ON THE INVOLUTES FOR DUAL SPLIT QUATERNIONIC CURVES
Year 2015,
Volume: 3 Issue: 2, 190 - 201, 01.10.2015
Cumali Ekıcı
,
Hatice Tozak
Abstract
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.
References
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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.
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[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.
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[3] Blaschke, W., Diferensiyel Geometri Dersleri, _Istanbul Universitesi Yaynlar, 1949.
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[4] Boyer, C., A History of Mathematics, New York: Wiley, 1968.
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[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.
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[6] Clifford, W. K., Preliminary skecth of biquaternions, Proceedings of London Math. Soc. 4,
(1873), 361-395.
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[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
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[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.
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[9] do Carmo, M.P., Dierential Geometry of Curves and Surfaces, 1976.
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[10] Hacsalihoglu, H. H., Hareket Geometrisi ve Kuaterniyonlar Teorisi, Gazi Universitesi, Fen-
Edebiyat Fakultesi Yayinlari 2, 1983.
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[11] Inoguchi, J., Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo
Journal of Mathematics 21(1): (1998), 141-152.
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[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.
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[13] Kobayashi, S. and Nomizu, K., Foundations of dierential geometry, Vol. I, John Wiley Sons
Inc. Lcccn: (1963), 63-19209.
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[14] Kuhnel, W., Dierential Geometry, Curves-Surfaces-Manifolds, American Mathematical Society,
2002.
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[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.
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[16] Nizamoglu, S., Surfaces reglees paralleles, Ege Univ. Fen Fak. Derg., 9 (Ser. A), (1986), 37-48.
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[17] O'Neill, B., Semi Riemannian Geometry with Applications to Relativity, Academic Press,
Inc. New York, 1983.
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[18] O'Neill, B., Elementary Dierential Geometry, Academic Press, Inc. New York, 2006.
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[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).
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[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.
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[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.
Year 2015,
Volume: 3 Issue: 2, 190 - 201, 01.10.2015
Cumali Ekıcı
,
Hatice Tozak
References
-
[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.