[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
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.
[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).
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.