Scholar 

  3. Babaarslan, M., Yayli, Y.: Differential equation of the loxodrome on a helicoidal surface. J. Navig. 68(5), 962–970 (2015)

    Article  Google Scholar 

  4. Beneki, ChrC, Kaimakamis, G., Papantoniou, B.J.: Helicoidal surfaces in three-dimensional Minkowski space. J. Math. Anal. Appl. 275(2), 586–614 (2002)

    Article  MathSciNet  Google Scholar 

  5. Ji, F., Hu, Z.H.: Helicoidal surfaces under the cubic screw motion in Minkowski 3-space. J. Math. Anal. Appl. 318(2), 634–647 (2006)

    Article  MathSciNet  Google Scholar 

  6. Ji, F., Kim, Y.H.: Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space. J. Math. Anal. Appl. 368(2), 623–635 (2010)

    Article  MathSciNet  Google Scholar 

  7. Kos, S., Filjar, R., Hess, M.: Differential equation of the loxodrome on a rotational surface. Proceedings of the 2009 international technical meeting of The Institute of Navigation, Anaheim, pp. 958–960 (2009)

  8. Lee, S., Varnado, J.H.: Timelike surfaces of revolution with constant mean curvature in Minkowski 3-space. Differ. Geom. Dyn. Syst. 9, 82–102 (2007)

    MathSciNet  MATH  Google Scholar 

  9. López, R.: Differential geometry of curves and surfaces in Lorentz–Minkowski space. Int. Electron. J. Geom. 7(1), 44–107 (2014)

    Article  MathSciNet  Google Scholar 

  10. Noble, C.A.: Note on loxodromes. Bull. Am. Math. Soc. 12(3), 116–119 (1905)

    Article  MathSciNet  Google Scholar 

  11. O’Neill, B.: Semi-Riemannian geometry with application to relativity. Academic Press, New York (1983)

    MATH  Google Scholar 

  12. Ratcliffe, J.G.: Foundations of hyperbolic manifolds, Graduate texts in mathematics, vol. 149, 2nd edition (2006)

  13. Ugail, H., Márquez, M.C., Yilmaz, A.: On Bézier surfaces in three-dimensional Minkowski space. Comput. Math. Appl. 62(8), 2899–2912 (2011)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This paper was supported by the Scientific Research Project Coordination Unit of Bozok University under project 2015FBE/T159.

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Authors and Affiliations

  1. Department of Mathematics, Bozok University, 66100, Yozgat, Turkey

    Murat Babaarslan & Mustafa Kayacik

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  1. Murat Babaarslan
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  2. Mustafa Kayacik
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Correspondence to Murat Babaarslan.

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Babaarslan, M., Kayacik, M. Differential Equations of the Space-Like Loxodromes on the Helicoidal Surfaces in Minkowski 3-Space. Differ Equ Dyn Syst 28, 495–512 (2020). https://doi.org/10.1007/s12591-016-0343-5

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