##

4.2.1 2D Rigid Bodies:

Section 3.2.2 expressed how to transform a rigid body in
by a homogeneous transformation matrix, , given by
(3.35). The task in this chapter is to characterize the
set of all possible rigid-body transformations. Which manifold will
this be? Here is the answer and brief explanation. Since any
can be selected for translation, this alone yields a
manifold
. Independently, any rotation,
, can be applied. Since yields the same rotation as 0,
they can be identified, which makes the set of 2D rotations into a
manifold,
. To obtain the manifold that corresponds to
all rigid-body motions, simply take
. The answer to the question is that the C-space is a
kind of cylinder.

Now we give a more detailed technical argument. The main purpose is
that such a simple, intuitive argument will not work for the 3D case.
Our approach is to introduce some of the technical machinery here for
the 2D case, which is easier to understand, and then extend it to the
3D case in Section 4.2.2.

**Subsections**
Steven M LaValle
2020-08-14