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Mannheim Curves in 3-Dimensional Euclidean Space

Year 2020, , 86 - 89, 31.12.2020
https://doi.org/10.47897/bilmes.818723

Abstract

In this paper, we consider the Mannheim curve and the slant helix together. We called this curve as a Mannheim slant helix shortly. First we calculate the (first) curvature 𝜿(𝒔), and the curvature of the tangent indicatrix of the Mannheim curve, in terms of the arc-lenght parameter of the curve. Also, we proved that if the Mannheim curve is also slant helix, i.e. if it is Mannheim slant helix, then the partner curve is general helix. Moreover, we show the striction curve of the ruled surface such that the base curve is Mannheim curve, and the rulings are the normal vector field of the Mannheim curve, is the Mannheim partner curve. Finally, we show the ruled surface such that the base curve is Mannheim curve, and the rulings are the normal vector field of the Mannheim curve is non-developable while the torsion of the Mannheim partner curve 𝝉(𝒔)≠±∞ for all s.

References

  • [1] F. Wang and H. Liu “Mannheim partner curves in 3-space,” Proceedings of the eleventh international workshop on differential geometry ’11, pp. 25-31, 2007.
  • [2] K. Orbay and E. Kasap, “On Mannheim partner curves in E^3,” International Journal of Physical Sciences, vol. 4, no. 5, pp. 261-264, 2009.
  • [3] Y. Yaylı “Mannheim slant helix in Lorentz-Minkowski space,” International meeting on Lorentzian Geometry,’IX, p. 23, 2018.
  • [4] S. Honda and M. Takahashi, “Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space,” Turkish Journal of Mathematics, vol. 44, no. 3, pp. 883-899, 2020.
  • [5] S. Izumiya and N. Takeuchi “New special curves and developable surfaces,” Turkish Journal of Mathematics, vol. 28, no. 2, pp. 153-163, 2004.

Mannheim Curves in 3-Dimensional Euclidean Space

Year 2020, , 86 - 89, 31.12.2020
https://doi.org/10.47897/bilmes.818723

Abstract

In this paper, we consider the Mannheim curve and the slant helix together. We called this curve as a Mannheim slant helix shortly. First we calculate the (first) curvature 𝜿(𝒔), and the curvature of the tangent indicatrix of the Mannheim curve, in terms of the arc-lenght parameter of the curve. Also, we proved that if the Mannheim curve is also slant helix, i.e. if it is Mannheim slant helix, then the partner curve is general helix. Moreover, we show the striction curve of the ruled surface such that the base curve is Mannheim curve, and the rulings are the normal vector field of the Mannheim curve, is the Mannheim partner curve. Finally, we show the ruled surface such that the base curve is Mannheim curve, and the rulings are the normal vector field of the Mannheim curve is non-developable while the torsion of the Mannheim partner curve 𝝉(𝒔)≠±∞ for all s.

References

  • [1] F. Wang and H. Liu “Mannheim partner curves in 3-space,” Proceedings of the eleventh international workshop on differential geometry ’11, pp. 25-31, 2007.
  • [2] K. Orbay and E. Kasap, “On Mannheim partner curves in E^3,” International Journal of Physical Sciences, vol. 4, no. 5, pp. 261-264, 2009.
  • [3] Y. Yaylı “Mannheim slant helix in Lorentz-Minkowski space,” International meeting on Lorentzian Geometry,’IX, p. 23, 2018.
  • [4] S. Honda and M. Takahashi, “Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space,” Turkish Journal of Mathematics, vol. 44, no. 3, pp. 883-899, 2020.
  • [5] S. Izumiya and N. Takeuchi “New special curves and developable surfaces,” Turkish Journal of Mathematics, vol. 28, no. 2, pp. 153-163, 2004.
There are 5 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Emre Öztürk 0000-0001-6638-3233

Publication Date December 31, 2020
Acceptance Date December 31, 2020
Published in Issue Year 2020

Cite

APA Öztürk, E. (2020). Mannheim Curves in 3-Dimensional Euclidean Space. International Scientific and Vocational Studies Journal, 4(2), 86-89. https://doi.org/10.47897/bilmes.818723
AMA Öztürk E. Mannheim Curves in 3-Dimensional Euclidean Space. ISVOS. December 2020;4(2):86-89. doi:10.47897/bilmes.818723
Chicago Öztürk, Emre. “Mannheim Curves in 3-Dimensional Euclidean Space”. International Scientific and Vocational Studies Journal 4, no. 2 (December 2020): 86-89. https://doi.org/10.47897/bilmes.818723.
EndNote Öztürk E (December 1, 2020) Mannheim Curves in 3-Dimensional Euclidean Space. International Scientific and Vocational Studies Journal 4 2 86–89.
IEEE E. Öztürk, “Mannheim Curves in 3-Dimensional Euclidean Space”, ISVOS, vol. 4, no. 2, pp. 86–89, 2020, doi: 10.47897/bilmes.818723.
ISNAD Öztürk, Emre. “Mannheim Curves in 3-Dimensional Euclidean Space”. International Scientific and Vocational Studies Journal 4/2 (December 2020), 86-89. https://doi.org/10.47897/bilmes.818723.
JAMA Öztürk E. Mannheim Curves in 3-Dimensional Euclidean Space. ISVOS. 2020;4:86–89.
MLA Öztürk, Emre. “Mannheim Curves in 3-Dimensional Euclidean Space”. International Scientific and Vocational Studies Journal, vol. 4, no. 2, 2020, pp. 86-89, doi:10.47897/bilmes.818723.
Vancouver Öztürk E. Mannheim Curves in 3-Dimensional Euclidean Space. ISVOS. 2020;4(2):86-9.


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