5.3 Exponential Map

The exponential map from $\mathrm{\mathfrak{sim}}(2)$ to $\mathrm{Sim}(2)$ has a closed form:

$\displaystyle v=\left(\begin{array}{c} x\\ y\\ \theta\\ \lambda \end{array}\right)$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{4}$	(59)
$\displaystyle \mathbf{R}$	$\displaystyle \equiv$	$\displaystyle \left(\begin{array}{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{array}\right)$	(60)
$\displaystyle A$	$\displaystyle \equiv$	$\displaystyle \frac{\sin\theta}{\theta}$	(61)
$\displaystyle B$	$\displaystyle \equiv$	$\displaystyle \frac{1-\cos\theta}{\theta^{2}}$	(62)
$\displaystyle C$	$\displaystyle \equiv$	$\displaystyle \frac{\theta-\sin\theta}{\theta^{3}}$	(63)
$\displaystyle \alpha$	$\displaystyle \equiv$	$\displaystyle \frac{\lambda^{2}}{\lambda^{2}+\theta^{2}}$	(64)
$\displaystyle s$	$\displaystyle \equiv$	$\displaystyle e^{\lambda}$
$\displaystyle X$	$\displaystyle \equiv$	$\displaystyle \alpha\left(\frac{1-s^{-1}}{\lambda}\right)+\left(1-\alpha\right)\left(A-\lambda B\right)$	(65)
$\displaystyle Y$	$\displaystyle \equiv$	$\displaystyle \alpha\left(\frac{s^{-1}-1+\lambda}{\lambda^{2}}\right)+\left(1-\alpha\right)\left(B-\lambda C\right)$	(66)
$\displaystyle \mathbf{V}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{cc} X & -\theta Y\\ \theta Y & X \end{array}\right)$	(67)
$\displaystyle \exp\left(\mathrm{alg}\left(v\right)\right)=\exp\left(\begin{arra... ...0 & -\theta & x\\ \theta & 0 & y\\ \hline 0 & 0 & -\lambda \end{array}\right)$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{V}\cdot\left(\... ...}{c} x\\ y \end{array}\right)\\ \hline \mathbf{0} & s^{-1} \end{array}\right)$	(68)

The elements of $\mathbf{V}$ should be calculated with Taylor series when $\theta$ or $\lambda$ is small.