4.1 Representation

Elements of the rotation group in two dimensions, $\mathrm{SO}(2)$ , are represented by 2D rotation matrices. Composition and inversion in the group correspond to matrix multiplication and inversion. Because rotation matrices are orthogonal, inversion is equivalent to transposition.

$\displaystyle \mathbf{R}$	$\displaystyle \in$	$\displaystyle \mathrm{SO}(2)$	(90)
$\displaystyle \mathbf{R}^{-1}$	$\displaystyle =$	$\displaystyle \mathbf{R}^{T}$	(91)

The Lie algebra, $\mathrm{so}(2)$ , is the set of $2\times2$ skew-symmetric matrices. The single generator of $\mathrm{so}(2)$ corresponds to the derivative of 2D rotation, evaluated at the identity:

$\displaystyle G=\left(\begin{array}{cc} 0 & -1\\ 1 & 0 \end{array}\right)$

(92)

An element of $\mathrm{so}(2)$ is then any scalar multiple of the generator:

$\displaystyle \theta$	$\displaystyle \in$	$\displaystyle \mathbb{R}$	(93)
$\displaystyle \theta G$	$\displaystyle \in$	$\displaystyle \mathrm{so}(2)$	(94)

We will simply write $\mathrm{\theta\in so}(2)$ , and use $\theta_{\times}$ to represent the skew symmetric matrix $\theta G$ .

Ethan Eade 2012-02-16