3.3 Exponential Map

The exponential map from $\mathrm{\mathfrak{so}}(2)$ to $\mathrm{SO}(2)$ is simply a 2D rotation:

$$\exp\left(\mathrm{alg}\left(\theta\right)\right)=\exp\left(\begin... ...rray}{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{array}\right)$$ (39)

The logarithm is trivially computed from an element of $\mathrm{SO}(2)$.