Nonlinear Projections - Transformations and Projections in Computer Graphics - page 181

Figure 4.35: The Five Platonic Solids.
A polyhedron whose faces are congruent convex regular polygons is known as a Pla-
tonic solid. These figures were known in antiquity, and Euclid has already proved
that there are only five of them, the tetrahedron (a pyramid of four triangles), the
cube, the octahedron (eight faces, each a triangle), the dodecahedron (12 faces, each a
pentagon), and the icosahedron (20 triangles). Many properties and pictures of these
solids can be found in [Steinhaus 83].
One of the most original works of art depicting Platonic solids is the wood engraving
Stars by M. C. Escher. It takes a while to disentangle the many details in this picture
and locate the intersecting octahedra, tetrahedra, cubes, and other figures. The only
items that stand out immediately are the chameleons, placed by the artist inside the
polyhedra to attract nonmathematically-oriented viewers and capture their attention.
The principle of projection is always the same. We imagine an observer located
somewhere inside the surface, at the center or at some other preferred point, looking at
the three-dimensional scene outside and painting it on the surface. The surface is then
opened or unrolled to become a flat panoramic projection. In practice, only the cylinder
and the cube are commonly used for panoramic projections. It is rare to find a pyramidal
or a conic panoramic projection because opening and flattening such surfaces results in
a two-dimensional picture that looks foreign and unfamiliar and is often di cult to
visualize, perceive, and enjoy, even though it does not create any distortions.
Figure 4.36 (courtesy of Dick Termes) is a typical example. It shows a panorama
of the interior of St. Peter's Basilica in Rome projected on a dodecahedron. It is
immediately obvious that in spite of the high precision of the drawing and the many
details that are easy to observe, it is di cult, perhaps even impossible, to place the 12
individual pentagons of the projection in the viewer's mind and grasp them as a single
coherent work of art. Such a projection is best viewed after it is cut out, folded, and
glued together to actually form a dodecahedron (notice the matching tabs designed to
help in this process). The details of this process and how such pictures are taken are
described on page 191.
Conic Panoramic Projection
Given a cone of height H and radius R , we imagine an observer located at the center
of the base of the cone. Such an observer sees the hemisphere of space above him and
projects it on the cone, which is later cut and laid flat. It is also possible to place the
observer at the center of the cone, where he can see the entire 360 ◦ of space around him,
but this results in even more distortion because part of the lower hemisphere is seen by
the observer through the lower sides of the cone, while the rest of this hemisphere is
seen through the flat bottom.