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Constant Mean Curvature Surfaces Along a Spacelike Curve

Year 2022, Volume: 43 Issue: 3, 454 - 459, 30.09.2022

Ergin Bayram

https://doi.org/10.17776/csj.1063743

Abstract

We construct constant mean curvature surfaces along a given spacelike curve in 3 dimensional Minkowski space. We parametrically present these surfaces using the famous Frenet frame of the curve in question. We give the sufficient conditions for the so called marching scale functions, which are the coefficients of the Frenet frame fields. We show that it is possible to obtain such surfaces for any given spacelike curve. Finally, the validity of the presented method is supported with illustrative examples.

Keywords

Spacelike curve Constant mean curvature surfaces Minkowski 3-space

References

  • [1] O'Neill B., Semi-Riemannian geometry with applications to relativity, Academic Press Lim., London, (1983).
  • [2] Willmore T.J., An introduction to differential geometry, Oxford University Press, Delhi, (1959).
  • [3] Carmo M.P., Differential geometry of curves and surfaces, Englewood Cliffs, Prentice Hall, (1976).
  • [4] Wang G.J., Tang K., Tai C.L., Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Design, 36 (2004) 447-459.
  • [5] Kasap E., Akyıldız F.T., Surfaces with common geodesic in Minkowski 3-space, Appl. Math. Comput, 177 (2006) 260-270.
  • [6] Li C.Y., Wang R.H., Zhu C.G., Parametric representation of a surface pencil with a common line of curvature, Comput. Aided Design, 43 (2011) 1110-1117.
  • [7] Bayram E., Güler F., Kasap E., Parametric representation of a surface pencil with a common asymptotic curve, Comput. Aided Design, 44 (2012) 637-643.
  • [8] Bayram E., Bilici M., Surface family with a common involute asymptotic curve, Int. J. Geom. Methods Modern Phys., 13 (2016) 1650062.
  • [9] Güler F., Bayram E., Kasap E., Offset surface pencil with a common asymptotic curve, Int. J. Geom. Methods Modern Phys., 15 (2018) 1850195.
  • [10] Şaffak Atalay G., Kasap E., Surfaces family with common null asymptotic, Appl. Math. Comput., 260 (2015) 135-139.
  • [11] Yüzbaşı Z.K., On a family of surfaces with common asymptotic curve in the Galilean space G3, J. Nonlinear Sci. Appl., 9 (2016), 518-523.
  • [12] Şaffak Atalay G.,Bayram E., Kasap E., Surface family with a common asymptotic curve in Minkowski 3-space, Journal of Science and Arts, 2 (43) (2018) 357-368.
  • [13] Yoon D.W., Yüzbaşı Z.K., Bektaş M., An approach for surfaces using an asymptotic curve in Lie Group, Journal of Advanced Physics, 6 (4) (2017) 586-590.
  • [14] Bayram E., Surface pencil with a common adjoint curve, Turkish Journal of Mathematics, 44 (2020) 1649 -1659.
  • [15] Coşanoğlu H., Bayram E., Surfaces with constant mean curvature along a curve in 3-dimensional Euclidean space, Süleyman Demirel University, Journal of Natural and Applied Sciences, 24 (3) (2020) 533-538.
  • [16] Mert T., Karlığa B., Timelike surfaces with constant angle in de-Sitter space, Boletim da Sociedade Paranaense de Matemática, 35 (2017), 79-93.
  • [17] Mert T., Atçeken M., Some special ruled surfaces in hyperbolic 3-space, Konuralp Journal of Mathematics, 9 (2), (2021), 298-309.
  • [18] Walrave J., Curves and surfaces in Minkowski space, PhD. Thesis, K. U. Leuven Faculteit Der Wetenschappen, 1995.
  • [19] Lopez R., Differential geometry of curves and surfaces in Lorentz-Minkowski space, Int. El. J. of Geometry, 7 (1) (2014) 44-107.

Year 2022, Volume: 43 Issue: 3, 454 - 459, 30.09.2022

Ergin Bayram

https://doi.org/10.17776/csj.1063743

Abstract

References

  • [1] O'Neill B., Semi-Riemannian geometry with applications to relativity, Academic Press Lim., London, (1983).
  • [2] Willmore T.J., An introduction to differential geometry, Oxford University Press, Delhi, (1959).
  • [3] Carmo M.P., Differential geometry of curves and surfaces, Englewood Cliffs, Prentice Hall, (1976).
  • [4] Wang G.J., Tang K., Tai C.L., Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Design, 36 (2004) 447-459.
  • [5] Kasap E., Akyıldız F.T., Surfaces with common geodesic in Minkowski 3-space, Appl. Math. Comput, 177 (2006) 260-270.
  • [6] Li C.Y., Wang R.H., Zhu C.G., Parametric representation of a surface pencil with a common line of curvature, Comput. Aided Design, 43 (2011) 1110-1117.
  • [7] Bayram E., Güler F., Kasap E., Parametric representation of a surface pencil with a common asymptotic curve, Comput. Aided Design, 44 (2012) 637-643.
  • [8] Bayram E., Bilici M., Surface family with a common involute asymptotic curve, Int. J. Geom. Methods Modern Phys., 13 (2016) 1650062.
  • [9] Güler F., Bayram E., Kasap E., Offset surface pencil with a common asymptotic curve, Int. J. Geom. Methods Modern Phys., 15 (2018) 1850195.
  • [10] Şaffak Atalay G., Kasap E., Surfaces family with common null asymptotic, Appl. Math. Comput., 260 (2015) 135-139.
  • [11] Yüzbaşı Z.K., On a family of surfaces with common asymptotic curve in the Galilean space G3, J. Nonlinear Sci. Appl., 9 (2016), 518-523.
  • [12] Şaffak Atalay G.,Bayram E., Kasap E., Surface family with a common asymptotic curve in Minkowski 3-space, Journal of Science and Arts, 2 (43) (2018) 357-368.
  • [13] Yoon D.W., Yüzbaşı Z.K., Bektaş M., An approach for surfaces using an asymptotic curve in Lie Group, Journal of Advanced Physics, 6 (4) (2017) 586-590.
  • [14] Bayram E., Surface pencil with a common adjoint curve, Turkish Journal of Mathematics, 44 (2020) 1649 -1659.
  • [15] Coşanoğlu H., Bayram E., Surfaces with constant mean curvature along a curve in 3-dimensional Euclidean space, Süleyman Demirel University, Journal of Natural and Applied Sciences, 24 (3) (2020) 533-538.
  • [16] Mert T., Karlığa B., Timelike surfaces with constant angle in de-Sitter space, Boletim da Sociedade Paranaense de Matemática, 35 (2017), 79-93.
  • [17] Mert T., Atçeken M., Some special ruled surfaces in hyperbolic 3-space, Konuralp Journal of Mathematics, 9 (2), (2021), 298-309.
  • [18] Walrave J., Curves and surfaces in Minkowski space, PhD. Thesis, K. U. Leuven Faculteit Der Wetenschappen, 1995.
  • [19] Lopez R., Differential geometry of curves and surfaces in Lorentz-Minkowski space, Int. El. J. of Geometry, 7 (1) (2014) 44-107.
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Ergin Bayram 0000-0003-2633-0991

Publication Date September 30, 2022
Submission Date January 27, 2022
Acceptance Date August 1, 2022
Published in Issue Year 2022Volume: 43 Issue: 3

Cite

APA Bayram, E. (2022). Constant Mean Curvature Surfaces Along a Spacelike Curve. Cumhuriyet Science Journal, 43(3), 454-459. https://doi.org/10.17776/csj.1063743
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