2.1 Representation

Elements of the 3D rotation group, $\mathrm{SO}(3)$ , are represented by 3D rotation matrices. Composition and inversion in the group correspond to matrix multiplication and inversion. Because rotation matrices are orthogonal, inversion is equivalent to transposition.

$\displaystyle \mathbf{R}$	$\displaystyle \in$	$\displaystyle \mathrm{SO}(3)$	(1)
$\displaystyle \mathbf{R}^{-1}$	$\displaystyle =$	$\displaystyle \mathbf{R}^{T}$	(2)

The Lie algebra, $\mathrm{so}(3)$ , is the set of $3\times3$ skew-symmetric matrices. The generators of $\mathrm{so}(3)$ correspond to the derivatives of rotation around the each of the standard axes, evaluated at the identity:

$\displaystyle G_{1}=\left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & -1\\ 0 & 1 & 0... ...=\left(\begin{array}{ccc} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{array}\right)$

(3)

An element of $\mathrm{so}(3)$ is then represented as a linear combination of the generators:

$\displaystyle \boldsymbol{\omega}$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{3}$	(4)
$\displaystyle \boldsymbol{\omega}_{1}G_{1}+\boldsymbol{\omega}_{2}G_{2}+\boldsymbol{\omega}_{3}G_{3}$	$\displaystyle \in$	$\displaystyle \mathrm{so}(3)$	(5)

We will simply write $\mathrm{\boldsymbol{\omega}\in so}(3)$ as a 3-vector of the coefficients, and use $\boldsymbol{\omega}_{\times}$ to represent the corresponding skew symmetric matrix.

Ethan Eade 2012-02-16