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AN EXAMINATION ON THE MANNHEIM FRENET RULED SURFACE BASED ON NORMAL VECTOR FIELDS IN E$^{3}$

Year 2016, Volume: 4 Issue: 2, 223 - 229, 01.10.2016

Abstract

In this paper we consider six special Frenet ruled surfaces along to the Mannheim pairs $\left\{ \alpha ^{\ast },\alpha \right\} $. First we define and find the parametric equations of Frenet ruled surfaces which are called $% Mannheim$ $Frenet$ $ruled$ $surface$, along Mannheim curve $\alpha ,$ in terms of the Frenet apparatus of Mannheim curve $\alpha .$ Later, \ we find only one matrix gives us all nine positions of normal vector fields of these six Frenet ruled surfaces and $Mannheim$ $Frenet$ $ruled$ $surface$ in terms of Frenet apparatus of Mannheim curve $\alpha $ too. Further using that matrix we have some results such as; normal ruled surface and $Mannheim$ $% normal$ $ruled$ $surface$ of Mannheim curve $\alpha $ have perpendicular normal vector fields along the curve $\varphi _{2}\left( s\right) =\alpha +% \frac{\tan \theta }{k_{1}\tan \theta -k_{2}}V_{2},$ \ under the condition $% \tan \theta \neq \frac{k_{2}}{k_{1}}$.

References

  • [1] Do Carmo, M. P., DiFFerential Geometry of Curves and Surfaces, Prentice-Hall, isbn 0-13-212589-7, 1976.
  • [2] Ergut M., Korpinar T. and Turhan E., On Normal Ruled Surfaces of General Helices In The Sol Space Sol3, TWMS J. Pure Appl. Math., 4(2), 125-130, 2013.
  • [3] Graves L.K., Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc., 252, 367{392, 1979.
  • [4] Kilicoglu, S. Some Results on Frenet Ruled Surfaces Along the Evolute-Involute Curves, Based on Normal Vector Fields in E3. Proceedings of the Seventeenth International Conference on Geometry, Integrability and Quantization, 296{308, Avangard Prima, So a, Bulgaria, 2016. doi:10.7546/giq-17-2016-296-308. http://projecteuclid.org/euclid.pgiq/1450194164.
  • [5] Kilicoglu, S., On the Involute B-scrolls in the Euclidean Three-space E3. XIIIth Geometry Integrability and Quantization, Varna, Bulgaria: So a, 205-214, 2012.
  • [6] Kilicoglu S., Senyurt S., Hacisalihoglu H. H., An examination on the positions of Frenet ruled surfaces along Bertrand pairs and  according to their normal vector elds in E3 Applied Mathematical Sciences, Vol. 9, 2015, no. 142, 7095-7103 http://dx.doi.org/10.12988/ams.2015.59605
  • [7] Liu H. andWang F., Mannheim partner curves in 3-space, Journal of Geometry, 2008, 88(1-2), 120-126(7).
  • [8] Orbay K. and Kasap E., On Mannheim partner curves in E3 ,International Journal of Physical Sciences, 2009, 4 (5), 261-264.
  • [9] Orbay K, Kasap E, Aydemir _I. Mannheim O sets of Ruled Surfaces. Mathematical Problems in Engineering. Volume 2009, Article ID 160917, 9 pages doi:10.1155/2009/160917
  • [10] Senyurt S. and Kilicoglu, S., On the di erential geometric elements of the involute ~D scroll, Adv. Appl. Clifford Algebras Springer Basel, doi:10.1007/s00006-015-0535-z, 25(4), 977-988, 2015.
Year 2016, Volume: 4 Issue: 2, 223 - 229, 01.10.2016

