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The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices

Year 2022, Volume: 15 Issue: 2, 266 - 276, 31.10.2022
https://doi.org/10.36890/iejg.1127216

Abstract

The Lie group over the hyper-dual matrices and its corresponding Lie algebra are first introduced in this study. One of Euler's strategies called the Euler-Rodrigues formula is applied to the matrices of hyper-dual rotations. The fundamental relationship between the hyper-dual numbers and the dual numbers allows us to apply the formula on dual lines and two intersecting real lines in the three-dimensional dual and Euclidean spaces, respectively.

References

  • [1] Aslan, S., Bekar, M., Yaylı, Y.: Hyper-dual split quaternions and rigid body motion, Journal of Geometry and Physics. 158, 103876 (2020).
  • [2] Bottema, O., Roth, B.: Theoretical Kinematics, North-Holland Publishing Company, New York, (1979).
  • [3] Cantún-Avila, K. B., González-Sánchez, D., Díaz-Infante, S., Peñuñuri, F.: Optimizing functionals using differential evolution, Engineering Applications of Artificial Intelligence. 97, 104086 (2021).
  • [4] Chasles, M.: Note sur les propriétés générales du système de deux corps semblables entr’eux, Bulletin des Sciences, Mathématiques, Astronomiques, Physiques et Chemiques. 4, 321–326 (1830).
  • [5] Clifford, W.K.: Preliminary sketch of biquaternions, Proc. London Mathematical Society. 4, 381-395 (1873).
  • [6] Cohen, A., Shoham, M.: Application of hyper-dual numbers to multi-body kinematics, Journal of Mechanisms and Robotics. 8, 011015 (2016).
  • [7] Cohen, A., Shoham, M.: Application of hyper-dual numbers to rigid bodies equations of motion, J. Mech. Mach. Theory. 111, 76–84 (2017).
  • [8] Cohen, A., Shoham, M.: Hyper dual quaternions representation of rigid bodies kinematics, J. Mech. Mach. Theory. 150, 103861 (2020).
  • [9] Dai, J.S.: Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections, Mech. Mach. Theory. 92, 144–152 (2015).
  • [10] Euler, L.: Formulae generales pro translatione quacunque corporum rigidorum, Novi Comm. Acad. Sci. Imp. Petrop. 20, 189-207 (1776).
  • [11] Fike, J., Alonso, J.: The development of hyper-dual numbers for exact second derivative calculations,49th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition. (2011).
  • [12] Fischer, I.: Dual-number methods in kinematics, statics and dynamics. CRC press, (1998).
  • [13] Griewank, A.: On automatic differentiation, Mathematical Programming: recent developments and applications, 6(6), 83-107 (1989).
  • [14] Gromov, N.A.: Possible quantum kinematics. J. Math. Phys. 47(1), 013502 (2006).
  • [15] Hall, B.: Lie groups, Lie algebras, and representations: an elementary introduction (Vol. 222). Springer, (2015).
  • [16] Imoto, Y., Yamanaka, N., Uramoto, T., Tanaka, M., Fujikawa, M., Mitsume, N.: Fundamental theorem of matrix representations of hyper-dual numbers for computing higher-order derivatives, JSIAM Letters. 12, 29-32 (2020).
  • [17] Kahveci, D., Gök, ˙I., Yaylı, Y.: Some variations of dual Euler–Rodrigues formula with an application to point–line geometry, Journal of Mathematical Analysis and Applications. 459(2), 1029-1039 (2018).
  • [18] Kisil, V.V.: Geometry of Mobius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL2(R), Imperial College Press, London, (2012). [19] McCarthy, J.M.: Introduction to Theoretical Kinematics, MIT Press, (1990).
  • [20] Müller, A.: Coordinate mappings for rigid body motions. Journal of Computational and Nonlinear Dynamics. 12(2), (2017).
  • [21] Palais, B., Palais, R.: Euler’s fixed point theorem: The axis of a rotation, Journal of Fixed Point Theory and Applications. 2(2), 215-220 (2007).
  • [22] Palais, B., Palais, R.: Chasles’ fixed point theorem for Euclidean motions. Journal of Fixed Point Theory and Applications, 12(1-2), 27-34 (2012).
  • [23] Pottmann, H., Wallner, J.: Computational Line Geometry, Springer-Verlag, (2001).
  • [24] Ramis, Ç., Yaylı, Y.: Dual Split Quaternions and Chasles’ Theorem in 3-Dimensional Minkowski Space E31, Advances in Applied Clifford Algebras, 23(4), 951-964 (2013).
  • [25] Rall, L.B.: Automatic differentiation - techniques and applications, Springer Lecture Notes in Computer Science, Vol.120, (1981).
  • [26] Ravani, B., Wang, J.W.: Computer aided geometric design of line constructs, ASME Journal of Mechanical Design. 113 (3), 363–371 (1991).
  • [27] Study, E.: Geometry der Dynamen, Leipzig, (1901).
  • [28] Yu, W., Blair, M.: DNAD, a simple tool for automatic differentiation of Fortran codes using dual numbers, Comput. Phys. Comm. 184, 1446–1452 (2013).
  • [29] Zhang, Y., Ting, K.L.: On point–line geometry and displacement, Mech. Mach. Theory 39, 1033-1050 (2004).
Year 2022, Volume: 15 Issue: 2, 266 - 276, 31.10.2022
https://doi.org/10.36890/iejg.1127216

