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.

Steven M LaValle 2020-08-14