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

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

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

Steven M LaValle 2020-08-14