Surfaces Family with Common Smarandache Geodesic Curve According to Bishop Frame in Euclidean Space

Year 2016, Volume: 4 Issue: 1, 164 - 174, 15.04.2016

Gülnur Şaffak Atalay , Emin Kasap

https://doi.org/10.36753/mathenot.421425

Cited By: 8

Abstract

In this paper, we analyzed the problem of consructing a family of surfaces from a given some special
Smarandache curves in Euclidean 3-space. Using the Bishop frame of the curve in Euclidean 3-space,
we express the family of surfaces as a linear combination of the components of this frame, and derive
the necessary and sufficient conditions for coefficents to satisfy both the geodesic and isoparametric
requirements. Finally, examples are given to show the family of surfaces with common Smarandache
geodesic curve.

Keywords

Bishop frame

References

• [1] B. O’Neill, Elementary Differential Geometry, Academic Press Inc., New York, 1966.
• [2] M.P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1976.
• [3] B. O’Neill, Semi-Riemannian Geometry, Academic Press , New York, 1983.
• [4] M. Turgut, and S. Yilmaz, Smarandache Curves in Minkowski Space-time, International Journal of Mathematical Combinatorics, Vol.3, pp.51-55.
• [5] Ali, Ahmad.T. , Special Smarandache Curves in Euclidean Space, International Journal of Mathematical Combinatorics, Vol.2, pp.30-36, 2010.
• [6] Çetin, M. , Tunçer Y. , Karacan, M.K. , Smarandache Curves According to Bishop Frame in Euclidean Space. arxiv : 1106. 3202v1 [math. DG] , 16 Jun 2011.
• [7] Bektaş, Ö. and Yüce, S. , Smarandache Curves According to Darboux Frame in Euclidean Space. Romanian Journal of Mathematics and Computer Science, 2013, Volume 3, Issue 1, p.48-59.
• [8] Bayrak, N. , Bektaş, Ö. and Yüce, S. , Smarandache Curves in Minkowski Space. arxiv : 1204. 5656v1 [math. HO] , 25 Apr 2012.
• [9] Taşköprü, K. ,and Tosun. M. , Smarandache Curves According to Sabban Frame on . Boletim da Sociedade Paraneanse de Matematica, vol,32, no.1, pp.51-59,2014.
• [10] Çetin, M. , and Kocayiğit, H. , On the Quaternionic Smarandache Curves in Euclidean 3-Space. Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 3, 139 – 150.
• [11] Deng, B. , 2011. Special Curve Patterns for Freeform Architecture Ph.D. thesis, Eingereicht an der Technischen Universitat Wien, Fakultat für Mathematik und Geoinformation von.
• [12] G. J. Wang, K. Tang, C. L. Tai, Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Des. 36 (5)(2004) 447-459.
• [13] E. Kasap, F.T. Akyildiz, K. Orbay, A generalization of surfaces family with common spatial geodesic, Applied Mathematics and Computation, 201 (2008) 781-789.
• [14] C.Y. Li, R.H. Wang, C.G. Zhu, Parametric representation of a surface pencil with a common line of curvature, Comput. Aided Des. 43 (9)(2011) 1110-1117.
• [15] L. R. Bishop, “There is more than one way to Frame a Curve”, Amer. Math. Monthly 82(3) (1975) 246-251.B. O’Neill, Semi-Riemannian Geometry, Academic Press , New York, 1983.