Lie Algebra and Lie Groups

Last updated Dec 22, 2023 Edit Source

Special Orthogonal Group: the group of rotational matrices $$ \mathrm{SO}(3) = { \mathbf{R} \in \mathbb{R}^{3 \times 3} | \mathbf{RR}^T = \mathbf{I}, \det(\mathbf{R})=1 }. $$ Special Euclidean Group: the group of transformation matrices $$ \mathrm{SE}(3) = \left{ \mathbf{T} = \begin{bmatrix} \mathbf{R} & \mathbf{t} \\ {{\mathbf{0}^T}} & 1 \end{bmatrix} \in \mathbb{R}^{4 \times 4} | \mathbf{R} \in \mathrm{SO}(3), \mathbf{t} \in \mathbb{R}^3\right} $$

These are groups because they satisfy all the required properties, but furthermore, because the elements of the set are continuous (smooth), they actually also form what we call a Lie Group

Let $\mathbf{R}(t)$ be the rotation of a camera that changes continuously over time, we know that it satisfies: $$ \begin{equation} \mathbf{R}(t)\mathbf{R}(t)^T = \mathbf{I} \end{equation} $$ Take derivative yields (by chain rule): $$ \dot{\mathbf{R}}(t)\mathbf{R}(t)^T+\mathbf{R}(t)\dot{\mathbf{R}}(t)^T=0 $$ which equals