Transformation matrices in RISC OS

Rotation and scaling

A general rotation plus scaling can be defined by multiplying a vector with a transformation matrix:

This can also be written as the following equations:

The matrix will rotate any vector over an angle , and enlarge it with a factor . Note that by convention the angle is positive for a rotation from the positive X axis towards the positive Y axis (i.e. anti-clockwise).

So, for example, a +90° rotation with no scaling, we have , and , which results in the matrix:

Note: In order to make fractions possible, the RISC OS transformation matrices have values in fixed-point notation, with 16 bits after the 'decimal' point. So a value of '1' must be given as .

If we want no rotation and a magnification with a factor of 2, we have , so and , and . This gives the matrix:

As an example of combined rotation and scaling, we will try a -90° rotation with a reduction in size with a factor 2. This gives , and :

In fixed-point notation the value of 0.5 can be represented as and -0.5 as .

Note that any angle is potentionally possible. Also note that it is not mandatory to make the scaling factor the same for all directions. If we use the matrix:

then the result is magnified (after rotation) in the direction with a factor and in the direction with a factor .

Adding translations

Equations (1.1) and (1.2) can be modified to incorporate translations by adding an extra term:

This is sometimes written in matrix form as:

Note: for the transformations in RISC OS, the terms e and f are also written in fixed-point notation, only now with 8 bits after the 'decimal' point. So a value of 1 should be given as .