10.1 Description

$\mathrm{Sim}(3)$ is the group of similarity transformations in 3D space, the semi-direct product $\mathrm{SE}(3)\rtimes\mathbb{R}^{*}$ . It has seven degrees of freedom: three for translation, three for rotation, and one for scale. Subgroups include $\mathrm{Sim}(2)$ and $\mathrm{SE}(3)$ .

$\displaystyle \mathbf{R}$	$\displaystyle \in$	$\displaystyle \mathrm{SO}(3)$	(127)
$\displaystyle \mathbf{t}$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{3}$	(128)
$\displaystyle s$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{+}$	(129)
$\displaystyle X$	$\displaystyle =$	$\displaystyle \mathrm{\left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{t}\\ ... ...e \mathbf{0} & s^{-1} \end{array}\right)\in Sim}(3)\subset\mathbb{R}^{4\times4}$	(130)
$\displaystyle X^{-1}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R}^{T} & -s\mathbf{R}^{T}\mathbf{t}\\ \hline \mathbf{0} & s \end{array}\right)$	(131)