Now that the group of rotations, , is characterized, the next
step is to allow both rotations and translations. This corresponds to
the set of all
transformation matrices of the
This should look like a generalization of (3.52) and
(3.56), which were for and , respectively.
The part of the matrix achieves rotation of an -dimensional
, and the part achieves translation of the same
body. The result is a group, , which is called the
special Euclidean group. As a topological space, is
, because the rotation matrix and
translation vectors may be chosen independently. In the case of
, this means is homeomorphic to
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
To be more precise,
should be used in the place of to
indicate that could be any space homeomorphic to
; however, this notation will mostly be avoided.
Steven M LaValle