Studying on a skew ruled surface by using the geodesic Frenet trihedron of its generator

Fathi M. Hamdoon, A. K. Omran

Abstract


In this article, we study skew ruled surfaces by using the geodesic Frenet trihedron of its generator.  We obtained some conditions on this surface to ensure that this ruled surface is flat, II-flat, minimal, II-minimal and Weingarten surface.  Moreover, the  parametric equations of asymptotic and geodesic lines on this ruled surface are determined and illustrated through example using the program of mathematica.

Keywords


Ruled surfaces, Geodesic Frenet trihedron, Geodesic and Asymptotic lines.

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References


Abdel-All, N. H. and Hamdoon, F. M., Extension B-scroll surfaces in Lorentz 3-dimensional space, Rivista Di Math., Parma 6(3) (2000), 57–67.

Abdel-All, N. H. and Hamdoon, F. M., Timelike ruled surfaces immersed in Lorentz space, International conference Centennial Vranceanu Bucharest, July (2000) (Vranceanu proceedings March 2001).

Abdel-All, N. H., Abdel-Baky, R. A. and Hamdoon, F. M., Ruled surfaces with timelike rullings, Appl. Math. comput., 147 (2004), 241–253.

Abd-Ellah, H. N., Translation L/W-surfaces in Euclidean 3-space E3, Journal of the Egyptian Mathematical Society, 23 (2015) 513–517.

Ali, A., Abdel Aziz, H. and Sorour, A., Ruled surfaces generated by some special curves in Euclidean 3-Space, Journal of the Egyptian Mathematical Society, 21 (2013), 285–294.

Carmo, M. P., Differential geometry of curves and surfaces, Prentice- Hall, Englewood Cliffs, NJ, (1976).

Dillen, F. and Sodsiri, W., Ruled surfaces of Weingarten type in Minkowski 3-space, J. Geom., 83 (2005) 10–21.

Kim, Y. H. and Yoon, D. W., On Non-Developable Ruled Surfaces in Lorentz-Minkoweski 3- Space, Taiwanese Journal of mathematics, 1(11) (2007), 197–214.

Kim, Y. H. and Yoon, D. W., Classification of ruled surfaces in Minkowski 3-space, J. Geometry Phys., 49 (2004) 89–100.

O’Neill, B., Elementary Differential geometry, Academic Press ,(1966).

Onder, M and Huseyin, H., Some Results and Characterizations for Mannheim Offsets of Ruled Surfaces, Bol. Soc. Paran. Mat., 34 (2015), 85–95.

Orbay, K. and Kasap, E., Mannheim offsets of ruled surfaces, Mathematical proplems in engineering, (2009).

Ravani, B. and Ku, T., Bertrand offsets of ruled and developable surfaces, Computer-Aided Design, 2(23) (1991), 145–152.

Weatherburn, C. E., Differential Geometry of Three Dimensions, Cambridge, UK, (1930).




DOI: http://dx.doi.org/10.11568/kjm.2016.24.4.613

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