9.1 Description

$\mathrm{SE}(3)$ is the group of rigid transformations in 3D space, the semi-direct product $\mathrm{SO}(3)\ltimes\mathbb{R}^{3}$ . It has six degrees of freedom: three for translation and three for rotation. Subgroups include $\mathrm{SE}(2)$ and $\mathrm{SO}(3)$ .

$\displaystyle \mathbf{R}$	$\displaystyle \in$	$\displaystyle \mathrm{SO}(3)$	(111)
$\displaystyle \mathbf{t}$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{3}$	(112)
$\displaystyle X$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{t}\\ \hline \mathbf{0} & 1 \end{array}\right)\in\mathrm{SE}(3)\subset\mathbb{R}^{4\times4}$	(113)
$\displaystyle X^{-1}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R}^{T} & -\mathbf{R}^{T}\mathbf{t}\\ \hline \mathbf{0} & 1 \end{array}\right)$	(114)