6.1 Representation

Similarity transformations are combinations of rigid transformation and scaling. The group of similarity transforms in 3D space, $\mathrm{Sim}(3)$ , has a nearly identical representation to $\mathrm{SE}(3$ ), with an additional scale factor:

$\displaystyle \mathbf{R}$	$\displaystyle \in$	$\displaystyle \mathrm{SO}(3),\:\mathbf{t}\in\mathbb{R}^{3},\; s\in\mathbb{R}$	(133)
$\displaystyle T$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{t}\\ \hline \mathbf{0} & s^{-1} \end{array}\right)\in\mathrm{Sim}(3)$	(134)

$\displaystyle T_{1},T_{2}$	$\displaystyle \in$	$\displaystyle \mathrm{Sim}(3)$	(135)
$\displaystyle T_{1}\cdot T_{2}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R}_{1} & \mathbf{t}_{1}\\ ... ...bf{R}_{2} & \mathbf{t}_{2}\\ \hline \mathbf{0} & s_{2}^{-1} \end{array}\right)$	(136)
	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R}_{1}\mathbf{R}_{2} & \mat... ...mathbf{t}_{1}\\ \hline \mathbf{0} & (s_{1}\cdot s_{2})^{-1} \end{array}\right)$	(137)
$\displaystyle T_{1}^{-1}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R}_{1}^{T} & -s_{1}\mathbf{R}_{1}^{T}\mathbf{t}\\ \hline \mathbf{0} & s_{1} \end{array}\right)$	(138)

$\displaystyle \mathbf{x}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{cccc} x & y & z & w\end{array}\right)^{T}\in\... ...bb{RP}^{3}\quad(\lambda\mathbf{x}\simeq\mathbf{x}\:\forall\lambda\in\mathbb{R})$
$\displaystyle T\cdot\mathbf{x}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{t}\\ \hline \mathbf{0} & s^{-1} \end{array}\right)\cdot\mathbf{x}$	(139)
	$\displaystyle =$	$\displaystyle \left(\begin{array}{c} \mathbf{R}\left(\begin{array}{ccc} x & y & z\end{array}\right)^{T}+w\mathbf{t}\\ s^{-1}w \end{array}\right)$	(140)
	$\displaystyle \simeq$	$\displaystyle \left(\begin{array}{c} s\left(\mathbf{R}\left(\begin{array}{ccc} x & y & z\end{array}\right)^{T}+w\mathbf{t}\right)\\ w \end{array}\right)$	(141)

In the typical case with

, this corresponds to rigid transformation followed by scaling.

The generators of the Lie algebra $\mathrm{sim(3)}$ are identical to those of $\mathrm{se(3)}$ (Eq. 58), with the addition of a generator corresponding to scale change:

$\displaystyle G_{7}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 \end{array}\right)$

(142)

$\displaystyle \left(\begin{array}{cc} \mathbf{u} & \boldsymbol{\omega}\end{array}\lambda\right)^{T}$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{7}$	(143)
$\displaystyle \mathbf{u}_{1}G_{1}+\mathbf{u}_{2}G_{2}+\mathbf{u}_{3}G_{3}+\bold... ...1}G_{4}+\boldsymbol{\omega}_{2}G_{5}+\boldsymbol{\omega}_{3}G_{6}+\lambda G_{7}$	$\displaystyle \in$	$\displaystyle \mathrm{sim}(3)$	(144)

For convenience, we write $\left(\begin{array}{cc} \mathbf{u} & \boldsymbol{\omega}\end{array}\lambda\right)^{T}\in\mathrm{sim}(3)$ , with multiplication against the generators implied.