Is there a characteristic length of a rigid-body displacement?☆

The quest for a norm of a rigid-body displacement led to the idea of approximating the displacement by a rotation in a higher-dimensional space. Recently, this approximation has been attempted using the singular value and the polar decompositions of the homogeneous transformation matrix representing the displacement. This paper is an attempt to shed light on the question at the title of the paper. It is shown here that the singular-value decomposition is not needed to derive the approximation error, the polar decomposition of the homogeneous transformation matrix being sufficient. We show, moreover, that the approximation error is (a) a function solely of the translation part of the displacement and (b) a monotonically decreasing function of the normalizing length, termed here the characteristic length. Hence, the approximation does not have a minimum that would allow us to define naturally the characteristic length. While such a natural definition is not possible, we propose here a heuristic approach to engineer such a length.