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 |