Applying coordinate transformations to ESRI world files

The ESRI World file is a plain text file with numbers on six lines. These numbers actually are just the coefficients of an Affine transformation matrix. The world file helps to place the raster image on a 2-dimensional plane. Since the world file is just a matrix, it is a simple matter to perform coordinate transformations (translation, rotation, and scaling) to the raster image by applying the various transformation matrices with the world file matrix.

An example of a world file is shown below.

0.124966541755889
-0.216448399567377
-0.216438882466538
-0.124961047055157
2132.25596524947
1263.76014101498

If we label the lines as the following for discussion:
A
D
B
E
C
F

Then the Affine transformation matrix of the raster image is

A C E
B D F
0 0 1

For more information on the world file format, visit the Wikipedia site http://en.wikipedia.org/wiki/World_file.

Translation transformation
To apply a translation transformation to the raster image, it is just a matrix multiplication of the translation matrix with the world file matrix, as shown below. dx and dy are the x offset and y offset of the translation.

| 1 0 dx |   | A C E |
| 0 1 dy | * | B D F |
| 0 0 1  |   | 0 0 1 |

Scaling transformation
To apply a scaling transformation to the raster image, just multiply the world file matrix with the scaling transformation matrix. s is the scale factor.

Note: if the scaling origin is not at (0,0), then apply the translation transformation to move the origin to the scaling origin (x,y) first. After the scaling is done, the apply an inverse translation translation to move the origin back.

| s 0 0 |   | A C E |
| 0 s 0 | * | B D F |
| 0 0 1 |   | 0 0 1 |

Rotation transformation
Similarly, to apply a rotation transformation to the raster image, multiply the rotation transformation matrix with the world file matrix. r is the rotation angle in radians (positive clockwise). 

Note: if the rotation origin is not at (0,0), then apply the translation transformation to move the origin to the rotation origin (x,y) first. After the rotation is done, the apply an inverse translation translation to move the origin back.

|  cos(r) sin(r) 0 |   | A C E |
| -sin(r) cos(r) 0 | * | B D F |
|  0      0      1 |   | 0 0 1 |