Before revision:  Here we generalize this basic problem , sometimes referred to as the " Procrustes Problem, " and present algebraic solutions that exhibit properties that are inaccessible to traditional numerical methods. We begin with the 4D data problem , a problem one dimension higher than the conventional 3D problem, but one that is also solvable by quaternion methods, we then study the 3D and 2D data problems as special cases.
After revision:  Here we generalize this basic problem , sometimes referred to as the " Procrustes Problem, " is often phrased in terms of minimizing a root-mean-square deviation or RMSD corresponding to a Euclidean distance measure relating the two sets of matched coordinates. We focus on quaternion eigensystem methods that have been exploited to solve this problem , a problem one dimension higher than the conventional 3D problem, but one that is also solvable by quaternion methods, we then study the 3D and 2D data problems as special cases.