4.1 Description

$\mathrm{SE}(2)$ is the group of rigid transformations in the 2D plane, the semi-direct product $\mathrm{SO}(2)\ltimes\mathbb{R}^{2}$ . It has three degrees of freedom: two for translation and one for rotation. Subgroups include $\mathrm{SO}(2)$ .

$\displaystyle \mathbf{R}$	$\displaystyle \in$	$\displaystyle \mathrm{SO}(2)$	(45)
$\displaystyle \mathbf{t}$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{2}$	(46)
$\displaystyle X=\left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{t}\\ \hline \mathbf{0} & 1 \end{array}\right)$	$\displaystyle \in$	$\displaystyle \mathrm{SE}(2)\subset\mathbb{R}^{3\times3}$	(47)
$\displaystyle X^{-1}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R}^{T} & -\mathbf{R}^{T}\mathbf{t}\\ \hline \mathbf{0} & 1 \end{array}\right)$	(48)