Abstract

References

  • [1] Do Carmo, M. P., DiFFerential Geometry of Curves and Surfaces, Prentice-Hall, isbn 0-13-212589-7, 1976.
  • [2] Ergut M., Korpinar T. and Turhan E., On Normal Ruled Surfaces of General Helices In The Sol Space Sol3, TWMS J. Pure Appl. Math., 4(2), 125-130, 2013.
  • [3] Graves L.K., Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc., 252, 367{392, 1979.
  • [4] Kilicoglu, S. Some Results on Frenet Ruled Surfaces Along the Evolute-Involute Curves, Based on Normal Vector Fields in E3. Proceedings of the Seventeenth International Conference on Geometry, Integrability and Quantization, 296{308, Avangard Prima, So a, Bulgaria, 2016. doi:10.7546/giq-17-2016-296-308. http://projecteuclid.org/euclid.pgiq/1450194164.
  • [5] Kilicoglu, S., On the Involute B-scrolls in the Euclidean Three-space E3. XIIIth Geometry Integrability and Quantization, Varna, Bulgaria: So a, 205-214, 2012.
  • [6] Kilicoglu S., Senyurt S., Hacisalihoglu H. H., An examination on the positions of Frenet ruled surfaces along Bertrand pairs and  according to their normal vector elds in E3 Applied Mathematical Sciences, Vol. 9, 2015, no. 142, 7095-7103 http://dx.doi.org/10.12988/ams.2015.59605
  • [7] Liu H. andWang F., Mannheim partner curves in 3-space, Journal of Geometry, 2008, 88(1-2), 120-126(7).
  • [8] Orbay K. and Kasap E., On Mannheim partner curves in E3 ,International Journal of Physical Sciences, 2009, 4 (5), 261-264.
  • [9] Orbay K, Kasap E, Aydemir _I. Mannheim O sets of Ruled Surfaces. Mathematical Problems in Engineering. Volume 2009, Article ID 160917, 9 pages doi:10.1155/2009/160917
  • [10] Senyurt S. and Kilicoglu, S., On the di erential geometric elements of the involute ~D scroll, Adv. Appl. Clifford Algebras Springer Basel, doi:10.1007/s00006-015-0535-z, 25(4), 977-988, 2015.
There are 10 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

SEYDA Kılıcoglu

Publication Date October 1, 2016
Submission Date December 27, 2015
Published in Issue Year 2016 Volume: 4 Issue: 2

Cite

APA Kılıcoglu, S. (2016). AN EXAMINATION ON THE MANNHEIM FRENET RULED SURFACE BASED ON NORMAL VECTOR FIELDS IN E$^{3}$. Konuralp Journal of Mathematics, 4(2), 223-229.
AMA Kılıcoglu S. AN EXAMINATION ON THE MANNHEIM FRENET RULED SURFACE BASED ON NORMAL VECTOR FIELDS IN E$^{3}$. Konuralp J. Math. October 2016;4(2):223-229.
Chicago Kılıcoglu, SEYDA. “AN EXAMINATION ON THE MANNHEIM FRENET RULED SURFACE BASED ON NORMAL VECTOR FIELDS IN E$^{3}$”. Konuralp Journal of Mathematics 4, no. 2 (October 2016): 223-29.
EndNote Kılıcoglu S (October 1, 2016) AN EXAMINATION ON THE MANNHEIM FRENET RULED SURFACE BASED ON NORMAL VECTOR FIELDS IN E$^{3}$. Konuralp Journal of Mathematics 4 2 223–229.
IEEE S. Kılıcoglu, “AN EXAMINATION ON THE MANNHEIM FRENET RULED SURFACE BASED ON NORMAL VECTOR FIELDS IN E$^{3}$”, Konuralp J. Math., vol. 4, no. 2, pp. 223–229, 2016.
ISNAD Kılıcoglu, SEYDA. “AN EXAMINATION ON THE MANNHEIM FRENET RULED SURFACE BASED ON NORMAL VECTOR FIELDS IN E$^{3}$”. Konuralp Journal of Mathematics 4/2 (October 2016), 223-229.
JAMA Kılıcoglu S. AN EXAMINATION ON THE MANNHEIM FRENET RULED SURFACE BASED ON NORMAL VECTOR FIELDS IN E$^{3}$. Konuralp J. Math. 2016;4:223–229.
MLA Kılıcoglu, SEYDA. “AN EXAMINATION ON THE MANNHEIM FRENET RULED SURFACE BASED ON NORMAL VECTOR FIELDS IN E$^{3}$”. Konuralp Journal of Mathematics, vol. 4, no. 2, 2016, pp. 223-9.
Vancouver Kılıcoglu S. AN EXAMINATION ON THE MANNHEIM FRENET RULED SURFACE BASED ON NORMAL VECTOR FIELDS IN E$^{3}$. Konuralp J. Math. 2016;4(2):223-9.
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