Similar Ruled Surfaces with Variable Transformations in Minkowski 3-Space E_1^3

Abstract

In this study, we consider the notion of similar ruled surface for timelike andspacelike ruled surfaces in Minkowski 3-space E3. We obtain some properties of thesespecial surfaces in E31and show that developable ruled surfaces in E1form a family ofsimilar ruled surfaces if and only if the striction curves of the surfaces are similar curveswith variable transformation. Moreover, we obtain that cylindrical surfaces and conoidsform two families of similar ruled surfaces in E1

Keywords

• Timelike and spacelike ruled surfaces,Frenet frame,similar curve.

References

1. [1] Beem, J.K. and Ehrlich, P.E., (1981), Global Lorentzian Geometry, Marcel Dekker, New York.
2. [2] El-Sabbagh, M.F. and Ali, A.T., (2013), Similar Curves with Variable Transformations, Konuralp J. Math., 1(2), pp. 80-90.
3. [3] Kim, Y.H. and Yoon, W.D., (2000), Ruled surfaces with pointwise 1-type Gauss map, Journal of Geometry and Physics, 34, pp. 191-205.
4. [4] K¨u¸c¨uk, A., (2004), On the developable timelike trajectory ruled surfaces in Lorentz 3-space IR3 1, App. Math. and Comp., 157, pp. 483-489.
5. [5] K¨u¸c¨uk, A. and G¨ursoy, O., (2004), On the invariants of Bertrand trajectory surface offsets, App. Math. and Comp., 151, pp. 763-773.
6. [6] Liu, H. and Wang, F., (2008), Mannheim partner curves in 3-space, Journal of Geometry, 88(1-2) pp. 120-126.
7. [7] O’Neill, B., (1983), Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London.
8. [8] Orbay, K., Kasap, E. and Aydemir, ˙I., (2009), Mannheim Offsets of Ruled Surfaces, Mathematical Problems in Engineering, Article ID 160917.

9. [9] Onder, M., (2014), Similar Ruled Surfaces with Variable Transformations in Euclidean 3-space, Konu- ¨ ralp J. Math. 2(2), pp. 127-136.
10. [10] Onder, M. and Uˇgurlu, H.H., (2013), Frenet Frames and Invariants of Timelike Ruled Surfaces, Ain ¨ Shams Eng J. 4, pp. 507-513.
11. [11] Papaioannou, S.G. and Kiritsis, D., (1985), An application of Bertrand curves and surfaces to CAD/CAM, Computer Aided Design, 17(8), pp. 348-352.
12. [12] Pottmann, H., L, W. and Ravani, B., (1996), Rational ruled surfaces and their offsets, Graphical Models and Image Processing, 58(6), pp. 544-552.
13. [13] Ravani, B. and Ku, T.S., (1991), Bertrand Offsets of ruled and developable surfaces, Comp. Aided Geom. Design, 23(2), pp. 145-152.
14. [14] Struik, D.J., (1988), Lectures on Classical Differential Geometry, 2nd ed. Addison Wesley, Dover.
15. [15] Turgut, A. and Hacısalihoˇglu, H.H., (1997), Timelike ruled surfaces in the Minkowski 3-space, Far East J. Math. Sci. 5(1), pp. 83-90.
16. [16] Uˇgurlu, H.H. and Onder, M., (2009), On Frenet Frames and Frenet Invariants of Skew Spacelike Ruled ¨ Surfaces in Minkowski 3-space, VII. Geometry Symposium, Kırehir, Turkey.
17. [17] Wang, F. and Liu, H., (2007), Mannheim partner curves in 3-Euclidean space, Mathematics in Practice and Theory, 37(1), pp. 141-143