2.6 Group Action on

Given the matrix representation of $\mathrm{G}$ in $\mathbb{R}^{n\times n}$ , there is a natural action on the vector space $\mathbb{R}^{n}$ (equivalently the projective space $\mathbb{P}^{n-1}$ ) by multiplication:

$\displaystyle X$	$\displaystyle \in$	$\displaystyle \mathrm{G}$	(29)
$\displaystyle X:\mathbb{R}^{n}$	$\displaystyle \rightarrow$	$\displaystyle \mathbb{R}^{n}$	(30)
$\displaystyle X\left(\mathbf{v}\right)$	$\displaystyle =$	$\displaystyle X\cdot\mathbf{v}$	(31)

For the groups described below, this group action by matrix multiplication yields a transformation on points or lines in 2D or 3D Euclidean or projective space. For example, the group action of an element of $\mathrm{SE}(2)$ on $\mathbb{P}^{2}$ (the 2D plane as homogeneous coordinates in $\mathbb{R}^{3}$ ) is a rotation and translation of the plane coordinates.

The Jacobian of this action by the group differentials around the identity is trivially computed using the

generators of the algebra:

$\displaystyle \mathbf{p}$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{n}$	(32)
$\displaystyle \mathbf{c}$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{k}$	(33)
$\displaystyle f\left(\mathbf{c},p\right)$	$\displaystyle \equiv$	$\displaystyle \exp\left(\mathrm{alg}\left(\mathbf{c}\right)\right)\cdot\mathbf{p}$	(34)
$\displaystyle \frac{\partial f}{\partial\mathbf{c}}$	$\displaystyle \big\vert _{\mathbf{c}=0}=$	$\displaystyle \left(\begin{array}{c\vert c\vert c\vert c} G_{1}\cdot\mathbf{p} ... ...f{p} & \cdots & G_{k}\cdot\mathbf{p}\end{array}\right)\in\mathbb{R}^{n\times k}$	(35)