Special Euclidean group

Now that the complicated part of representing $SO(3)$ has been handled, the representation of $SE(3)$ is straightforward. The general form of a matrix in $SE(3)$ is given by (4.16), in which $R \in SO(3)$ and $v \in {\mathbb{R}}^3$. Since $SO(3) { \;\cong\; }{\mathbb{RP}}^3$, and translations can be chosen independently, the resulting C-space for a rigid body that rotates and translates in ${\mathbb{R}}^3$ is

$${\cal C}= {\mathbb{R}}^3 \times {\mathbb{RP}}^3 ,$$ (4.31)

which is a six-dimensional manifold. As expected, the dimension of ${\cal C}$ is exactly the number of degrees of freedom of a free-floating body in space.