Abstract

References

  • [1] Aslan, S., Bekar, M., Yaylı, Y.: Hyper-dual split quaternions and rigid body motion, Journal of Geometry and Physics. 158, 103876 (2020).
  • [2] Bottema, O., Roth, B.: Theoretical Kinematics, North-Holland Publishing Company, New York, (1979).
  • [3] Cantún-Avila, K. B., González-Sánchez, D., Díaz-Infante, S., Peñuñuri, F.: Optimizing functionals using differential evolution, Engineering Applications of Artificial Intelligence. 97, 104086 (2021).
  • [4] Chasles, M.: Note sur les propriétés générales du système de deux corps semblables entr’eux, Bulletin des Sciences, Mathématiques, Astronomiques, Physiques et Chemiques. 4, 321–326 (1830).
  • [5] Clifford, W.K.: Preliminary sketch of biquaternions, Proc. London Mathematical Society. 4, 381-395 (1873).
  • [6] Cohen, A., Shoham, M.: Application of hyper-dual numbers to multi-body kinematics, Journal of Mechanisms and Robotics. 8, 011015 (2016).
  • [7] Cohen, A., Shoham, M.: Application of hyper-dual numbers to rigid bodies equations of motion, J. Mech. Mach. Theory. 111, 76–84 (2017).
  • [8] Cohen, A., Shoham, M.: Hyper dual quaternions representation of rigid bodies kinematics, J. Mech. Mach. Theory. 150, 103861 (2020).
  • [9] Dai, J.S.: Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections, Mech. Mach. Theory. 92, 144–152 (2015).
  • [10] Euler, L.: Formulae generales pro translatione quacunque corporum rigidorum, Novi Comm. Acad. Sci. Imp. Petrop. 20, 189-207 (1776).
  • [11] Fike, J., Alonso, J.: The development of hyper-dual numbers for exact second derivative calculations,49th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition. (2011).
  • [12] Fischer, I.: Dual-number methods in kinematics, statics and dynamics. CRC press, (1998).
  • [13] Griewank, A.: On automatic differentiation, Mathematical Programming: recent developments and applications, 6(6), 83-107 (1989).
  • [14] Gromov, N.A.: Possible quantum kinematics. J. Math. Phys. 47(1), 013502 (2006).
  • [15] Hall, B.: Lie groups, Lie algebras, and representations: an elementary introduction (Vol. 222). Springer, (2015).
  • [16] Imoto, Y., Yamanaka, N., Uramoto, T., Tanaka, M., Fujikawa, M., Mitsume, N.: Fundamental theorem of matrix representations of hyper-dual numbers for computing higher-order derivatives, JSIAM Letters. 12, 29-32 (2020).
  • [17] Kahveci, D., Gök, ˙I., Yaylı, Y.: Some variations of dual Euler–Rodrigues formula with an application to point–line geometry, Journal of Mathematical Analysis and Applications. 459(2), 1029-1039 (2018).
  • [18] Kisil, V.V.: Geometry of Mobius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL2(R), Imperial College Press, London, (2012). [19] McCarthy, J.M.: Introduction to Theoretical Kinematics, MIT Press, (1990).
  • [20] Müller, A.: Coordinate mappings for rigid body motions. Journal of Computational and Nonlinear Dynamics. 12(2), (2017).
  • [21] Palais, B., Palais, R.: Euler’s fixed point theorem: The axis of a rotation, Journal of Fixed Point Theory and Applications. 2(2), 215-220 (2007).
  • [22] Palais, B., Palais, R.: Chasles’ fixed point theorem for Euclidean motions. Journal of Fixed Point Theory and Applications, 12(1-2), 27-34 (2012).
  • [23] Pottmann, H., Wallner, J.: Computational Line Geometry, Springer-Verlag, (2001).
  • [24] Ramis, Ç., Yaylı, Y.: Dual Split Quaternions and Chasles’ Theorem in 3-Dimensional Minkowski Space E31, Advances in Applied Clifford Algebras, 23(4), 951-964 (2013).
  • [25] Rall, L.B.: Automatic differentiation - techniques and applications, Springer Lecture Notes in Computer Science, Vol.120, (1981).
  • [26] Ravani, B., Wang, J.W.: Computer aided geometric design of line constructs, ASME Journal of Mechanical Design. 113 (3), 363–371 (1991).
  • [27] Study, E.: Geometry der Dynamen, Leipzig, (1901).
  • [28] Yu, W., Blair, M.: DNAD, a simple tool for automatic differentiation of Fortran codes using dual numbers, Comput. Phys. Comm. 184, 1446–1452 (2013).
  • [29] Zhang, Y., Ting, K.L.: On point–line geometry and displacement, Mech. Mach. Theory 39, 1033-1050 (2004).
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Çağla Ramis 0000-0002-2809-8324

