On Cayley’s Factorization of 4D Rotations and Applications

Abstract

A 4D rotation can be decomposed into a left- and a right-isoclinic rotation. This decomposition, known as Cayley’s factorization of 4D rotations, can be performed using the Elfrinkhof–Rosen method. In this paper, we present a more straightforward alternative approach using the corresponding orthogonal subspaces, for which orthogonal bases can be defined. This yields easy formulations, both in the space of \({4 \times 4}\) real orthogonal matrices representing 4D rotations and in the Clifford algebra \({\mathcal{C}_{4,0,0}}\). Cayley’s factorization has many important applications. It can be used to easily transform rotations represented using matrix algebra to different Clifford algebras. As a practical application of the proposed method, it is shown how Cayley’s factorization can be used to efficiently compute the screw parameters of 3D rigid-body transformations.