Visualizing Rotations and Composition of Rotations with Rodrigues' Vector

Abstract

The purpose of this paper is to show that the mathematical treatment of three dimensional rotations can be simplified, and its geometrical understanding improved, by using the Rodrigues' vector representation. We present a novel geometrical interpretation of the Rodrigues' vector. Based on this interpretation and simple geometrical considerations, we derive Euler Rodrigues' formula, Cayley's rotation formula, and the composition law for finite rotations. The level of this discussion should be suitable for undergraduate physics or engineering courses where rotations are discussed.