Surfaces family with common Smarandache asymptotic curve
DOI:
https://doi.org/10.5269/bspm.v34i1.24392Abstract
In this paper, we analyzed the problem of constructing a family of surfaces from a given some special Smarandache curves in Euclidean 3-space. Using the Frenet 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 coefficients to satisfy both the asymptotic and isoparametric requirements. Finally, examples are given to show the family of surfaces with common Smarandache curve.
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, A.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. Bektas, Ö. and Yüce, S. , Smarandache Curves According to Darboux Frame in Euclidean Space. arxiv : 1203. 4830v1 [math. DG] , 20 Mar 2012.
8. Bayrak, N. , Bektas, Ö. and Yüce, S. , Smarandache Curves in Minkowski Space. arxiv : 1204. 5656v1 [math. HO] , 25 Apr 2012.
9. Tasköprü, 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 Kocayigit, H. , On the Quaternionic Smarandache Curves in Euclidean 3-Space. Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 3, 139 – 150.
11. W. Klingenberg, A Course in Differential Geometry, Springer-Verlag, New York, 1978.
12. D.J. Struik, Lectures on Classical Differential Geometry, Dover Publications Inc., New York, 1961.
13. 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.
14. E. Kasap, F.T. Akyildiz, K. Orbay, A generalization of surfaces family with common spatial geodesic, Applied Mathematics and Computation, 201 (2008) 781-789.
15. 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.
16. Bayram E. , Güler F. , Kasap E. Parametric representation of a surface pencil with a common asymptotic curve. Comput. Aided Des. (2012).
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, A.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. Bektas, Ö. and Yüce, S. , Smarandache Curves According to Darboux Frame in Euclidean Space. arxiv : 1203. 4830v1 [math. DG] , 20 Mar 2012.
8. Bayrak, N. , Bektas, Ö. and Yüce, S. , Smarandache Curves in Minkowski Space. arxiv : 1204. 5656v1 [math. HO] , 25 Apr 2012.
9. Tasköprü, 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 Kocayigit, H. , On the Quaternionic Smarandache Curves in Euclidean 3-Space. Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 3, 139 – 150.
11. W. Klingenberg, A Course in Differential Geometry, Springer-Verlag, New York, 1978.
12. D.J. Struik, Lectures on Classical Differential Geometry, Dover Publications Inc., New York, 1961.
13. 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.
14. E. Kasap, F.T. Akyildiz, K. Orbay, A generalization of surfaces family with common spatial geodesic, Applied Mathematics and Computation, 201 (2008) 781-789.
15. 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.
16. Bayram E. , Güler F. , Kasap E. Parametric representation of a surface pencil with a common asymptotic curve. Comput. Aided Des. (2012).
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2014-09-22
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