On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space

Şeyda Kılıçoğlu

doi:doi:10.11648/j.pamj.s.2015040102.15

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On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space

Şeyda Kılıçoğlu

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 1-2)

Received: 21 November 2014 Accepted: 24 November 2014 Published: 12 January 2015

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Abstract

Nil geometry is one of the eight geometries of Thurston's conjecture. In this paper we study in Nil 3-space and the Nil metric with respect to the standard coordinates (x,y,z) is gNil₃=(dx)²+(dy)²+(dz-xdy)² in IR³. In this paper, we find out the explicit parametric equation of a general helix. Further, we write the explicit equations Frenet vector fields, the first and the second curvatures of general helix in Nil 3-Space. The parametric equation the Normal and Binormal ruled surface of general helix in Nil 3-space in terms of their curvature and torsion has been already examined in [12], in Nil 3-Space.

Published in	Pure and Applied Mathematics Journal (Volume 4, Issue 1-2) This article belongs to the Special Issue Applications of Geometry
DOI	10.11648/j.pamj.s.2015040102.15
Page(s)	19-23
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Nil Space, Helix, Curvatures

References

[1]	Yildirim, A. and Hacısalihoglu, H.H., On BCV-Sasakian Manifold. International Mathematical Forum, 2011,Vol. 6, no. 34, 1669 - 1684.
[2]	Struik, D. J., Lectures on Classical Differential Geometry, Dover, New-York, 1988.
[3]	Turhan, E. and Körpınar, T. “ Parametric equations of general helices in the sol space Sol3,” Bol. Soc. Paran. Mat., 2013, 31(1), 99-104.
[4]	E. Wilson, "Isometry groups on homogeneous nilmanifolds", Geometriae Dedicata 12 (1982) 337—346.
[5]	Fastenakels, J., Munteanu, M.I. and J. van der Veken,“Constant angle surfaces in the Heisenberg Group,” Acta Mathematica Sinica, English Series Apr.,March 15, 2011, Vol. 27, No. 4, pp. 747–756.?
[6]	Kula, L. and Yayli, Y., “On slant helix and its spherical indicatrix,” Applied Mathematics and Computation, 169,600-607, 2005.
[7]	Lancret, M. A., Memoire sur les courbes `a double courbure, Memoires presentes alInstitut 1(1806), 416-454.
[8]	Ergüt, M., Turhan, E. and Körpınar, T., “On the Normal ruled surfaces of general helices in the Sol space Sol3”, TWMS J. Pure Appl. Math. V. 4, N. 2, 2013, 125-130.
[9]	Ekmekci, N. and Ilarslan, K., “Null general helices and submanifolds,” Bol. Soc. Mat. Mexicana, 2003, 9(2), 279-286.
[10]	Izumiya, S. and Takeuchi, N. “Special curves and Ruled surfaces,”. Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, 2003, Volume 44, No. 1, 203-212.
[11]	Kılıçoğlu, S., “On the Involutive B-scrolls in the Euclidean three-space E3.”, XIIIth. Geometry Integrability and Quantization, Varna, Bulgaria: Sofia 2012, pp 205-214
[12]	Kılıçoğlu, S. and Hacısalihoglu, H. H., “On the parametric equations of the Normal and Binormal ruled surface of general helices in Nil Space Nil₃,” unpublished.
[13]	Thurston, W., “Three-Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry.”, Bull. Amer. Math. Soc., 1982, 6, 357-381.
[14]	Ou, Y. and Wang, Z., “Linear biharmonic maps into Sol, Nil and Heisenberg Spaces.”, Mediterr. j. math., 2008, 5, 379-394.

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Şeyda Kılıçoğlu. (2015). On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space. Pure and Applied Mathematics Journal, 4(1-2), 19-23. https://doi.org/10.11648/j.pamj.s.2015040102.15

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Şeyda Kılıçoğlu. On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space. Pure Appl. Math. J. 2015, 4(1-2), 19-23. doi: 10.11648/j.pamj.s.2015040102.15

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AMA Style

Şeyda Kılıçoğlu. On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space. Pure Appl Math J. 2015;4(1-2):19-23. doi: 10.11648/j.pamj.s.2015040102.15

