A note on the rotational superposition problem

A rotation axis vector with magnitude tan (θ/2) for a rotation angle θ and a closely related unit vector of dimension 4 are used to show that : (i) the quadratic residual (weighted sum of squares of coordinate differences) that results when one vector set is rotated relative to another is a quadratic form of order 4, (ii) the stationary values of the residual are given by the eigenvalues of a matrix of order 4, (iii) the minimum residual is given by the largest eigenvalue, (iv) the rotations required to obtain such residuals are uniquely defined by the corresponding eigenvectors, and (v) the stationary values are related by the operations of 222 symmetry. No precautions against the generation of improper rotations are required. In addition, an equivalent solution based on a scalar iteration is presented, together with some relationships of general interest.