Singularity-free computation of quaternions from rotation matrices in E4 and E3

Abstract

A real orthogonal matrix representing a rotation in E4 can be decomposed into the commutative product of a left-isoclinic and a right-isoclinic rotation matrix. The double quaternion representation of rotations in E4 follows directly from this decomposition. In this paper, it is shown how this decomposition can be performed without divisions. This avoids the common numerical issues attributed to the computation of quaternions from rotation matrices. The map from the 4×4 rotation matrices to the set of double unit quaternions is a 2-to-1 covering map. Thus, this map cannot be smoothly inverted. As a consequence, it is erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception is herein clari¿ed. When particularized to three dimensions, it is shown how the resulting formulation outperforms, from the numerical point of view, the celebrated Shepperd’s method.