On the composition of finite rotations in $\mathbb{E}^4$

TL;DR

This paper develops composition rules for finite rotations in four-dimensional Euclidean space, providing conditions for when the composition of two simple rotations results in another simple rotation, analogous to Rodrigues' formula.

Contribution

It introduces new composition rules for rotations in D, including a necessary and sufficient condition for the composition of two simple rotations to remain simple.

Findings

Derived composition rules for D rotations

Established a criterion for the composition of simple rotations

Analogous to Rodrigues formula in higher dimensions

Abstract

We achieve compositions rules for the geometric parameters of the composed rotations, which is in a certain sense analogous to the well known Rodrigues formula. We also obtain a necessary and sufficient condition for a composition of two simple rotations in $\mathbb{E}^4$ to be a simple rotation.