Infinitesimal Rotations

The compact definition $\mathbf{w}=\vartheta \mathbf{k}$, together with the Rodrigues formula, makes it possible to express infinitesimal rotations in a very simple-to-compute form.

Indeed, if $\vartheta$ is taken to be infinitesimal, formula (A.13) can be approximated as

$$\mathbf{R} \approx \mathbf{I} + \sin \vartheta [\mathbf{k}]_... ...& w_y \\ w_z & 1 & -w_x \\ -w_y & w_x & 1 \\ \end{bmatrix}$$ (A.17)

The skew-symmetric matrices constitute the Lie algebra $\mathfrak{so}(3)$ associated with the rotation group A.4.