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BERTRAND PARTNER D-CURVES IN THE MINKOWSKI3-SPACE E3-1

Year 2014, Volume: 2 Issue: 1, 68 - 82, 01.06.2014

Abstract

In this paper, we consider the idea of Bertrand curves for curveslying on surfaces in the Minkowski 3-space E3. By considering the Darbouxframe, we define these curves as Bertrand partner D-curves and give the general characterizations for those curves. Then, we find the relations betweenthe geodesic curvatures, the normal curvatures and the geodesic torsions ofthese associated curves in some special cases

References

  • Burke, J.F., Bertrand curves associated with a pair of curves. Mathematics Magazine. 34 (1960), no.1, 60-62.
  • G¨org¨ul¨u, E., ¨Ozdamar, E., A generalizations of the Bertrand curves as general inclined curves in En. Communications de la Fac. Sci. Uni. Ankara, Series A1. 35 (1986), 53-60.
  • Ersoy, S., ˙Inalcık, A., On the generalized timelike Bertrand curves in 5-dimensional Lorentzian space. Differential Geometry-Dynamical Systems. 13 (2011), 78-88.
  • Izumiya, S., Takeuchi, N., Special Curves and Ruled surfaces. Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry. 44 (2003), no. 1, 203-212.
  • Izumiya, S., Takeuchi, N., Generic properties of helices and Bertrand curves. Journal of Geometry. 74 (2002), 97-109.
  • Kasap, E., Kuruoˇglu, N., The Bertrand offsets of ruled surfaces in R3. Acta Mathematica 1 Vietnamica. 31 (2006), no. 1, 39-48.
  • O’Neill, B., Elemantery Differential Geometry. Academic Press Inc. New York, 1966.
  • O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London, 1983.
  • ¨Oˇgrenmi¸s, A.O., ¨Oztekin, H., Erg¨ut, M., Bertrand curves in Galilean space and their char- acterizations. Kragujevac J. Math. 32 (2009), 139-147.
  • Ratcliffe, J.G., Foundations of Hyperbolic Manifolds. Springer, 2006.
  • Ravani, B., Ku, T.S., Bertrand Offsets of ruled and developable surfaces. Comp. Aided Geom. Design. 23 (1991), no. 2, 145-152.
  • Struik, D.J., Lectures on Classical Differential Geometry. 2nded. Addison Wesley, Dover, 19 Uˇgurlu, H.H., C¸ alı¸skan, A., Darboux Ani D¨onme Vekt¨orleri ile Spacelike ve Timelike Y¨uzeyler Geometrisi. Celal Bayar ¨Universitesi Yayınları, Yayın No: 0006, 2012.
  • Walrave, J., Curves and Surfaces in Minkowski space. PhD. thesis, K.U. Leuven, Fac. of Science, Leuven, 1995.
  • Whittemore, J.K., Bertrand curves and helices. Duke Math. J. 6 (1940), no. 1, 235-245.
  • Celal Bayar University, Department of Mathematics, Faculty of Arts and Sciences, Muradiye Campus, Muradiye, Manisa, Turkey. E-mail address: mehmet.onder@cbu.edu.tr E-mail address: mustafa.kazaz@cbu.edu.tr Gazi University, Gazi Faculty of Education, Department of Secondary Education Science and Mathematics Teaching, Mathematics Teaching Program, Ankara, Turkey. E-mail address: hugurlu@gazi.edu.tr

Year 2014, Volume: 2 Issue: 1, 68 - 82, 01.06.2014

Abstract

References

  • Burke, J.F., Bertrand curves associated with a pair of curves. Mathematics Magazine. 34 (1960), no.1, 60-62.
  • G¨org¨ul¨u, E., ¨Ozdamar, E., A generalizations of the Bertrand curves as general inclined curves in En. Communications de la Fac. Sci. Uni. Ankara, Series A1. 35 (1986), 53-60.
  • Ersoy, S., ˙Inalcık, A., On the generalized timelike Bertrand curves in 5-dimensional Lorentzian space. Differential Geometry-Dynamical Systems. 13 (2011), 78-88.
  • Izumiya, S., Takeuchi, N., Special Curves and Ruled surfaces. Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry. 44 (2003), no. 1, 203-212.
  • Izumiya, S., Takeuchi, N., Generic properties of helices and Bertrand curves. Journal of Geometry. 74 (2002), 97-109.
  • Kasap, E., Kuruoˇglu, N., The Bertrand offsets of ruled surfaces in R3. Acta Mathematica 1 Vietnamica. 31 (2006), no. 1, 39-48.
  • O’Neill, B., Elemantery Differential Geometry. Academic Press Inc. New York, 1966.
  • O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London, 1983.
  • ¨Oˇgrenmi¸s, A.O., ¨Oztekin, H., Erg¨ut, M., Bertrand curves in Galilean space and their char- acterizations. Kragujevac J. Math. 32 (2009), 139-147.
  • Ratcliffe, J.G., Foundations of Hyperbolic Manifolds. Springer, 2006.
  • Ravani, B., Ku, T.S., Bertrand Offsets of ruled and developable surfaces. Comp. Aided Geom. Design. 23 (1991), no. 2, 145-152.
  • Struik, D.J., Lectures on Classical Differential Geometry. 2nded. Addison Wesley, Dover, 19 Uˇgurlu, H.H., C¸ alı¸skan, A., Darboux Ani D¨onme Vekt¨orleri ile Spacelike ve Timelike Y¨uzeyler Geometrisi. Celal Bayar ¨Universitesi Yayınları, Yayın No: 0006, 2012.
  • Walrave, J., Curves and Surfaces in Minkowski space. PhD. thesis, K.U. Leuven, Fac. of Science, Leuven, 1995.
  • Whittemore, J.K., Bertrand curves and helices. Duke Math. J. 6 (1940), no. 1, 235-245.
  • Celal Bayar University, Department of Mathematics, Faculty of Arts and Sciences, Muradiye Campus, Muradiye, Manisa, Turkey. E-mail address: mehmet.onder@cbu.edu.tr E-mail address: mustafa.kazaz@cbu.edu.tr Gazi University, Gazi Faculty of Education, Department of Secondary Education Science and Mathematics Teaching, Mathematics Teaching Program, Ankara, Turkey. E-mail address: hugurlu@gazi.edu.tr
There are 15 citations in total.

Details

Primary Language English
Authors

MUSTAFA Kazaz This is me

H.HÜSEYİNUĞURLU This is me

MEHMETÖNDER This is me

SEDAORAL This is me

Submission Date March 9, 2015
Publication Date June 1, 2014
Published in Issue Year 2014 Volume: 2 Issue: 1

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