5.3 Adjoint

$\displaystyle \boldsymbol{\delta}=\left(\begin{array}{cc} \mathbf{u} & \theta\end{array}\right)^{T}$	$\displaystyle \in$	$\displaystyle \mathrm{se(2)},\; C=\left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{t}\\ \hline \mathbf{0} & 1 \end{array}\right)\mathbf{\in}\mathrm{SE(2)}$	(129)
$\displaystyle \mathrm{Adj}_{C}\cdot\boldsymbol{\delta}$	$\displaystyle =$	$\displaystyle C\cdot\left(\sum_{i=1}^{3}\boldsymbol{\delta}_{i}G_{i}\right)\cdot C^{-1}$	(130)
	$\displaystyle =$	$\displaystyle \left(\begin{array}{c} \mathbf{R}\mathbf{u}+\theta\left(\begin{ar... ...athbf{t}_{2}\\ -\mathbf{t}_{1} \end{array}\right)\\ \theta \end{array}\right)$	(131)
$\displaystyle \implies\mathrm{Adj}_{C}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R} & \begin{array}{c} \math... ...\end{array}\\ \hline \mathbf{0} & 1 \end{array}\right)\in\mathbb{R}^{3\times3}$	(132)

Note that moving a tangent vector via the adjoint mixes the rotation component into the translation component.