Nonlinear Projections - Transformations and Projections in Computer Graphics - page 179

Figure 4.33: Grids and Art for 1, 2, and 3 Vanishing Points (courtesy of Dick Termes).
introduce distortions in any artwork based on them. The former grid shows straight
lines bending and converging to four points. The vanishing points on the left and right
sides are familiar. They result in the familiar two-point perspective. The extra two
vanishing points, at the top and bottom of the grid, force all the vertical lines to bend
and introduce distortions in this way. The result is an image (see example to the right
of the grid) that becomes more distorted as the eye moves up or down away from the
center of the image. This type of distortion has its own artistic value but it is not
immediately clear to which of the projections discussed in this chapter it corresponds.
A closer look at the four-point grid of Figure 4.34, however, shows its resemblance
to Figure 4.23b, which corresponds to the cylindrical panoramic projection. Thus, a
complete 360 ◦ cylindrical projection, such as the one depicted by Figure 4.25c, can be
obtained by placing four four-point grids side by side. This type of grid is referred to
by Dick Termes as a continuous four-point perspective.
Initially, the five-point grid of Figure 4.34 looks unfamiliar and strange. It is not
trivial to guess the type of distortion that results from bending lines in five different
directions, toward the four extreme points on the periphery as well as toward the center.
However, a glance at Figure 4.10 should convince the reader that the effect of five-
point perspective is similar (perhaps even identical) to the angular fisheye projection
(page 151) as well as the spherical panoramic projection of Section 4.6. All the horizontal
and vertical lines, except those passing through the middle of the figure, are curved.
This drawing shows only half a sphere (180 ◦ vertically and horizontally), but it points
the way toward depicting a complete sphere on a flat surface. Simply place two five-point