Chasles' fixed point theorem for Euclidean motions (Q1950306)

The authors give a simple proof of the following 19th century theorem by Chasles: Let \(\mathcal{E}\) be a proper motion of \(\mathbb{R}^3\), so \(\mathcal{E}=T_vR\) where \(R\) is a rotation and \(T_v\) is the translation by a vector \(v\). Assume that \(R\) is not the identity. A ``line'' is defined as the translation \(\ell\) of a one-dimensional subspace \(A\). It is given by \((\omega,p)\) where \(\pm\omega\) is the intersection of \(A\) with \(S^2\) and \(p\) is the unique point on \(\ell\) closest to the origin. An orientation of \(\ell\) amounts to the choice of either \(\omega\) or \(-\omega\). The authors give a simple proof that \(\mathcal{E}\) leaves fixed a unique oriented line \((\omega,p)\). A twist around \(\ell\) is the composition \(T_a\rho\) of a rotation \(\rho\) about \(\ell\) followed by a translation parallel to \(\ell\). In these terms a proper motion (i.e., an element of the semidirect product of the translation group and \(\mathbf{SO}(3)\)) is a twist.