On Euler's rotation theorem

TL;DR

This paper provides a natural geometric proof of Euler's rotation theorem in three dimensions, classifies rigid motions in space, and demonstrates the method with explicit examples, emphasizing elementary and intuitive arguments.

Contribution

It offers a geometric construction-based proof of Euler's rotation theorem in 3D and extends the classification of space rigid motions with explicit examples.

Findings

A natural geometric proof of Euler's rotation theorem in 3D.

Complete classification of space rigid motions, including orientation-preserving and not.

Explicit example classifying the composition of two isometries.

Abstract

It is well known that a rigid motion of the Euclidean plane can be written as the composition of at most three reflections. It is perhaps not so widely known that a similar result holds for Euclidean space in any number of dimensions. The purpose of the present article is, firstly, to present a natural proof of this result in dimension 3 by explicitly constructing a suitable sequence of reflections, and, secondly, to show how a careful analysis of this construction provides a quick and pleasant geometric path to Euler's rotation theorem, and to the complete classification of rigid motions of space, whether orientation preserving or not. Finally, we present an example where we use the general scheme of our proofs to classify the composition of two explicitly given orientation preserving isometries. We believe that our presentation will highlight the elementary nature of the results and hope that readers, perhaps especially those more familiar with the usual linear algebra approach, will appreciate the simplicity and geometric flavour of the arguments.