Unphysical properties of the rotation tensor estimated by least squares optimization with specific application to biomechanics

Analysis of the transformation of one data set into another is a ubiquitous problem in many fields of science. Many works approximate the transformation of a reference cluster of n vectors X_i (i = 1, 2,...,n) into another cluster of n vectors x_i by a translation and a rotation using a least squares optimization to obtain the rotation tensor Q. The objective of this work is to prove that this rotation tensor Q exhibits unphysical dependence on the shape and orientation of the reference cluster. In contrast, when the transformation is approximated by a translation and a general non-singular tensor F, which includes deformations, then the associated rotation tensor R does not exhibit these unphysical properties. An example in biomechanics quantifies the errors of these unphysical properties.