A transformation (a.k.a., map, function) over a domain takes the elements to elements , where the range (a.k.a., image) of is defined as
Note that when transformations are specified with respect to a coordinate system, it is important to specify whether the rotation takes place on the coordinate system, with space and objects embedded in it being viewed as fixed (a so-called alias transformation), or on the space itself relative to a fixed coordinate system (a so-called alibi transformation).
Examples of transformations are summarized in the following table.
|dilation||center of dilation, scale decrease factor|
|expansion||center of expansion, scale increase factor|
|reflection||mirror line or plane|
|rotation||center of rotation, rotation angle|
|shear||invariant line and shear factor|
|stretch (1-way)||invariant line and scale factor|
|stretch (2-way)||invariant lines and scale factors|