Research Article
BibTex RIS Cite
Year 2021, Volume: 14 Issue: 2, 292 - 304, 29.10.2021
https://doi.org/10.36890/iejg.888373

Abstract

References

  • [1] Agrawal, O. P.: Hamilton operators and dual-number-quaternions in spatial kinematics. J. Mech. Mach. Theory. 22(6), 569-575 (1987).
  • [2] Clifford, W. K.: Preliminary sketch of biquaternions. Proc London Mathematical Society. 4(64), 381-395 (1873).
  • [3] Cohen, A., Shoham, M.: Application of hyper-dual numbers to multi-body kinematics. J. Mech. Rob. 8, (2015). doi: 10.1115/1.4030588.
  • [4] Cohen, A., Shoham, M.: Application of hyper-dual numbers to rigid bodies equations of motion. J. Mech. Mach. Theory. 111, 76-84 (2017).
  • [5] Cohen, A., Shoham, M.: Principle of transference-An extension to hyper-dual numbers. J. Mech. Mach. Theory. 125, 101-110 (2018).
  • [6] Fike, J. A.: Numerically exact derivative calculations using hyper-dual numbers, 3rd Annual Student Joint Workshop in Simulation-Based Engineering and Design, (2009).
  • [7] Fike, J. A., Alonso, J. J.: The development of hyper-dual numbers for exact second-derivative calculations. 49th AIAA Aerodpace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. 4-7 (2011).
  • [8] Fike, J. A., Alonso, J. J.: Automatic differentiation through the use of hyper-dual numbers for second derivatives. in: Lecture Notes in Computational Science and Engineering. 87(201), 163-173 (2011).
  • [9] Fike, J. A., Jongsma, S., Alonso, J. J., van der Weida, E.: Optimization with gradient and hessian information calculated using hyper-dual numbers. 29 AIAA Applied Aerodynamics Conference. (2011).
  • [10] Kotelnikov, A. P.: Screw calculus and some applications to geometry and mechanics. Annal Imp. Univ. Kazan, Russia, (1895).
  • [11] Study, E.: Geometry der Dynamen. Leipzig. (1901).
  • [12] Veldkamp, G. R.: On the use of dual numbers, vectors and matrices in instantaneous, spatial kinematics. J. Mech. Mach. Theory. 11(2), 141-156 (1976).
  • [13] Yuca, G., Yaylı, Y.: Dual Transformation Between S ^O(3) and S ^O(2; 1) and Its Geometric Applications. Proc. Natl. Acad. Sci. India, Sect. A Phys. Sci. 88, 267–273 (2018).
  • [14] Yuca, G., Yaylı, Y.: Dual Transformations and Instantaneous Screw Axes. International Journal of Mathematics Trends and Technology. 15 (2), (2014).

Kinematic Applications of Hyper-Dual Numbers

Year 2021, Volume: 14 Issue: 2, 292 - 304, 29.10.2021
https://doi.org/10.36890/iejg.888373

Abstract

Hyper-dual numbers are a new number system that is an extension of dual numbers. A hyper-dual number can be written uniquely as an ordered pair of dual numbers. In this paper, some basic algebraic properties of hyper-dual numbers are given using their ordered pair representaions of dual numbers. Moreover, the geometric interpretation of a unit hyper-dual vector is given in module as a dual line. And a geometric interpretation of a subset of unit hyper-dual sphere (the set of all unit hyper-dual vectors) is given as two intersecting perpendicular lines in 3-dimensional real vector space.

