#### Special Euclidean group

Now that the complicated part of representing has been handled, the representation of is straightforward. The general form of a matrix in is given by (4.16), in which and . Since , and translations can be chosen independently, the resulting C-space for a rigid body that rotates and translates in is

 (4.31)

which is a six-dimensional manifold. As expected, the dimension of is exactly the number of degrees of freedom of a free-floating body in space.

Steven M LaValle 2020-08-14