On Cayley’s Factorization of 4d Rotations and Applications

A 4D rotation can be decomposed into a left-isoclinic and a right-isoclinic rotation. This decomposition, known as Cayley’s factorization of 4R rotations, can be performed using Elfrinkhof-Rosen method. In this paper, we present a more straightforward alternative approach based on the fact that there is an orthogonal basis, in the sense of Hilbert-Schmidt, for the space of 4×4 real orthonormal matrices representing isoclinic rotations. Cayley’s factorization has many important applications. It can actually be seen as a unifying procedure to obtain the double quaternion representation of 4D rotations, the quaternion representation of 3D rotations, and the dual quaternion representation of 3D rigid-body transformations. Hence its interest in different Geometric 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.