The Group of Rigid Motions

Three-dimensional rigid transformations belong to the group

$$SE(3),$$ (A.41)

defined as

$$\mathbf{T} = \begin{bmatrix} \mathbf{R} & \mathbf{t}\\ \mathbf{0}^{T} & 1 \end{bmatrix},$$ (A.42)

where $\mathbf{R}\in SO(3)$ and $\mathbf{t}\in\mathbb{R}^{3}$.

While $SO(3)$ describes rotations exclusively, $SE(3)$ represents rotations and translations simultaneously.

The poses of cameras, sensors, and vehicles are generally represented by elements of $SE(3)$.

Pose optimization in calibration, visual odometry, SLAM, and bundle adjustment problems is often formulated directly in the tangent space associated with this group.