Special Euclidean group

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 form

 (4.16)

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 body in , 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 homeomorphic to , 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

 (4.17)

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 2020-08-14