Abstract
The paper provides a consistent study on the projective construction of low-dimensional motion groups starting with \(\mathrm {SO}(3)\) and then gradually extending to the Galilean and Lorentzian settings. In the case of spatial rotations one simply needs to consider \({{\mathbb {R}}}{{\mathbb {P}}}^3\) with an additional group structure inherited from quaternion multiplication, which allows for associating particular types of curves in \({\mathbb {E}}^3\) with rigid body kinematics, based on the corresponding Maurer–Cartan form. A similar construction in complex projective space yields the relativistic version of the above approach, while a dual extension leads respectively to the study of nonhomogeneous isometries. The text provides also plenty of examples as well as a brief discussion on possible further generalizations.
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Notes
we are dealing with an affine space, so canonical coordinates impose nontrivial restriction.
that is, factorizing the Clifford group of \(\mathrm{Cliff}_{1,3}^\circ \cong \mathrm{Cliff}_{0,3}^{}\) by its center, which is now \({\mathbb {C}}^\times \!\).
related in the decomposable case to the Maxwell electromagnetic tensor in vacuum.
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Acknowledgements
I am grateful to Professor José Vargas for his valuable remarks and the whole organizing team of the Alterman Conference and School on Clifford Algebras and Kähler Calculus (Milano 2018) for inviting me to give a talk at the event.
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This article is part of the Topical Collection on 2018 Alterman Conference/School on Geometric Algebra/Kahler Calculus, edited by Rodolfo Fiorini and Jose’ Vargas.
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Brezov, D.S. Projective View on Motion Groups I: Kinematics and Relativity. Adv. Appl. Clifford Algebras 29, 47 (2019). https://doi.org/10.1007/s00006-019-0962-3
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DOI: https://doi.org/10.1007/s00006-019-0962-3