useful identities

The following is a random collection of identities and facts I encounter semi-regularly in my research.

Lie Groups

Lie groups are ubiquitous in robotics due to the need to handle rotations. If you are looking for a comprehensive introduction to the subject, and are classic resources.

Fact 1: For the $SO(2)$ group of 2D rotations, the following holds, \begin{align} v^\wedge \mathbf{u} &= \begin{bmatrix} 0 & -v\\ v & 0 \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} \\ &= \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} v \\ &= \mathbf{P} \mathbf{u}v, \end{align} Where \(\mathbf{P}=\begin{bmatrix} 0 & -1 \\\ 0 & 1\end{bmatrix}\). This contrasts with the $SO(3)$ case, where \(\mathbf{v}^\wedge \mathbf{u} = -\mathbf{u}^\wedge \mathbf{v}\). This ties into the $\ \odot\ $ operator as described in in the section covering homogeneous points.