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@article{10.11648/j.pamj.s.2015040102.15,
  author = {Şeyda Kılıçoğlu},
  title = {On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {1-2},
  pages = {19-23},
  doi = {10.11648/j.pamj.s.2015040102.15},
  url = {https://doi.org/10.11648/j.pamj.s.2015040102.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040102.15},
  abstract = {Nil geometry is one of the eight geometries of Thurston's conjecture. In this paper we study in Nil 3-space and the Nil metric with respect to the standard coordinates (x,y,z) is gNil₃=(dx)²+(dy)²+(dz-xdy)² in IR³. In this paper, we find out the explicit parametric equation of a general helix. Further, we write the explicit equations Frenet vector fields, the first and the second curvatures of general helix in Nil 3-Space. The parametric equation the Normal and Binormal ruled surface of general helix in Nil 3-space in terms of their curvature and torsion has been already examined in [12], in Nil 3-Space.},
 year = {2015}
}

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TY  - JOUR
T1  - On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space
AU  - Şeyda Kılıçoğlu
Y1  - 2015/01/12
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.s.2015040102.15
DO  - 10.11648/j.pamj.s.2015040102.15
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JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
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PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.s.2015040102.15
AB  - Nil geometry is one of the eight geometries of Thurston's conjecture. In this paper we study in Nil 3-space and the Nil metric with respect to the standard coordinates (x,y,z) is gNil₃=(dx)²+(dy)²+(dz-xdy)² in IR³. In this paper, we find out the explicit parametric equation of a general helix. Further, we write the explicit equations Frenet vector fields, the first and the second curvatures of general helix in Nil 3-Space. The parametric equation the Normal and Binormal ruled surface of general helix in Nil 3-space in terms of their curvature and torsion has been already examined in [12], in Nil 3-Space.
VL  - 4
IS  - 1-2
ER  -

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Şeyda Kılıçoğlu

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APA Style

Şeyda Kılıçoğlu. (2015). On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space. Pure and Applied Mathematics Journal, 4(1-2), 19-23. https://doi.org/10.11648/j.pamj.s.2015040102.15

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ACS Style

Şeyda Kılıçoğlu. On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space. Pure Appl. Math. J. 2015, 4(1-2), 19-23. doi: 10.11648/j.pamj.s.2015040102.15

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AMA Style

Şeyda Kılıçoğlu. On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space. Pure Appl Math J. 2015;4(1-2):19-23. doi: 10.11648/j.pamj.s.2015040102.15

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@article{10.11648/j.pamj.s.2015040102.15,
  author = {Şeyda Kılıçoğlu},
  title = {On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {1-2},
  pages = {19-23},
  doi = {10.11648/j.pamj.s.2015040102.15},
  url = {https://doi.org/10.11648/j.pamj.s.2015040102.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040102.15},
  abstract = {Nil geometry is one of the eight geometries of Thurston's conjecture. In this paper we study in Nil 3-space and the Nil metric with respect to the standard coordinates (x,y,z) is gNil₃=(dx)²+(dy)²+(dz-xdy)² in IR³. In this paper, we find out the explicit parametric equation of a general helix. Further, we write the explicit equations Frenet vector fields, the first and the second curvatures of general helix in Nil 3-Space. The parametric equation the Normal and Binormal ruled surface of general helix in Nil 3-space in terms of their curvature and torsion has been already examined in [12], in Nil 3-Space.},
 year = {2015}
}

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TY  - JOUR
T1  - On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space
AU  - Şeyda Kılıçoğlu
Y1  - 2015/01/12
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.s.2015040102.15
DO  - 10.11648/j.pamj.s.2015040102.15
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 19
EP  - 23
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.s.2015040102.15
AB  - Nil geometry is one of the eight geometries of Thurston's conjecture. In this paper we study in Nil 3-space and the Nil metric with respect to the standard coordinates (x,y,z) is gNil₃=(dx)²+(dy)²+(dz-xdy)² in IR³. In this paper, we find out the explicit parametric equation of a general helix. Further, we write the explicit equations Frenet vector fields, the first and the second curvatures of general helix in Nil 3-Space. The parametric equation the Normal and Binormal ruled surface of general helix in Nil 3-space in terms of their curvature and torsion has been already examined in [12], in Nil 3-Space.
VL  - 4
IS  - 1-2
ER  -

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