Before revision:  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. In addition, we consider data that are themselves quaternions isomorphic to orthonormal triads describing 3 coordinate frames (amino acids in proteins possess such frames). Adopting a reasonable approximation to the exact quaternion-data minimization problem, we find a novel closed form "quaternion RMSD" (QRMSD) solution for the optimal rotation from a quaternion data set to a reference set.
After revision:  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 spatial alignment problem. We then explore the structure of the less-studied orientation data context, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. We conclude with a brief discussion of the combined 3D translation-orientation data alignment problem. Appendices are devoted to a quaternion data set to a reference set.