AIMS Mathematics, 2020, 5(6): 5422-5438. doi: 10.3934/math.2020348

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Non-lightlike Bertrand W curves: A new approach by system of differential equations for position vector

1 Department of Mathematics and Science Education, Necmettin Erbakan University, Konya, 42090, TURKEY
2 Department of Mathematics, Necmettin Erbakan University, Konya, 42090, TURKEY

In this study, the characterization of position vectors belonging to non-lightlike Bertrand W curve mate with constant curvature are obtained depending on differentiable functions. The position vector of Bertrand W curve is stated by a linear combination of its Frenet frame with differentiable functions. There exist also different cases for the curve depending on the value of curvature and torsion. The relationships between Frenet apparatuas of these curves are stated separately for each case. Finally, the timelike and spacelike Bertrand curve mate visualized of given curves as examples, separately.
  Article Metrics


1. F. K. Aksoyak, İ. Gök, K. İlarslan, Generalized null Bertrand curves in Minkowski space-time, Ann. Alexandru Ioan Cuza Univ. Math., 60 (2014), 489-502.    

2. E. Arrondo, J. Sendra, R. Sendrab, Genus formula for generalized offset curves, J. Pure Appl. Algebra, 136 (1999), 199-209.    

3. B-Y. Chen, D-S. Kim, Y. H. Kim, New characterization of W-curves, Publ. Math. Debrecen, 69 (2006), 457-472.

4. S. L. Devadoss, J. O'Rourke, Discrete and computational geometry, Princeton University Press, (2011), 128-129.

5. Q. Ding, J. Inoguchi, Schrödinger flows, binormal motion for curves and second AKNS-hierarchies, Chaos Solitons Fractals, 21 (2004), 669-677.    

6. M. P. Do Carmo, Differential geometry of curves and surfaces, PRENTICE HALL, ISBN 10: 0132125897, New Jersey, 1976.

7. M. Erdoǧdu, Parallel frame of nonlightlike curves in Minkowski space-time, Int. J. Geom. M., 12 (2015), 1550109.

8. A. Yavuz, M. Erdoǧdu, Characterization of timelike curves: A new approach by system of differential equations for the position vector, Cumhuriyet J. Sci., In progress. 2020.

9. M. Erdoǧdu, A. Yavuz, On characterization and backlund transformation of null cartan curves, Turk J. Math., In progress. 2020.

10. K. İlarslan, Ö. Boyacıoǧlu, Position vectors of a space-like W-curve in Minkowski space $\mathbb{E}% _{1}^{3}$}, Bull. Korean Math. Soc., 44 (2007), 429-438.

11. K. İlarslan, N. K. Aslan, On space bertrand curves in Minkowski 3-space, Konuralp J. Math., 5 (2017), 214-222.

12. J. Inoguchi, Timelike surfaces of constant mean curvaturein Minkowski 3-space, Tokyo J. Math., 21 (1998), 141-152.    

13. J. Inoguchi, Biharmonic curves in Minkowski 3-space, Int. J. Math. Math. Sci., 2003 (2003), 1365- 1368.

14. P. Lucas, J. A. Ortega-Yagues, Bertrand curves in non-flat 3-dimensional (Riemannian or Lorentzian) space forms, Bull. Korean Math. Soc., 50 (2013), 1109-1126.    

15. T. Maekawa, An overview of offset curves and surfaces, Comput. Aided Des., 31 (1999), 165-173.    

16. A. Maǧden, S. Yılmaz, Y. Ünlütürk, Characterization of special time-like curves in Lorentzian plane, Int. J. Geom. Methods M., 14 (2017).

17. W. F. Newton, Theoretical and practical graphics, Macmillan, 1898.

18. M. Özdemir, A. A. Ergin, Rotations with unit timelike quaternions in Minkowski 3-space, J. Geom. Phys., 56 (2006), 322-336.    

19. M. Özdemir, A. A. Ergin, Parallel frames of non-lightlike curves, Mo. J. Math. Sci., 20 (2008), 127-137.

20. H. B. Öztekin, M. Bektas, Representation formulae for Bertrand curves in the Minkowski 3-space, Sci. Magna, 6 (2010), 89.

21. M. Petrovic-Torgasev, E. Sucurovic, W-curves in Minkowski spacetime, Novi. Sad. J. Math., 32 (2002), 55-65.

22. B. Pham, Offset curves and surfaces: A brief survey, Comput. Aided Des., 24 (1992), 223-239.    

23. W. K. Schief, On the integrability of Bertrand curves and Razzaboni surfaces, J. Geom. Phys., 45 (2003), 130-150.    

24. J. R. Sendra, F. Winkler, S. Pérez-Diaz, Rational algebraic curves: A computer algebra approach, Springer Science Business Media, 2007.

25. N. Tanrıöver, Some characterizations of Bertrand curves in Lorentzian n-space Ln, Int. J. Geom. M., 13 (2016), 1650064.

26. J. Walrave, Curves and surfaces in Minkowski space, Ph. D. thesis, K. U. Leuven, Fac. Sci. Leuven, 1995.

27. A. Yavuz, M. Erdoǧdu, A different approach by system of differential equations for the characterization position vector of spacelike curves, Punjab Univ. J. Math., In progress. 2020.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved