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

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

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