BLASCHKE APPROACH TO EULER-SAVARY FORMULAE FOR ONE PARAMETER DUAL HYPERBOLIC SPHERICAL MOTION

Year 2016, Volume: 4 Issue: 2, 95 - 115, 01.10.2016

ZEHRA Ekıncı H. HUSEYIN Ugurlu

Abstract

In this paper, we have introduced one parameter dual hyperbolic spherical motions in the dual Lorentzian space. This examination is given using Blaschke frame of axodes corresponding to the curves on the unit dual hyperbolic sphere. By considering Disteli axis on the Blaschke frame we have obtained Euler Savary formulae for one parameter dual hyperbolic spherical motions. At the end of this study, by obtaining orthogonal rotation matrices in the sense of dual Lorentzian type, we have found real and dual invariants of fixed and moving axodes.

Keywords

Euler-Savary formulae, Lorentzian geometry, Motion geometry

References

• [1] Abdel-All N.H., Abdel-Baky R. A., Hamdoon F. M.,Ruled Surfaces with Timelike Rulings, App. Math. And Comp., 147 (2004) 241-253.
• [2] Abdel-Baky, R.A., Al-Solamy, F. R., A New Geometrical Approach to One-Parameter Spatial Motion, J. Eng. Math., 60 (2008) 149-172.
• [3] Abdel-Baky, R.A., Al-Ghefari, R.A.,On the One Parameter Dual Spherical Motions, Comp. Aided Geom. Design, 28 (2011) 23-37.
• [4] Angeles, J., The Application of Dual Algebra to Kinematic Analysis, In J. Angeles, E. Zakhariev (eds): Computational Methods in Mechanical Systems, volume 161, pages 3-31, Heidelberg, Springer- Verlag, 1998.
• [5] Aydogmus, O.H.,Lorentz Uzay Hareketleri ve Lie Gruplar, Ankara Univerisitesi Fen Bilimleri Enstitusu, Yuksek Lisans Tezi, 2007.
• [6] Birman, G.S., Nomizu, K., Trigonometry in Lorentzian geometry, Amer. Math. Montly, 91(9) (1984) 543-549.
• [7] Blaschke, W., Differential Geometrie and Geometrischke Grundlagen ven Einsteins Relativitasttheorie Dover", New York, 1945.
• [8] Gungor, M.A., Lorentz Uzaynda Bir Prametreli Dual Hareketler, Sakarya Universitesi Fen Bilimleri Enstitusu, Doktora Tezi, 2006.
• [9] Hacisalihoglu, H.H., Hareket Geometrisi ve Kuaterniyonlar Teorisi", Gazi Universitesi Fen- Edb. Fakultesi, 1983.
• [10] Karger A., Space Kinematics and Lie Groups", Gordon and Breach Science Publishers, New York, 1985
• [11] Kim, Y. H., Yoon, W. D.,Classication of Ruled Surfaces in Minkowski 3-space, J. of Geom. and Phiysics, 49 (2004) 89-100.
• [12] Kotel'nikov, A.P., \Screw Calculus and Some Applications to Geometry and Mechanics" Annals of the Imperial of Kazan, 1895.
• [13] Lopez, R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space", Mini-Course taught at the IME-USP, Brazil, 2008.
• [14] Muller, H.R., Kinematik Dersleri" (ceviri), Ankara Universitesi Fen Fakultesi yayinlari 27, 1963.
• [15] O'Neill, B., Semi Riemannian Geometry", Academic Press, New York-London, 1983.
• [16]  Onder, M., Ugurlu, H.H., Caliskan, A., The Euler{Savary Analogue Equations of a Point Trajectory in Lorentzian Spatial Motion, Proc. Natl.Acad. Sci., India, Sect. APhys. Sci., 83(2) (2013) 119-127.
• [17]  Onder, M., Reel ve Dual Uzaylarda Regle Yuzeylerin Mannheim Ofsetleri, Celal Bayar Universitesi Fen Bilimleri Enstitusu, Doktora Tezi, 2012.
• [18] Ratcliffe, J.G., \Foundations of Hyperbolic Manifolds", Springer, New York, 2006.
• [19] Schaaf, J.A., Curvature Theory of Line Trajectories in Spatial Kinematics, University of California, PhD Thesis, Davis, 1988
• [20] Study, E., \Geometrie der Dynamen", Verlag Teubner, Leipzig, 1903.
• [21] Senol, A., Dual Kuresel Timelike ve Spacelike Egrilerin Geometrisi ve  Ozel Regle Yuzeyler, Celal Bayar Universitesi Fen Bilimleri Enstitusu, Yuksek Lisans Tezi, 2000.
• [22] Tosun, M., Gungor, M. A., Hacsalihoglu, H. H., Okur, I., A Study on the one Parameter Lorentzian Spherical Motions, Acta. Math. Univ. Comenianae, Vol: LXXV, 1 (2006) 85-93.
• [23] Turgut, A., \3-Boyutlu Mikowski Uzay{nda Spacelike ve Timelike Regle Yuzeyler", A.U. Fen Bilimleri Enstitusu, Doktora Tezi,1995.
• [24] Ugurlu, H.H., Caliskan, A., Kilic, O., Instantaneous Lorentzian Spatial Kinematics and the Invariants of the axodes, IV. International Geometry Symposium 17-21 July 2006, Zonguldak.
• [25] Ugurlu, H.H., Caliskan, A,.The Study Mapping for Directed Spacelike and Timelike Lines in Minkowski 3-Space IR3 1, Mathematical&Computational Applications, 1(2) (1996) 142-148.
• [26] Ugurlu H.H.,  Onder M., Instantaneous Rotation vectors of Skew Timelike Ruled Surfaces in Minkowski 3-space, VI. Geometry Symposium, 01-04 July 2008,Bursa, Turkey.
• [27] Ugurlu H. H.,  Onder M.,On Frenet Frames and Frenet Invariants of Skew Spacelike Ruled Surfaces in Minkowski 3-space, VII. Geometry Symposium, 7-10 July 2009, Kirsehir, Turkey.
• [28] Ugurlu, H.H., Caliskan, A., \Darboux Ani Donme Vektorleri ile Spacelike ve Timelike Yuzeyler Geometrisi", Celal Bayar Universitesi Yayinlari, Yayin No: 0006, 2012.
• [29] Ugurlu, H.H., Caliskan, A., Kilic, O., \  Oklid ve Lorentz Uzaylar{nda Dogrular Geometrisi", Lecture Notes, In press.
• [30] Veldkamp, G.R., On the use of dual numbers, vectors, and matrices in instantaneous spatial kinematics, Mech. and Mach. Theory, 11 (1976) 141-156.
• [31] Yakut, N.N., \Reel ve Dual Uzaylarda Aci Kavrami", CBU Fen Bilimleri Enstitusu, Yuksek Lisans Tezi, 2012.
• [32] Yang AT., Application of quaternion algebra and dual numbers to the analysis of spatial mechanism. Doctoral Dissertation, Colombia University, 1963.
• [33] Yayli, Y., Caliskan, A., Ugurlu, H.H., The E. Study Maps of Circle on Dual Hyperbolic and Lorentzian Unit Spheres ~H 2 0 and ~ S2 1 , Mathematical Proceedings of the Royal Irish Academy, 102A (1) (2002) 37-47.