5.1 Description

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

$\displaystyle X=\left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{t}\\ \hline \mathbf{0} & s^{-1} \end{array}\right)$	$\displaystyle \in$	$\displaystyle \mathrm{Sim}(2)\subset\mathbb{R}^{3\times3}$	(56)
$\displaystyle X^{-1}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R}^{T} & -s\mathbf{R}^{T}\mathbf{t}\\ \hline \mathbf{0} & s \end{array}\right)$	(57)

Ethan Eade 2012-02-16