Nonlinear Projections - Transformations and Projections in Computer Graphics - page 203

“What's the good of Mercator's north poles and equators, tropics, zones, and merid-
ian lines?” so the Bellman would cry: and the crew would reply “They are merely
conventional signs!”

—Lewis Carroll, The Hunting of the Snark (1876)

Figure 4.54 shows the standard Mercator projection of the Earth (where the cylin-
der is tangent to the equator), and Figure 4.55 is the oblique 45 ◦ Mercator projection
introduced by Charles Peirce in 1894.
Cylindrical equal-area projection . When the cylinder is aligned with the rotation
axis of the globe, any cylindrical projection results in uniformly spaced, parallel merid-
ians and parallel latitudes. However, the latitudes don't have to be spaced uniformly,
and their spacings can be adjusted to preserve areas. There is essentially only one way
to design a cylindrical equal-area projection, and it was first described by Johann H.
Lambert in 1772.
In a cylindrical projection, the x coordinate for longitude θ on the unrolled cylinder
is the length of the arc between θ and θ 0 .Thus x = R ( θ θ 0 ). We have to adjust the

space between consecutive latitudes such that any area on the cylinder will equal the
corresponding area on the sphere, and this is easy to achieve by comparing areas on the
sphere and the cylinder. The total surface area of a sphere is 2 πR 2 , so the area below
latitude φ is 2 πR 2 sin φ . The area of a cylinder below a certain height y is 2 πRy ,so
equating the expressions 2 πR 2 sin φ and 2 πRy results in y = R sin φ .
Table 4.52 lists y values for R = 1 and for latitudes from 0 to 90 ◦ and the stretch
factor for each. This factor is the extra amount the y coordinate has to be moved
relative to its “natural” position. For example, for φ =30 ◦ , the natural position for
the y coordinate is 0.3, but it has moved to 0.5, a stretch factor of 1.67. Figure 4.53
illustrates how each latitude is raised (the dashed lines in the Northern Hemisphere) in
order to preserve areas. The figure illustrates the fact that such a projection is useful
in the equatorial regions but useless in the polar regions, where the small gaps between
consecutive latitudes make it impossible to distinguish shapes, borders, and distances.

−

70

90

60

φ ◦ φ ◦

y y

Stretch Stretch

50

70

40
30

0 0 0.00 50 0.77 1.53

10 0.17 1.74 60 0.87 1.44

50

20 0.34 1.71 70 0.94 1.34

20

30

30 0.50 1.67 80 0.98 1.23

10

10

40 0.64 1.61 90 1.00 1.11

Table 4.52 Cylindrical Equal-Area Projection. Figure 4.53 Cylindrical Equal-Area Projection.

Lambert's design for an equal-area projection can be varied and is used by several
similar equal-area cylindrical projections. These vary the standard parallels, the general
map proportions, and the ways of distorting shapes. These projections can be converted
back to Lambert's by rescaling both the width and height.
Cylindrical equidistant projection . Perhaps the most familiar feature of the cylin-
drical projections discussed so far is the straight, parallel, and equidistant meridians.