###
12.4.1 Generalized points

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 (*
p*
, *
q*
, *
r*
, *
s*
, *
u*
, and *
v*
) of the 3 x 3 matrix contain all the transformation information. These elements are stored in a list in the graphics state slot ```
transform
```

.

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.

LispWorks CAPI User Guide (Unix version) - 22 Dec 2009