Korean J. Math.  Vol 24, No 4 (2016)  pp.613-626
DOI: https://doi.org/10.11568/kjm.2016.24.4.613

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

Fathi M. Hamdoon, A. K. Omran


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.


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

Subject classification

53A04, 53A05.


Full Text:



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. (Google Scholar)

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). (Google Scholar)

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

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

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. (Google Scholar)

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

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

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. (Google Scholar)

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

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

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

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

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

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


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