It would be convenient to apply both rotation and translation together in a single operation. Suppose we want to apply a rotation matrix , and follow it with a translation by . Algebraically, this is

Although there is no way to form a single 3 by 3 matrix to accomplish both operations, it can be done by increasing the matrix dimensions by one. Consider the following 4 by 4

in which fills the upper left three rows and columns. The notation is used to denote that the matrix is a

The same result as in (3.20) can be obtained by performing multiplication with (3.23) as follows:

Because of the extra dimension, we extended the point by one dimension, to obtain . Note that (3.23) represents rotation

Steven M LaValle 2020-11-11