Yusuf Yaylı 0000-0003-4398-3855

İrem Zengin 0000-0003-1642-5559

Early Pub Date July 23, 2022
Publication Date October 31, 2022
Acceptance Date September 29, 2022
Published in Issue Year 2022 Volume: 15 Issue: 2

Cite

APA Ramis, Ç., Yaylı, Y., & Zengin, İ. (2022). The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices. International Electronic Journal of Geometry, 15(2), 266-276. https://doi.org/10.36890/iejg.1127216
AMA Ramis Ç, Yaylı Y, Zengin İ. The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices. Int. Electron. J. Geom. October 2022;15(2):266-276. doi:10.36890/iejg.1127216
Chicago Ramis, Çağla, Yusuf Yaylı, and İrem Zengin. “The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices”. International Electronic Journal of Geometry 15, no. 2 (October 2022): 266-76. https://doi.org/10.36890/iejg.1127216.
EndNote Ramis Ç, Yaylı Y, Zengin İ (October 1, 2022) The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices. International Electronic Journal of Geometry 15 2 266–276.
IEEE Ç. Ramis, Y. Yaylı, and İ. Zengin, “The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices”, Int. Electron. J. Geom., vol. 15, no. 2, pp. 266–276, 2022, doi: 10.36890/iejg.1127216.
ISNAD Ramis, Çağla et al. “The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices”. International Electronic Journal of Geometry 15/2 (October 2022), 266-276. https://doi.org/10.36890/iejg.1127216.
JAMA Ramis Ç, Yaylı Y, Zengin İ. The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices. Int. Electron. J. Geom. 2022;15:266–276.
MLA Ramis, Çağla et al. “The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices”. International Electronic Journal of Geometry, vol. 15, no. 2, 2022, pp. 266-7, doi:10.36890/iejg.1127216.
Vancouver Ramis Ç, Yaylı Y, Zengin İ. The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices. Int. Electron. J. Geom. 2022;15(2):266-7.