[1] In the study of globular protein conformations, one customarily measures the similarity in three-dimensional structure by the RMSD of the Cα atomic coordinates after optimal rigid body superposition.
The size of this fluctuation can be measured, for example using Mössbauer spectroscopy or nuclear magnetic resonance, and can provide important physical information.
Coutsias, et al. presented a simple derivation, based on quaternions, for the optimal solid body transformation (rotation-translation) that minimizes the RMSD between two sets of vectors.
[4] The quaternion solution to compute the optimal rotation was published in the appendix of a paper of Petitjean.
Note that, for the case of ligands (contrary to proteins, as described above), their structures are most commonly not superimposed prior to the calculation of the RMSD.
RMSD is also one of several metrics that have been proposed for quantifying evolutionary similarity between proteins, as well as the quality of sequence alignments.