Nonlinear Projections - Transformations and Projections in Computer Graphics - page 189

segment (#58) is drawn, perpendicularly bisecting the first one, as shown in part 10.
Once this is done, it is easy to construct the two segments 62 and 64 of part 11 and
end up with an equilateral triangle on the picture. The picture is then trimmed as in
part 12, with small tabs that are later used to paste this picture to several (up to three)
other ones. Part 13 shows how the 20 triangles resulting from this process are mounted
in one horizontal strip that can later be converted to an actual icosahedron (part 14).
Each face of a dodecahedron is a pentagon, and each side of a cube is a square, but
the details of removing overlapping parts and trimming each picture in these cases are
similar to the triangular case. Figure 4.36 is an example of a panorama constructed on
a dodecahedron.
From around 1930 on, therefore, the standard photographic image on 35 mm film was
15.6 mm high by 20.8 mm wide, a proportion of roughly four by three. The same
proportion of height to width (the aspect ratio) is obtained on the screen when such
a frame is projected, and this shape of image (ratio 1:1.33) came to be called the
“Academy ratio.” But substantial variations are possible even on conventional 35 mm
film. Masks or caches can cut the height of each frame, and thus increase the aspect
ratio of the projected image: alternatively, special lenses can be used which “squeeze”
a wider image on to the film (through a procedure called anamorphosis , often used by
Renaissance painters) and “unsqueeze” it again when the film is projected. A French
optical scientist called Chretien invented the anamorphic lens and its application to
the cinema in the 1920s; Autant-Lara experimented with it in a film version of Jack
London's To Make A Fire , but the Hypergonar, as Chretien called his invention, failed
to catch on, and development work on it stopped.
—David Bellos, Jacques Tati (1999)

4.11 Telescopic Projection

Seen through a microscope, small objects look bigger than they are. The telescope,
however, does not enlarge objects; it brings them closer. Objects close to the telescope
are brought a little closer, while objects located far away are moved much closer. This
short section discusses the mathematics of the telescopic projections, but it should
be emphasized that this is not a projection from three dimensions to two dimensions,
but rather a three-dimensional transformation. (This is also true for the microscopic
projection.) Nevertheless, these topics are discussed here because of their nonlinearity.

The diameter of the moon is 3,476 kilometers (2,160 miles). When we see the moon
through a telescope, its diameter seems only a few centimeters or a few meters, much
smaller than the real diameter. This shows that the telescope does not increase the
size of the object being viewed. Instead, it decreases the apparent distance of the
object.
Figure 3.4 is a perspective projection of a long row of telephone poles. The poles,
which are the same height and are equally spaced, seem to get smaller and closer together
as they get farther from the viewer.

This is a common effect of linear perspective.