Nonlinear Projections - Transformations and Projections in Computer Graphics - page 202

68 0

60 0

R cos φ

45 0

30 0

15 0

R

φ

(a) (b) (c) (d)

Figure 4.51: Three Types of Cylindrical Projections.

Mercator projection . A common variant of the cylindrical perspective projection
is the popular Mercator projection, developed by Gerhardus Mercator in 1569. Its
principle is to increase the distance between consecutive latitudes in proportion to the
increased distance between meridians. This effect is illustrated in Figure 4.51d. The
circumference of a globe of radius R at the equator is 2 πR and at latitude φ it is
2 πR cos φ . Thus, the width of a longitude at latitude φ [the distance between longitude
θ ◦ and ( θ +1) ◦ ] is smaller than the width of a longitude at the equator by a factor of cos φ .
In a cylindrical projection, the longitudes are shown as parallel lines, which means that
at latitude φ , the width of a longitude in the projection has been artificially increased by
afactorof1 / cos φ . The width of a meridian can be considered the horizontal scale, so
the principle of the Mercator projection is to also increase the vertical scale by the same
factor. In the basic cylindrical projection, the y coordinate depends on the latitude φ
as y = R tan φ . Now, we have to change the dependence such that a small change Δ φ
in φ changes y by a factor of R Δ φ/ cos φ . The basic equation of y as a function of φ is
therefore

Rdφ
cos φ ,

dy =

which integrates to yield

y ( φ )= R ln tan π

.

φ
2 4 −

Any integration constant is eliminated if we impose the condition that φ = 0 implies
y =0.
Now imagine a small region at latitude φ . Both its width and its height have been
increased by a factor of 1 / cos φ , so its area is increased by a factor of 1 / cos 2 φ , but its
shape hasn't changed. A large region tends to spread beyond a single latitude, so its
shape is distorted. Thus, the Mercator projection preserves the shapes of small regions
and makes it relatively easy to compute their true areas. Large regions are distorted and
also appear very large. Greenland, for example, appears bigger than South America,
even though the latter is nine times bigger than Greenland.