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.
| Transformation | Characterization |
| 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 |
| translation | displacement vector |
