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Generalized Dual Quaternions and Screw Motion in Generalized Space

Year 2022, Volume: 10 Issue: 1, 197 - 202, 15.04.2022

Ümit Ziya Savcı

Abstract

In this paper, we showed that the set of displacements of generalized space is a group under the composite operation. We obtained this screw axis of displacement in generalized space. Using this screw axis, we obtained Rodrigues equation in terms of spatial displacement in this space. Finally, the components of a dual generalized quaternion and the dual orthogonal matrix were obtained using Euler parameters in generalized space.

Keywords

Generalized space rigid motion kinematics matrix groups

References

  • [1] O.P. Agrawal, Hamilton Operators and Dual-number-quaternions in Spatial Kinematics, Mech. Mach. Theory, 22(1987), 569-575.
  • [2] B. Akyar, Dual Quaternions in Spatial Kinematics in an Algebraic Sense, Turk J. Math., 32(2008), 373-391.
  • [3] S.L. Altmann, Rotations, Quaternions, and Double Groups, Oxford University Press, Oxford, 1986.
  • [4] E. Ata, Y. Yıldırım, Different Polar Representation for Generalized and Generalized Dual Quaternions, Adv. Appl. Clifford Al., (28)(2010), 193-202.
  • [5] E. Ata, U¨ .Z. Savci Spherical Kinematics in 3-Dimensional Generalized Space, International Journal of Geometric Methods in Modern Physics, 18(3)(2021), 2150033.
  • [6] O. Bottema, B. Roth, Theoretical Kinematics, North-Holland Press, New York 1979.
  • [7] W.K. Clifford, Preliminary sketch of bi-quaternions, Proc. London Math.Soc., 4(1873), 381-395.
  • [8] J. Cockle, On Systems of Algebra Involving More than One Imaginary, Philos. Mag. (series 3), 35(1849), 434-435.
  • [9] K. Erdmann, A. Skowronski, Algebras of generalized quaternion type, Advances in Mathematics, 349(2019), 1036-1116.
  • [10] J.M. Herve, The mathematical group structure of the set of displacements, Mech. Mach. Theory, 29(1994), 73-81.
  • [11] M. Hiller, C. Woernle, A Unified Representation of Spatial Displacements, Mech. Mach. Theory, 19(1984), 477-486.
  • [12] M. Jafari, Y. Yaylı, Generalızed Quaternions and Rotation in 3-Space E3a b , TWMS J. Pure Appl. Math., 6(2)(2015), 224-232.
  • [13] L. Kula, Y. Yaylı, Dual Split Quaternions and Screw Motion in Minkowski $3-$space, Iranian Journal of Science & Technology, Transaction A, 30(2006), 245-258.
  • [14] A.P. Kotel’nikov, Screw calculus and some of its applications to geometry and mechanies, Annals of The Imperial University of Kazan, 1895.
  • [15] T.Y. Lam, Introduction to Quadratic Forms Over Fields, American Mathematical Society, USA 2005.
  • [16] J.M. McCarthy, An Introduction to Theoretical Kinematics, MIT Press, Cambridge, 1990.
  • [17] M. Ozdemir, A.A. Ergin, Rotations with unit timelike quaternions in Minkowski $3-$space, Journal of Geometry and Physics, 56(2006), 322-336.
  • [18] S. Ozkaldı, H. G¨undo˘gan, Split Quaternions and Screw Motions in $3-$dimensional Lorentzian Space, Adv. Appl. Clifford Al., 21(2011), 193-202.
  • [19] H. Pottman, J. Wallner, Computational line geometry, Springer-Verlag Berlin Heidelberg, New York, 2000.
  • [20] E. Study, Geometrie der Dynamen, Leipzig, Germany, 1903.

