A parameterized family of transformations

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 ${h}$ to accept two variables: a parameter vector, $q \in {\mathbb{R}}^n$, along with $a \in {\cal A}$. The resulting transformed point $a$ is denoted by ${h}(q,a)$, and the entire robot is transformed to ${h}(q,{\cal A}) \subset {\cal W}$.

The coming material will use the following shorthand notation, which requires the specific ${h}$ to be inferred from the context. Let ${h}(q,a)$ be shortened to $a(q)$, and let ${h}(q,{\cal A})$ be shortened to ${\cal A}(q)$. This notation makes it appear that by adjusting the parameter $q$, the robot ${\cal A}$ travels around in ${\cal W}$ as different transformations are selected from the predetermined family. This is slightly abusive notation, but it is convenient. The expression ${\cal A}(q)$ can be considered as a set-valued function that yields the set of points in ${\cal W}$ that are occupied by ${\cal A}$ when it is transformed by $q$. 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 ${h}$ or ${h^{-1}}$ is needed.