An (x, y) coordinate pair can be transformed to another coordinate system by scaling, rotation and translation. The first two can be implemented using 2 x 2 matrices to hold the coefficients:
If the point P is (a, b) and it is transformed to the point Q (a', b')
P ⇒ Q or (a, b) ⇒ (a', b')
a' = pa + rb , b' = qa + sb .
Translation can be included in this if the points P and Q are regarded as 3-vectors instead of 2-vectors, with the 3rd element being unity:
The coefficients u and v specify the translation.
So, the six elements (
) of the 3 x 3 matrix contain all the transformation information. These elements are stored in a list in the graphics state slot
Transforms can be combined by matrix multiplication to effect successions of translation, scaling and rotation operations.
Functions are provided in Graphics Ports which apply translation, scaling and rotation to a transform, combine transforms by pre- or post-multiplication, invert a transform, and so on.