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

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

(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