Exponential Map

The exponential map makes it possible to move from the tangent space to the manifold of rotations:

$$\mathbf{R} = \exp\!\left( [\boldsymbol{\omega}]_\times \right).$$ (A.34)

If

$$\boldsymbol{\omega}=\theta \mathbf{k},$$ (A.35)

where $\mathbf{k}$ is a unit axis and $\theta$ is the rotation angle, the preceding expression coincides with the Rodrigues formula already introduced in the chapter devoted to axis-angle parameterization.

In other words, the Rodrigues formula can be interpreted as the exponential map of the group $SO(3)$ A.4.