References

  • [1] Agrawal, O. P.: Hamilton operators and dual-number-quaternions in spatial kinematics. J. Mech. Mach. Theory. 22(6), 569-575 (1987).
  • [2] Clifford, W. K.: Preliminary sketch of biquaternions. Proc London Mathematical Society. 4(64), 381-395 (1873).
  • [3] Cohen, A., Shoham, M.: Application of hyper-dual numbers to multi-body kinematics. J. Mech. Rob. 8, (2015). doi: 10.1115/1.4030588.
  • [4] Cohen, A., Shoham, M.: Application of hyper-dual numbers to rigid bodies equations of motion. J. Mech. Mach. Theory. 111, 76-84 (2017).
  • [5] Cohen, A., Shoham, M.: Principle of transference-An extension to hyper-dual numbers. J. Mech. Mach. Theory. 125, 101-110 (2018).
  • [6] Fike, J. A.: Numerically exact derivative calculations using hyper-dual numbers, 3rd Annual Student Joint Workshop in Simulation-Based Engineering and Design, (2009).
  • [7] Fike, J. A., Alonso, J. J.: The development of hyper-dual numbers for exact second-derivative calculations. 49th AIAA Aerodpace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. 4-7 (2011).
  • [8] Fike, J. A., Alonso, J. J.: Automatic differentiation through the use of hyper-dual numbers for second derivatives. in: Lecture Notes in Computational Science and Engineering. 87(201), 163-173 (2011).
  • [9] Fike, J. A., Jongsma, S., Alonso, J. J., van der Weida, E.: Optimization with gradient and hessian information calculated using hyper-dual numbers. 29 AIAA Applied Aerodynamics Conference. (2011).
  • [10] Kotelnikov, A. P.: Screw calculus and some applications to geometry and mechanics. Annal Imp. Univ. Kazan, Russia, (1895).
  • [11] Study, E.: Geometry der Dynamen. Leipzig. (1901).
  • [12] Veldkamp, G. R.: On the use of dual numbers, vectors and matrices in instantaneous, spatial kinematics. J. Mech. Mach. Theory. 11(2), 141-156 (1976).
  • [13] Yuca, G., Yaylı, Y.: Dual Transformation Between S ^O(3) and S ^O(2; 1) and Its Geometric Applications. Proc. Natl. Acad. Sci. India, Sect. A Phys. Sci. 88, 267–273 (2018).
  • [14] Yuca, G., Yaylı, Y.: Dual Transformations and Instantaneous Screw Axes. International Journal of Mathematics Trends and Technology. 15 (2), (2014).
There are 14 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Selahattin Aslan 0000-0001-5322-3265

Publication Date October 29, 2021
Acceptance Date July 31, 2021
Published in Issue Year 2021 Volume: 14 Issue: 2

Cite

APA Aslan, S. (2021). Kinematic Applications of Hyper-Dual Numbers. International Electronic Journal of Geometry, 14(2), 292-304. https://doi.org/10.36890/iejg.888373
AMA Aslan S. Kinematic Applications of Hyper-Dual Numbers. Int. Electron. J. Geom. October 2021;14(2):292-304. doi:10.36890/iejg.888373
Chicago Aslan, Selahattin. “Kinematic Applications of Hyper-Dual Numbers”. International Electronic Journal of Geometry 14, no. 2 (October 2021): 292-304. https://doi.org/10.36890/iejg.888373.
EndNote Aslan S (October 1, 2021) Kinematic Applications of Hyper-Dual Numbers. International Electronic Journal of Geometry 14 2 292–304.
IEEE S. Aslan, “Kinematic Applications of Hyper-Dual Numbers”, Int. Electron. J. Geom., vol. 14, no. 2, pp. 292–304, 2021, doi: 10.36890/iejg.888373.
ISNAD Aslan, Selahattin. “Kinematic Applications of Hyper-Dual Numbers”. International Electronic Journal of Geometry 14/2 (October 2021), 292-304. https://doi.org/10.36890/iejg.888373.
JAMA Aslan S. Kinematic Applications of Hyper-Dual Numbers. Int. Electron. J. Geom. 2021;14:292–304.
MLA Aslan, Selahattin. “Kinematic Applications of Hyper-Dual Numbers”. International Electronic Journal of Geometry, vol. 14, no. 2, 2021, pp. 292-04, doi:10.36890/iejg.888373.
Vancouver Aslan S. Kinematic Applications of Hyper-Dual Numbers. Int. Electron. J. Geom. 2021;14(2):292-304.