In this paper, we define
dual curvature motion on the dual hyperbolic unit sphere H20 in the dual Lorentzian
space D31
with dual signature (+,+-) . We carry the obtained
results to the Lorentzian line space R31 by means of Study mapping.
Then we make an analysis of the orbits during the dual hyperbolic spherical
curvature motion. Finally, we find some line congruences, the family of ruled
surfaces and ruled surfaces in R31.
References
[1]
Study, E. (1903) Geometrie der Dynamen. Mathematiker Deutschland Publisher, Leibzig.
[2]
Guggenheimer, H.W. (1963) Differential Geometry. McGraw-Hill, New York.
[3]
Hacisalihoglu, H.H. (1972) On the Pitch of a Closed Ruled Surface. Mechanism and Machine Theory, 7, 291-305. http://dx.doi.org/10.1016/0094-114X(72)90039-0
[4]
Veldkamp, G.R. (1976) On the Use of Dual Number, Vector and Matrices in Instantaneous Spatial Kinematics. Mechanism and Machine Theory, 11, 141-156. http://dx.doi.org/10.1016/0094-114X(76)90006-9
[5]
Zha, X.F. (1997) A New Approach to Generation of Ruled Surfaces and Its Application in Engineering. The International Journal of Advanced Manufacturing Technology, 13, 155-163. http://dx.doi.org/10.1007/BF01305867
[6]
Ugurlu, H.H. and Caliskan, A. (1996) The Study Mapping for Directed Spacelike and Timelike Line in Minkowski 3-Space . Mathematical and Computational Applications, 1, 142-148.
[7]
Yayli, Y., Caliskan, A. and Ugurlu, H.H. (2002) The E Study Maps of Circles on Dual Hyperbolic and Lorentzian Unit Spheres and . Mathematical Proceedings of the Royal Irish Academy, 102A, 37-47.
[8]
Karger, A. and Novak, J. (1985) Space Kinematics and Lie Groups (Translated by Michal Basch). Gordon & Breach, New York.
[9]
Yapar, Z. (1989) On the Curvature Motion. Communications, Faculty of Sciences, University of Ankara, Series A1, 38, 103-114.
[10]
Ugurlu, H.H., Caliskan, A. and KiliC, O. (2001) On the Geometry of Spacelike Congruence. Communications Faculty of Sciences, University of Ankara, Series A1, 50, 9-24.
[11]
Kazaz, M., Ozdemir, A. and Ugurlu, H.H. (2009) Elliptic Motion on Dual Hyperbolic Unit Sphere . Mechanism and Machine Theory, 44, 1450-1459. http://dx.doi.org/10.1016/j.mechmachtheory.2008.11.006
[12]
Caliskan, A., Ugurlu, H.H. and KiliC, O. (2003) On the Geometry of Timelike Congruence. Hadronic Journal Supplement, 18, 311-328.
[13]
Akutagawa, K. and Nishikawa, S. (1990) The Gauss Map and Spacelike Surfaces with Prescribed Mean Curvature in Minkowski 3-Space. Tohoku Mathematical Journal, 42, 67-82. http://dx.doi.org/10.2748/tmj/1178227694
[14]
Birman, G.S. and Nomizu, K. (1984) Trigonometry in Lorentzian Geometry. American Mathematical Monthly, 91, 543-549. http://dx.doi.org/10.2307/2323737
[15]
O’Neil, B. (1983) Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London.
[16]
Yaglom, I.M. (1979) A Simple Non-Euclidean Geometry and Its Physical Basis. Springer-Verlag, New York.
