Nonlinear Projections - Transformations and Projections in Computer Graphics - page 170

a rectangle of width 360 and height 180 units and project P ∗ on the rectangle by simply
using its longitude and latitude as the x and y coordinates, respectively, on the rectangle.
Figure 4.26 illustrates the Earth in this projection, and the deformation is immediately
obvious. On the rectangle, the lines of latitude are the same length, so polar latitudes,
which on the sphere are short, have to be stretched.
Figure 4.26: Equidirectional Projection of a Sphere.
When the entire 360 ◦ space around an observer is projected onto the rectangle in
this way, the regions directly above and below the observer (which often are less impor-
tant) are stretched and feature much detail. The regions at the height of the observer
(the equator), however, lack detail, but are to scale. This projection is sometimes used
in map making and is referred to as equirectangular projection, rectangular projection,
plane chart, or plate carre.
The remainder of this section describes another, highly distorted version of spherical
panoramic projection. This version is another manifestation of the concept of curved
perspective.
What you see on these screens up here is a fantasy; a computer enhanced
hallucination!
—John Wood (as Stephen Falken) in WarGames (1983).
Imagine a transparent sphere of radius R centered on the origin, where an observer
is located, looking through the sphere in the z direction. The sphere is now truncated
by selecting a value θ in the range [0 ,π/ 2] and removing the parts of the sphere above
and below latitude θ . The remaining part is shaped like a barrel (Figure 4.27a). The