3.2.1 General Concepts

Before giving specific transformations, it will be helpful to define them in general to avoid confusion in later parts when intuitive notions might fall apart. Suppose that a rigid robot, $ {\cal A}$, is defined as a subset of $ {\mathbb{R}}^2$ or $ {\mathbb{R}}^3$. A rigid-body transformation is a function, $ {h}: {\cal A}\rightarrow {\cal W}$, that maps every point of $ {\cal A}$ into $ {\cal W}$ with two requirements: 1) The distance between any pair of points of $ {\cal A}$ must be preserved, and 2) the orientation of $ {\cal A}$ must be preserved (no ``mirror images'').

Using standard function notation, $ {h}(a)$ for some $ a \in {\cal A}$ refers to the point in $ {\cal W}$ that is ``occupied'' by $ a$. Let

$\displaystyle h({\cal A}) = \{h(a) \in {\cal W}\;\vert\; a \in {\cal A}\} ,$ (3.21)

which is the image of $ {h}$ and indicates all points in $ {\cal W}$ occupied by the transformed robot.

Steven M LaValle 2012-04-20