Year 2022, Volume: 10 Issue: 1, 197 - 202, 15.04.2022

Ümit Ziya Savcı

Abstract

References

  • [1] O.P. Agrawal, Hamilton Operators and Dual-number-quaternions in Spatial Kinematics, Mech. Mach. Theory, 22(1987), 569-575.
  • [2] B. Akyar, Dual Quaternions in Spatial Kinematics in an Algebraic Sense, Turk J. Math., 32(2008), 373-391.
  • [3] S.L. Altmann, Rotations, Quaternions, and Double Groups, Oxford University Press, Oxford, 1986.
  • [4] E. Ata, Y. Yıldırım, Different Polar Representation for Generalized and Generalized Dual Quaternions, Adv. Appl. Clifford Al., (28)(2010), 193-202.
  • [5] E. Ata, U¨ .Z. Savci Spherical Kinematics in 3-Dimensional Generalized Space, International Journal of Geometric Methods in Modern Physics, 18(3)(2021), 2150033.
  • [6] O. Bottema, B. Roth, Theoretical Kinematics, North-Holland Press, New York 1979.
  • [7] W.K. Clifford, Preliminary sketch of bi-quaternions, Proc. London Math.Soc., 4(1873), 381-395.
  • [8] J. Cockle, On Systems of Algebra Involving More than One Imaginary, Philos. Mag. (series 3), 35(1849), 434-435.
  • [9] K. Erdmann, A. Skowronski, Algebras of generalized quaternion type, Advances in Mathematics, 349(2019), 1036-1116.
  • [10] J.M. Herve, The mathematical group structure of the set of displacements, Mech. Mach. Theory, 29(1994), 73-81.
  • [11] M. Hiller, C. Woernle, A Unified Representation of Spatial Displacements, Mech. Mach. Theory, 19(1984), 477-486.
  • [12] M. Jafari, Y. Yaylı, Generalızed Quaternions and Rotation in 3-Space E3a b , TWMS J. Pure Appl. Math., 6(2)(2015), 224-232.
  • [13] L. Kula, Y. Yaylı, Dual Split Quaternions and Screw Motion in Minkowski $3-$space, Iranian Journal of Science & Technology, Transaction A, 30(2006), 245-258.
  • [14] A.P. Kotel’nikov, Screw calculus and some of its applications to geometry and mechanies, Annals of The Imperial University of Kazan, 1895.
  • [15] T.Y. Lam, Introduction to Quadratic Forms Over Fields, American Mathematical Society, USA 2005.
  • [16] J.M. McCarthy, An Introduction to Theoretical Kinematics, MIT Press, Cambridge, 1990.
  • [17] M. Ozdemir, A.A. Ergin, Rotations with unit timelike quaternions in Minkowski $3-$space, Journal of Geometry and Physics, 56(2006), 322-336.
  • [18] S. Ozkaldı, H. G¨undo˘gan, Split Quaternions and Screw Motions in $3-$dimensional Lorentzian Space, Adv. Appl. Clifford Al., 21(2011), 193-202.
  • [19] H. Pottman, J. Wallner, Computational line geometry, Springer-Verlag Berlin Heidelberg, New York, 2000.
  • [20] E. Study, Geometrie der Dynamen, Leipzig, Germany, 1903.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ümit Ziya Savcı 0000-0003-2772-9283

Publication Date April 15, 2022
Submission Date December 7, 2021
Acceptance Date March 22, 2022
Published in Issue Year 2022 Volume: 10 Issue: 1

Cite

APA Savcı, Ü. Z. (2022). Generalized Dual Quaternions and Screw Motion in Generalized Space. Konuralp Journal of Mathematics, 10(1), 197-202.
AMA Savcı ÜZ. Generalized Dual Quaternions and Screw Motion in Generalized Space. Konuralp J. Math. April 2022;10(1):197-202.
Chicago Savcı, Ümit Ziya. “Generalized Dual Quaternions and Screw Motion in Generalized Space”. Konuralp Journal of Mathematics 10, no. 1 (April 2022): 197-202.
EndNote Savcı ÜZ (April 1, 2022) Generalized Dual Quaternions and Screw Motion in Generalized Space. Konuralp Journal of Mathematics 10 1 197–202.
IEEE Ü. Z. Savcı, “Generalized Dual Quaternions and Screw Motion in Generalized Space”, Konuralp J. Math., vol. 10, no. 1, pp. 197–202, 2022.
ISNAD Savcı, Ümit Ziya. “Generalized Dual Quaternions and Screw Motion in Generalized Space”. Konuralp Journal of Mathematics 10/1 (April 2022), 197-202.
JAMA Savcı ÜZ. Generalized Dual Quaternions and Screw Motion in Generalized Space. Konuralp J. Math. 2022;10:197–202.
MLA Savcı, Ümit Ziya. “Generalized Dual Quaternions and Screw Motion in Generalized Space”. Konuralp Journal of Mathematics, vol. 10, no. 1, 2022, pp. 197-02.
Vancouver Savcı ÜZ. Generalized Dual Quaternions and Screw Motion in Generalized Space. Konuralp J. Math. 2022;10(1):197-202.
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