Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations

TL;DR

This paper derives the Euler-Rodrigues formula for 3D rotations from 4D rotation matrices, proving the relationship between quaternions and rotations, and extends the approach to rotoreflections.

Contribution

It provides a novel derivation of the Euler-Rodrigues formula from 4D rotation matrices and outlines a method to determine Euler parameters for 3D rotations.

Findings

Proves that left and right quaternions are inverses in 3D rotations.

Derives the Euler-Rodrigues formula from 4D rotation matrix specialization.

Extends the derivation to 3D rotoreflections.

Abstract

The general 4D rotation matrix is specialised to the general 3D rotation matrix by equating its leftmost top element (a00) to 1. Its associate matrix of products of the left-hand and right-hand quaternion components is specialised correspondingly. Inequalities involving the angles through which the coordinate axes in 3D space are displaced are used to prove that the left-hand and the right-hand quaternions are each other's inverses, thus proving the Euler-Rodrigues formula. A general procedure to determine the Euler parameters of a given 3D rotation matrix is sketched. By equating the leftmost top element to -1 instead of +1 in the general 4D rotation matrix, one proves the counterpart of the Euler-Rodrigues formula for 3D rotoreflections. Keywords: Euler--Rodrigues formula, Euler parameters, quaternions, four--dimensional rotations, three--dimensional rotations, rotoreflections