Nonlinear Projections - Transformations and Projections in Computer Graphics - page 209

This projection is often the one favored by climatologists to display climate data.
Sometimes it is used as a small inset inside another map (probably because of its pleasing
shape), and the National Geographic Society in the United States used it for printing
large wall maps of the world.
Conical projections . Projections that employ a cone as the developable sur-
face have limited applications because they result in a noticeable distortion of shapes.
Figure 4.59a portrays a cone of height h and radius R . Wedenotehalfitstopangle
by α and notice that α varies in the interval [0 , 90 ◦ ). It is immediately obvious that
l 2 = h 2 + R 2 and sin α = R/l . Part (b) of the figure shows the cone flattened, and the
problem is to compute its top angle β . The bottom part of the flattened cone is a cir-
cular arc whose length equals the circumferen ce 2 πR of the original cone bottom. Thus

βl =2 πR or β =2 πR/l =2 π sin α =2 πR/ √ h 2 + R 2 . For example, when α =45 ◦ ,we
get β 0 . 7071 = 255 ◦ .

≈ 2 π ·

l l

α

h

l

β

R

2 πR

(a) (b)

Figure 4.59: A Cone (a) Before and (b) After Flattening.

Clouds are not spheres, mountains are not cones, coastlines are not
circles, and bark is not smooth, nor does lightning travel in a straight
line.

—Beno ıt Mandelbrot, The Fractal Geometry of Nature (1982)

Figure 4.60 illustrates a simple equidistant conic projection of the Earth. This
projection is appropriate for small regions regardless of their shape. It is also acceptable
for large regions or even continents of predominant east-west extent. It illustrates the
main features of a conic projection which are as follows:
1. Meridians are straight equidistant lines converging at the apex of the cone
(normally a pole). The angular distance between meridians shrinks linearly as we move
toward the apex, and the shrink factor is referred to as the cone constant.
2. Parallels are concentric circular arcs whose center is the point of convergence
of the meridians. As a result, the parallels cross all the meridians at right angles and
distortion is constant along each parallel.
3. In addition, the particular conical projection of Figure 4.60 is neither conformal
nor equal-area, but such variations of the conical projection are possible.