Visualizing Rotations and Composition of Rotations with Rodrigues' Vector

TL;DR

This paper demonstrates how Rodrigues' vector simplifies the mathematical treatment and geometric understanding of 3D rotations, providing new interpretations and derivations of key rotation formulas suitable for educational purposes.

Contribution

It introduces a novel geometric interpretation of Rodrigues' vector and derives important rotation formulas using this perspective, enhancing understanding and teaching of 3D rotations.

Findings

Derived Euler-Rodrigues' formula using the new interpretation

Presented Cayley's rotation formula with geometric insights

Established a composition law for finite rotations

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.