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, , is defined as a subset of or . A rigid-body transformation is a function, , that maps every point of into with two requirements: 1) The distance between any pair of points of must be preserved, and 2) the orientation of must be preserved (no ``mirror images'').
Using standard function notation, for some refers to the point in that is ``occupied'' by . Let