Mapping Hacks

Georeference an Arbitrary Tourist Map #33

Chapter 3, Mapping Your World

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HACK

longitude. For each dimension, x and y, there’s a scale between one coordi-

nate system and the other, the skew between one and the other, and the off-

set of the image—e.g., where it begins, at its northwestern corner.

Figure 3-40 shows the linear algebra used in the affine transform.

a is the x scale

b is the y rotation/skew

c is the x rotation/skew

d is the y scale

e is the x offset

f is the y offset

The last three values are constants there to make the matrix algebra work

out, and in an affine transformation, they are always [0 0 1]. So the matrix

can be represented as a vector of numbers [abcdef]. This is the order that

the world file lists the coefficients in. If a map is oriented to True North,

then the rotational coefficients b and c from its world file will be zero.

This transformation matrix might be used in the following fashion. Suppose

that we have a map in a rectangular projection. If a user clicks on the map at

(x, y), what is the geographic location corresponding to that point? If we

have a world file expressed, for example, in UTM coordinates, then we can

calculate the easting and northing as (x', y') with the following formulas:

x' = ax + cy + e

y' = bx + dy + f

The actual target coordinate system doesn’t matter; one can use the same

technique to turn (x, y) into latitude and longitude, if the world file ...