On HD-Split Euler-Rodrigues Equations

Year 2025, Volume: 18 Issue: 1, 124 - 134, 24.04.2025

Bahar Doğan Yazıcı , Sıddıka Özkaldı Karakuş , Emine Gül

Abstract

In this work, we introduce \textbf{HD}- split Euler-Rodrigues equations. First, we include the basic concepts of dual numbers, dual vectors, \textbf{HD}- numbers, \textbf{HD}- vectors and \textbf{HD}- split vectors, which form the basis of the study. Then we obtain \textbf{HD}- split Euler-Rodrigues relations for \textbf{HD}- unit spacelike axes and \textbf{HD}- unit timelike axes. Thanks to these relations, we obtain \textbf{HD}- split rotation matrices and we examine the relationships with the E.Study transformation defined for \textbf{HD}- split vectors. We also reconstruct Euler's fixed point theorem with \textbf{HD}- split rotation matrices. Finally, we provide extensive and interesting examples that support the theory.

Keywords

HD-split vectors, HD-split Euler-Rodrigues, HD-split Euler Theorem

References

• Aslan, S., Bekar, M., Yaylı, Y.: Hyper-dual split quaternions and rigid body motion. J. Geom. Phys. 158, 103876 (2020).
• Aslan, S., Yaylı, Y.: Split quaternions and canal surfaces in Minkowski 3-space. Int. J. Geom. 5 (2), 51-61 (2016).
• Clifford, W.K.: Preliminary sketch of biquaternion. Proc. London Mathematical Society. 4, 381-395 (1873).
• Cohen, A., Shoham, M.: Application of hyper-dual numbers to multibody kinematics. J. Mech. Rob. 8 (1), 011005 (2015).
• Cohen, A., Shoham, M.: Application of hyper-dual numbers to rigid bodies equations of motion. Mech. Mach. Theory. 111, 76-84 (2017).
• Cohen, A., Shoham, M.: Hyper dual quaternions representation of rigid bodies kinematics. J. Mech. Mach. Theory. 150, 103861 (2020).
• Dai, J. S.: Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections. Mech. Mach. Theory. 92, 144–152 (2015).
• Euler, L.: Formulae generales pro translatione quacunque corporum rigidorum. Novi Comm. Acad. Sci. Imp. Petrop. 20, 189-207 (1776).
• Fike, J. A.: Numerically exact derivative calculations using hyper-dual numbers. In: 3rd Annual Student Joint Workshop in Simulation-BasedEngineering and Design, 2009.
• Fike, J.A., Alonso, J.J.: The development of hyper-dual numbers for exact second-derivative calculations. In: 49th AIAA Aerodpace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, pp. 4-7, 2011.
• Kahveci, D., Yaylı, Y., Gök, ˙I.: The geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space. Int. J. Geom. Methods Mod. Phys. 13(10), 1650116 (2016).
• Kahveci, D., Gök, İ., Yaylı, Y.: Some variations of dual Euler–Rodrigues formula with an application to point–line geometry. J. Math. Anal. Appl. 459(2), 1029-1039 (2018).
• Kazaz, M., Özdemir, A., U˘gurlu, H. H.: Elliptic motion on dual hyperbolic unit sphere H2 0. Mech. Mach. Theory. 44, 1450–1459 (2009).
• Kotelnikov, A. P.: Screw Calculus and Some Applications to Geometry and Mechanics. Annal. Imp. Univ. Kazan, 1895.
• Lopez, R.: Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int. Electron. J. Geom. 7 (1), 44–107 (2014).
• McCarthy, J. M.: Introduction to Theoretical Kinematics. MIT Press, 1990.
• O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic press, 1983.
• Özkaldı, S., Gündoğan, H.: Cayley formula, Euler parameters and rotations in 3-dimensional Lorentzian space. Adv. Appl. Clifford Algebras. 20, 367-377 (2010).
• Palais, B., Palais, R.: Euler’s fixed point theorem: The axis of a rotation. J. Fix. Point Theory A. 2(2), 215-220 (2007).
• Ramis, Ç., Yaylı, Y., Zengin, İ.: The application of Euler-Rodrigues formula over hyper-dual matrices. Int. Electron. J. Geom. 15(2), 266-276 (2022).
• Study, E.: Geometry der Dynamen. Leipzig, 1901.
• Yaylı, Y., Çalışkan, A., Uğurlu, H. H.: The E. Study mapping of circles on dual hyperbolic and Lorentzian unit spheres $\mathcal{H}_{0}^2$ and $\mathcal{S}_{1}^2$. Math. Proc. R. Ir. Acad. 102A(1), 37-47 (2002).