6.4 Adjoint Representation

6.4 Adjoint Representation

Let

$\displaystyle X=\left(\begin{array}{c\vert c} \mathbf{A} & \mathbf{t}\\ \hline... ...array}{cc\vert c} a & b & x\\ c & d & y\\ \hline 0 & 0 & 1 \end{array}\right)$	$\displaystyle \in$	$\displaystyle \mathrm{Aff}(2)$	(75)
$\displaystyle \mathbf{E}$	$\displaystyle \equiv$	$\displaystyle \left(\begin{array}{cc\vert cccc} 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 ... ...{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \end{array}\right)\in\mathbb{R}^{6\times6}$	(76)
$\displaystyle f$	$\displaystyle \equiv$	$\displaystyle \frac{1}{ad-bc}$	(77)
$\displaystyle \mathbf{\mathbf{C}}$	$\displaystyle \equiv$	$\displaystyle \left(\begin{array}{c\vert c} a\cdot\mathbf{A}^{-T} & b\cdot\math... ...thbf{A}^{-T} & d\cdot\mathbf{A}^{-T} \end{array}\right)\in\mathbb{R}^{4\times4}$	(78)
	$\displaystyle =$	$\displaystyle f\left(\begin{array}{cccc} ad & -ac & bd & -bc\\ -ab & a^{2} & -b^{2} & ab\\ cd & -c^{2} & d^{2} & -cd\\ -bc & ac & -bd & ad \end{array}\right)$	(79)
$\displaystyle \mathbf{T}$	$\displaystyle \equiv$	$\displaystyle \left(\begin{array}{cccc} x & y & 0 & 0\\ 0 & 0 & x & y \end{array}\right)\in\mathbb{R}^{2\times4}$	(80)

Then

$\displaystyle \mathrm{Adj}_{X}$

$\displaystyle =$

$\displaystyle \mathbf{E}^{T}\left(\begin{array}{c\vert c} \mathbf{A} & -\mathbf... ...hbf{C}}\\ \hline \mathbf{0} & \mathbf{\mathbf{C}} \end{array}\right)\mathbf{E}$

(81)

Writing out the product explicitly:

$\displaystyle \mathrm{Adj}_{X}={\scriptstyle \left(\begin{array}{cc\vert cccc} ... ...ab\right) & \frac{f}{2}\left(a^{2}-b^{2}-c^{2}+d^{2}\right) \end{array}\right)}$

(82)

Ethan Eade 2012-02-16