Definition

The set of all transformation matrices is the special Euclidean group: $$SE(3):= \left\{ \mathbf{T}=\begin{bmatrix}\mathbf{C} & \mathbf{r} \\ \mathbf{0}^{\top} & 1\end{bmatrix}\in\mathbb{R}^{4\times 4}\middle\vert\mathbf{C}\in SO(3)\text{ and }\mathbf{r}\in\mathbb{R}^{3} \right\}$$