FRENET FRAMES AND FRENET INVARIANTS OF SPACELIKE RULED SURFACES

Year 2017, Volume: 19 Issue: 57, 712 - 722, 01.09.2017

Mehmet Önder Hasan Hüseyin Uğurlu

Abstract

In this study, we introduce the Chasles theorem for spacelike ruled surfaces and give the Frenet frames and invariants of a spacelike ruled surface and of its directing cone. We show that a spacelike ruled surface and its directing cone have the same Frenet frame

Keywords

Minkowski 3space, Spacelike ruled surface, Frenet frame

SPACELİKE REGLE YÜZEYLERİN FRENET ÇATILARI VE FRENET İNVARYANTLARI

Abstract

1Bağımsız Araştırmacı, Delibekirli Mahallesi, Tepe Sokak, No: 63, 31440, Kırıkhan, Hatay 2Gazi Üniversitesi, Gazi Eğitim Fakültesi, Orta Öğretim Fen ve Matematik Alanları Eğitimi Bölümü, 06500, Ankara

Keywords

Minkowski 3- uzayı, Spacelike regle yüzey, Frenet çatısı

References

• Beem, J.K., Ehrlich, P.E. 1981. Global Lorentzian Dekker, New York. Marcel
• Dillen, F., Sodsiri, W. 2005. Ruled surfaces of Weingarten type in Minkowski 3-space: J. Geom., Vol. 83, No. DOI:10.1007/s00022-005-0002-4
• Ekici, C., Özüsağlam, E. 2012. On the Method of Determination of a Developable Timelike Ruled Surface: KJSE- Kuwait Journal of Science & Engineering, Vol. 39(1A), pp. 19-41.
• Ekici, C., Öztürk, H. 2013. On Timelike Ruled Surfaces in Minkowski 3- Space: Universal Journal of Applied Science, Vol. 1, No. 2, pp. 56-63. DOI: 10.13189/ujas.2013.010205
• Greub, W. 1975. Linear Algebra, Fourth ed., Springer-Verlag, New York.
• Guggenheimer, H. 1963. Differential Geometry, McGraw- Hill Book Comp. Inc. London, Lib. Cong. Cat. Card No. 68-12118.
• Karger, A. Novak, J. 1978. Space Kinematics and Lie Groups. STNL Publishers of Technical Lit., Prague, Czechoslovakia.
• Kim, Y.H., Yoon, W.D. 2004. Classification of ruled surfaces in Minkowski 3-space: Journal of Geometry and Physics, Vol. 49(1), pp. 89-100. 0440(03)00084-6.
• Küçük, A. 2004. On the developable timelike trajectory ruled surfaces in Lorentz 3-space IR: App. Math. Comput., Vol. 157(2), pp. 483-489. DOI: 10.1016/j.amc.2003.09.001.
• O’Neill, B. 1983. Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London.
• Önder, M., Uğurlu, H.H. 2013. Frenet Frames and Invariants of Timelike Ruled Surfaces: Ain Shams Eng J., Vol. 4, pp. 507-513. DOI: 10.1016/j.asej.2012.10.003.
• Peternel, M., Pottmann, H., Ravani, B. 1999. geometry of ruled surfaces: Comp. Aided Geom. Design, Vol. 31, pp. 17- 32. 4485(98)00077-3.
• Ratcliffe, J.G. 2006. Foundations of Hyperbolic Manifolds, Springer.
• Ravani, B., Ku, T.S. 1991. Bertrand Offsets of ruled and developable surfaces: Comp. Aided Geom. Design, Vol. 23, No. 2, pp. 147-152. DOI: 10.1016/0010-4485(91)90005-H
• Turgut, A,, Hacısalihoğlu, H.H. 1997. Timelike ruled surfaces in the Minkowski 3-space: Far East J. Math. Sci., Vol. 5, No. 1, pp. 83-90.
• Uğurlu, H.H., Çalışkan, A. 2012. Darboux Ani Dönme Vektörleri ile Spacelike ve Timelike Yüzeyler Geometrisi. Celal Bayar Üniversitesi Yayınları, Yayın No: 0006.
• Wang, D.L., Liu, J., Xiao, D.Z. 1997. Kinematic Differential Geometry of a Rigid Body in Spatial Motion I-A New Adjoin Approach and Instantaneous Properties of a Point Trajectory in Spatial Kinematics: Mech. and Mach. Theory, Vol. 32, No. 4, pp. 419-432. DOI: 114X(96)00075-4
• Wang, D.L., Liu, J., Xiao, D.Z. 1997. Kinematic Differential Geometry of a Rigid Body in Spatial Motion II-A New Instantaneous Properties of a Line Trajectory in Spatial Kinematics: Mech. and Mach. Theory, Vol. 32, No. 4, pp. 433-444. DOI: 10.1016/S0094- 114X(96)00076-6. and
• Motion: Mech. and Mach. Theory, Vol. , No. 4, pp. 445-457. DOI: 1016/S0094-114X(96)00077-8.