THE ORDER OF FINITE GENERATION OF SO(3) AND OPTIMIZATION OF ROTATION SEQUENCES

We generalize an old result due to Lowenthal (2011) and a more recent one due to Hamada (2014) on the order of finite generation of the rotation group SO(3) both for fixed and arbitrary compound transformations. Unlike the authors mentioned above, we consider decompositions into factors with more than two invariant axes and derive a simple estimate for certain subsets of rotations using intuitive geometric proofs. Particular examples of potential interest for the applications are considered with an emphasis on optimization. Possible generalizations are discussed as well, e.g., dimensional induction, the hyperbolic case and screw motion.