Cumali EKICI, Hatice TOZAK

ON THE INVOLUTES FOR DUAL SPLIT QUATERNIONIC CURVES

In this study, de nition of involute-evolute curves for semi-dual quaternionic curves in semi-dual spaces D42 known as dual split quaternion and D31 are given and also some well-known theorems for involute-evolute dual split quaternionic curves are obtained.

Keywords:

semi-dual quaternions, semi-dual space Serret-Frenet formula, involute-evolute curve couple,

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[1] Bharathi, K. and Nagaraj, M. Quaternion valued function of a real variable Serret-Frenet formula, Indian Journal of Pure and Applied Mathematics 18: (1987), 507-511.
[2] Bilici, M. and C alskan, M., On the Involutes of the Spacelike Curve with a Timelike Binormal in Minkowski 3-Space, International Mathematical Forum, 4 no 31 (2009), 1497-1509.
[3] Blaschke, W., Diferensiyel Geometri Dersleri, _Istanbul Universitesi Yaynlar, 1949.
[4] Boyer, C., A History of Mathematics, New York: Wiley, 1968.
[5] Bukcu, B. and Karacan, M.K., On the Involute and Evolute Curves of the Spacelike Curve with a Spacelike Binormal in Minkowski 3-space, Int. J. Math. Sciences, 2(5): (2007), 221-232.
[6] Clifford, W. K., Preliminary skecth of biquaternions, Proceedings of London Math. Soc. 4, (1873), 361-395.
[7] Çöken, A.C., Ekici, C., Kocayusufoglu, _I. and Gorgulu, A., Formulas for dual split quaternionic curves, Kuwait J. Sci. Eng.1A(36): (2009), 1-14
[8] Çöken, A.C. and Tuna, A., On the quaternionic inclined curves in the semi-Euclidean space E42 , Applied Mathematics and Computation 155(2): (2004), 373-389.
[9] do Carmo, M.P., Dierential Geometry of Curves and Surfaces, 1976.
[10] Hacsalihoglu, H. H., Hareket Geometrisi ve Kuaterniyonlar Teorisi, Gazi Universitesi, Fen- Edebiyat Fakultesi Yayinlari 2, 1983.
[11] Inoguchi, J., Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo Journal of Mathematics 21(1): (1998), 141-152.
[12] Kecilioglu, O. and Gundogan, H., Dual split quaternions and motions in Lorentz space R31 , Far East Journal of Mathematical Sciences (FJMS) 24(3): (2007), 425-437.
[13] Kobayashi, S. and Nomizu, K., Foundations of dierential geometry, Vol. I, John Wiley Sons Inc. Lcccn: (1963), 63-19209.
[14] Kuhnel, W., Dierential Geometry, Curves-Surfaces-Manifolds, American Mathematical Society, 2002.
[15] Lopez, R., Dierential geometry of curves and surfaces in Lorentz-Minkowski space, Mini- Course taught at the Instituto de Matematica e Estatistica (IME-USP), University of Sao Paulo, Brasil, 2008.
[16] Nizamoglu, S., Surfaces reglees paralleles, Ege Univ. Fen Fak. Derg., 9 (Ser. A), (1986), 37-48.
[17] O'Neill, B., Semi Riemannian Geometry with Applications to Relativity, Academic Press, Inc. New York, 1983.
[18] O'Neill, B., Elementary Dierential Geometry, Academic Press, Inc. New York, 2006.
[19] Ozylmaz, E. and Ylmaz, S., Involute-Evolute Curve Couples in the Euclidean 4-Space, Int. J. Open Problems Compt.Math., vol.2 No.2, (2009).
[20] Ozdemir, M. and Ergin, A. A., Rotations with unit timelike quaternions in Minkowski 3-space, Journal of Geometry and Physics 56: (2006), 322-336.
[21] Sivridag, A._I., Gunes, R. and Keles, S., The Serret-Frenet formulae for dual-valued functions of a single real variable, Mechanism and Machine Theory 29: (1994), 749-754.
[22] Study, E., Geometrie der Dynamen, Leipzig, Teubner, 1903.
[23] Turgut, M. and Yilmaz,S., On The Frenet Frame and A Characterization of space-like Involute-Evolute Curve Couple in Minkowski Space-time, Int. Math. Forum 3(16): (2008), 793-801.
[24] Ugurlu, H.H. and C alskan , A., The study mapping for directed space-like and time-like line in Minkowski 3-space R31 , Mathematical and ComputationalApplications 1(2): (1996), 142-148.
[25] Veldkamp, G. R., On the use of dual numbers, vectors and matrices in instantaneous spatial kinematics, Mechanism and Machine Theory 11: (1976), 141-156.
[26] Willmore, T.J., Riemannian Geometry, Published in the United States by Oxford University Press Inc., Newyork, 1993.