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A General Homogeneous Matrix Formulation to 3D Rotation Geometric Transformations

Feng Lu, Ziqiang Chen

TL;DR

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.

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.}

A General Homogeneous Matrix Formulation to 3D Rotation Geometric Transformations

TL;DR

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.

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.}

Paper Structure

This paper contains 3 sections, 1 theorem, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definition of 2D and 3D rotations
  3. Formulation to general 3D rotations

Key Result

Theorem 1

Suppose $\mathscr{T}_0$ is a geometric transformation in projective space which transforms an arbitrary point $X$ into $Y$; and the homogeneous coordinates of $X$ and $Y$ in reference coordinate systems $(I)$ and $(II)$ are $(x)$, $(y)$, $(x')$, $(y')$ respectively; the transformation matrices of $ The matrices of $\mathscr{T}_0$ in $(I)$ and $(II)$ are similar.

Figures (1)

  • Figure 1: Extended Desarguesian configuration for reflection

Theorems & Definitions (3)

  • Theorem 1
  • Definition 1: Rotation
  • Definition 2: Rotation