Nonlinear Projections - Transformations and Projections in Computer Graphics - page 140

4
Nonlinear Projections

In addition to the parallel and perspective projections, other projections may be de-
veloped that are useful for special applications or that create ornamental or artistic
effects. Such projections are termed nonlinear because they cannot be expressed by
linear transformations such as x ∗ = ax + cy + m and y ∗ = bx + dy + n . It seems that
the number of possible nonlinear projections is vast and is limited only by the imagi-
nations of those who try to develop new ones. This chapter discusses some of the more
common nonlinear projections, including the false perspective, the fisheye projection,
several 360 ◦ panoramic projections, the telescopic and microscopic projections, sphere
projections, and circle inversion (a special projection from two dimensions to two di-
mensions). These projections create aesthetically pleasing (and sometimes confusing)
effects and are mathematically simple and easy to derive. However, since they are
nonlinear, they generally cannot be represented by means of transformation matrices.
[Recall that multiplying a point ( x, y, z ) by a matrix results in a linear expression such
as ax + by + cz , but never in nonlinear constructs such as ax 2 .]

Back in the corridor of the building, posters of computer-generated fractal images
depicting the “arithmetic limits of iterative nonlinear equations” line the walls.
—Douglas Rushkoff, Cyberia: Life in the Trenches of Hyperspace (1994)

4.1 False Perspective

Equation (3.1) is the main expression for the linear perspective projection; it is dupli-
cated here:

x
1+( z/k ) ,

y
1+( z/k ) .

x ∗ = ∗ = (3 . 1)

It shows that the (two-dimensional) coordinates of the projected point P ∗ are obtained
by dividing by the z coordinate (the depth) of the original point P . False perspective (or