It will become important to study families of transformations, in which some parameters are used to select the particular transformation. Therefore, it makes sense to generalize to accept two variables: a parameter vector, , along with . The resulting transformed point is denoted by , and the entire robot is transformed to .

The coming material will use the following shorthand notation, which requires the specific to be inferred from the context. Let be shortened to , and let be shortened to . This notation makes it appear that by adjusting the parameter , the robot travels around in as different transformations are selected from the predetermined family. This is slightly abusive notation, but it is convenient. The expression can be considered as a set-valued function that yields the set of points in that are occupied by when it is transformed by . Most of the time the notation does not cause trouble, but when it does, it is helpful to remember the definitions from this section, especially when trying to determine whether or is needed.

Steven M LaValle 2020-08-14