Special Euclidean group

Now that the group of rotations, $SO(n)$, is characterized, the next step is to allow both rotations and translations. This corresponds to the set of all $(n+1) \times (n+1)$ transformation matrices of the form

$$\left\{ \begin{pmatrix}R & v 0 & 1 \end{pmatrix} \;\bigg\vert\; R \in SO(n) \mbox{ and } v \in {\mathbb{R}}^n \right\} .$$ (4.16)

This should look like a generalization of (3.52) and (3.56), which were for $n=2$ and $n=3$, respectively. The $R$ part of the matrix achieves rotation of an $n$-dimensional body in ${\mathbb{R}}^n$, and the $v$ part achieves translation of the same body. The result is a group, $SE(n)$, which is called the special Euclidean group. As a topological space, $SE(n)$ is homeomorphic to ${\mathbb{R}}^n \times SO(n)$, because the rotation matrix and translation vectors may be chosen independently. In the case of $n=2$, this means $SE(2)$ is homeomorphic to ${\mathbb{R}}^2 \times {\mathbb{S}}^1$, which verifies what was stated at the beginning of this section. Thus, the C-space of a 2D rigid body that can translate and rotate in the plane is

$${\cal C}= {\mathbb{R}}^2 \times {\mathbb{S}}^1 .$$ (4.17)

To be more precise, ${ \;\cong\; }$ should be used in the place of $=$ to indicate that ${\cal C}$ could be any space homeomorphic to ${\mathbb{R}}^2 \times {\mathbb{S}}^1$; however, this notation will mostly be avoided.