Research Article
BibTex RIS Cite

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.

Keywords

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

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., Di erential 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 di erential geometry, Vol. I, John Wiley Sons Inc. Lcccn: (1963), 63-19209.
  • [14] Kuhnel, W., Di erential Geometry, Curves-Surfaces-Manifolds, American Mathematical Society, 2002.
  • [15] Lopez, R., Di erential 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 Di erential 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.

Year 2015, Volume: 3 Issue: 2, 190 - 201, 01.10.2015

Cumali Ekıcı Hatice Tozak

Abstract

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., Di erential 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 di erential geometry, Vol. I, John Wiley Sons Inc. Lcccn: (1963), 63-19209.
  • [14] Kuhnel, W., Di erential Geometry, Curves-Surfaces-Manifolds, American Mathematical Society, 2002.
  • [15] Lopez, R., Di erential 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 Di erential 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.
There are 26 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Cumali Ekıcı

Hatice Tozak

Publication Date October 1, 2015
Submission Date July 10, 2014
Published in Issue Year 2015 Volume: 3 Issue: 2

Cite

APA Ekıcı, C., & Tozak, H. (2015). ON THE INVOLUTES FOR DUAL SPLIT QUATERNIONIC CURVES. Konuralp Journal of Mathematics, 3(2), 190-201.
AMA Ekıcı C, Tozak H. ON THE INVOLUTES FOR DUAL SPLIT QUATERNIONIC CURVES. Konuralp J. Math. October 2015;3(2):190-201.
Chicago Ekıcı, Cumali, and Hatice Tozak. “ON THE INVOLUTES FOR DUAL SPLIT QUATERNIONIC CURVES”. Konuralp Journal of Mathematics 3, no. 2 (October 2015): 190-201.
EndNote Ekıcı C, Tozak H (October 1, 2015) ON THE INVOLUTES FOR DUAL SPLIT QUATERNIONIC CURVES. Konuralp Journal of Mathematics 3 2 190–201.
IEEE C. Ekıcı and H. Tozak, “ON THE INVOLUTES FOR DUAL SPLIT QUATERNIONIC CURVES”, Konuralp J. Math., vol. 3, no. 2, pp. 190–201, 2015.
ISNAD Ekıcı, Cumali - Tozak, Hatice. “ON THE INVOLUTES FOR DUAL SPLIT QUATERNIONIC CURVES”. Konuralp Journal of Mathematics 3/2 (October 2015), 190-201.
JAMA Ekıcı C, Tozak H. ON THE INVOLUTES FOR DUAL SPLIT QUATERNIONIC CURVES. Konuralp J. Math. 2015;3:190–201.
MLA Ekıcı, Cumali and Hatice Tozak. “ON THE INVOLUTES FOR DUAL SPLIT QUATERNIONIC CURVES”. Konuralp Journal of Mathematics, vol. 3, no. 2, 2015, pp. 190-01.
Vancouver Ekıcı C, Tozak H. ON THE INVOLUTES FOR DUAL SPLIT QUATERNIONIC CURVES. Konuralp J. Math. 2015;3(2):190-201.
 Download Cover Image
Creative Commons License
The published articles in KJM are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.