Abstract
In this paper the generalization of the rotations on any lightcone in Minkowski 3-space \({\mathbb {R}}_{g}^{1,2}\) is given. The rotation motion on the lightcone is examined by means of a bilinear form and Lorentzian notions. We use the corresponding Rodrigues and Cayley formulas and benefit from the hyperbolic split quaternion product to obtain the corresponding rotation matrix.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brackett, R.W.: Robotic manipulators and the product of exponentials formula. Mathematical Theory and Networks and Systems. Proceeding of International Symposium, pp. 120–127. Springer, Berlin (1984)
Bükçü, B.: On the rotation matrices in semi-Euclidean space. Commun. Fac. Sci. Univ. Ank. Ser. A1(55), 7–13 (2006)
Erdoğdu, M., Özdemir, M.: Cayley formula in Minkowski space-time. Int. J. Geom. Methods Mod. Phys. 12, 05 (2015)
Gallier, J.H.: Geometric methods and applications, for computer science and engineering. Texts Appl. Math. 38, 680p (2011)
Kula, L., Karacan, M.K., Yaylı, Y.: Formulas for the exponential of semi symmetric matrix of order 4. Math. Comput. Appl. 10, 99–104 (2005)
Murray, M.R., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1994)
Nes̆ović, E.: On rotation about lightlike axis in three-dimensional Minkowski space. Adv. Appl. Clifford Algebras 26, 237–251 (2016)
Norris, A.N.: Euler-Rodrigues and Cayley formulae for rotation of elasticity tensors. Math. Mech. Solids 13, 465–498 (2008)
O’Neill, B.: Semi-Riemannian Geometry with Application to Relativity, Pure and Applied Mathematics. Academic Press, New York (1983)
Özdemir, M.: An alternative approach to elliptical motion. Adv. Appl. Clifford Algebras 26, 279–304 (2016)
Özdemir, M., Erdoğdu, M.: On the rotation matrix in Minkowski space-time. Rep. Math. Phys. 74, 27–38 (2014)
Özdemir, M., Erdoğdu, M., Simsek, H.: On the eigenvalues and eigenvectors of a lorentzian rotation matrix by using split quaternions. Adv. Appl. Clifford Algebras 24, 179–192 (2014)
Özdemir, M., Ergin, A.A.: Rotations with unit timelike quaternions in Minkowski 3-space. J. Geom. Phys. 56, 322–336 (2006)
Özkaldı, S., Gündoğan, H.: Cayley formula, Euler parameters and rotations in 3-dimensional Lorentzian Space. Adv. Appl. Clifford Algebras 20, 367–377 (2010)
Selig J.M.: Cayley Maps for SE(3). 12th IFToMM World Congress, Besancon, pp. 18–21 (2007)
Simsek, H., Özdemir, M.: Generating hyperbolical rotation matrix for a given hyperboloid. Linear Algebra Appl. 496, 221–245 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Rafał Abłamowicz
Rights and permissions
About this article
Cite this article
Simsek, H., Özdemir, M. Rotations on a Lightcone in Minkowski 3-Space. Adv. Appl. Clifford Algebras 27, 2841–2853 (2017). https://doi.org/10.1007/s00006-017-0771-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00006-017-0771-5