A General Homogeneous Matrix Formulation to 3D Rotation Geometric Transformations

TL;DR

This paper introduces a unified algebraic framework for 3D rotations using homogeneous matrices, accommodating arbitrary rotation axes and angles, and providing formulas suitable for numerical applications like avoiding gimbal lock.

Contribution

It proposes a general homogeneous matrix formulation for 3D rotations that applies to arbitrary axes, bridging theoretical definitions and practical applications.

Findings

Derived a general 3D rotation formula similar to Euler-Rodrigues

Presented matrix-vector form suitable for numerical computations

Addressed rotations not passing through the origin

Abstract

We present algebraic projective geometry definitions of 3D rotations so as to bridge a small gap between the applications and the definitions of 3D rotations in homogeneous matrix form. A general homogeneous matrix formulation to 3D rotation geometric transformations is proposed which suits for the cases when the rotation axis is unnecessarily through the coordinate system origin given their rotation axes and rotation angles. General three-dimensional rotation formula~\eqref{eqn:3D homogeneous roation} and~\eqref{eqn:3D rotation matrix vector Euclidean} similar to the Euler-Rodrigues formula were presented. The matrix-vector form of 3D rotation in Euclidean space is especially suited for numerical applications where gimbal lock is a concern.}