K. ARSLAN, P. ALEGR, Alfonso CARRİAZO, C. MURATHAN, G. ÖZTÜRK

Some special types of developable ruled surface

In this study we consider the focal curve C $gamma$ of a space curve $gamma$ and its focal curvatures. We characterize some special types of ruled surface, choosing one of the base curves or director curves as the focal curve of the space curve $gamma$ . Finally we construct new types of ruled surface and calculate their distinguished parameters. We give necessary and sufficient conditions for these types of ruled surface to become developable.

PDF

___

[1] Chen, B.Y. When does the position vector of a space curve always lie in its rectifying plane?, Am. Math Monthly 110, 147–152, 2003.
[2] Eisenhart, L.P. A Treatise on the Differential Geometry of Curves and Surfaces (Dover Pub. Inc., New York, 1909).
[3] Gluck, H. Higher curvatures of curves in Euclidean space, Am. Math. Monthly 73, 699–704, 1966.
[4] Gray, A. Modern Differential Geometry of Curves and Surfaces (CRC Press, Boca Raton, FL, 1993).
[5] Gursoy, O. Some results on closed ruled surfaces and closed space curves, Mech. Mach. Theory 27, 323–330, 1990.
[6] Hacısalihoğlu, H.H. Diferensiyel Geometri (Gazi Üniversitesi Basın-Yayın Yüksekokulu Basımevi, Ankara, 1983) (In Turkish).
[7] Izumiya, S. and Takeuchi, N. New special curves and developable surfaces, Turkish J. Math. 28, 153–163, 2004.
[8] Izumiya, S. and Takeuchi, N. Special curves and ruled surfaces, Cont. to Alg. and Geo. 44, 200–212, 2003.
[9] Izumiya, S. and Takeuchi, N. Geometry of Ruled Surfaces (Applicable mathematics in the Goldon Age (ed. J.C. Misra), 305–308, Narosa Pulishing House, New Delhi, 2003).
[10] Izumiya, S., Katsumi, H. and Yamashi, T. The rectifying developable and spherical Darboux image of a space curve, Geom. and Top. of caustics - Caustics’98 Banach Center Publications 50, 137–149, 1999.
[11] Koenderink, J. Solid Shape (MIT Press, Cambridge MA, 1990).
[12] Laugwitz, D. Differential and Riemannian Geometry (Academic Press, New York, 1965).
[13] Monterde, J. Curves with constant curvatures ratios, arXiv:math/0412323v1.
[14] Ozyilmaz, E. and Yayli, Y. On the closed space-like developable ruled surface, Hadronic J. 23 (4), 439–456, 2000.
[15] Uribe-Vargas, R. On vertices, focal curvatures and differential geometry of space curves, Bull. Brazilian Math. Soc. 36 (3), 285–307, 2005.