Chasles’ fixed point theorem for Euclidean motions

Abstract

Chasles’ theorem, a classic and important result of kinematics, states that every orientation-preserving isometry of \({\mathbb{R}^3}\) is a screw motion. We show that this is equivalent to the assertion that each proper Euclidean motion that is not a pure translation, acting on the space of oriented lines, has a unique fixed point (the axis of the screw motion). We use that formulation to derive a simple and novel constructive proof of Chasles’ theorem.