Perspective Projection - Transformations and Projections in Computer Graphics - page 85

x or y

P

P *

y

y *

z

k z

Figure 3.22: Perspective by Similar Triangles.

Step 3 . Equation(3.1)canbeemployedtoshowthattheprojectionruleofStep1
results in a projection that satisfies the three conditions above and can therefore be
called perspective. Condition 1 says that a distant object should appear small. The
object can become distant in three ways:
1. increasing the z coordinates of its points;
2. increasing the x or y coordinates;
3. increasing the value of k .
For large values of z , Equation (3.1) yields small values for x ∗ and y ∗ . Specifically

x ∗ = 0 y ∗ =0 . lim

z→∞

and lim

z→∞

For large values of x or y , imagine two points, P 1 =( x 1 ,y 1 ,z 1 )and P 2 =( x 2 ,y 1 ,z 1 ), on
the object that differ only in their x coordinates. They are projected to the two points
P 1 =( x 1 ,y 1 )and P 2 =( x 2 ,y 1 ), which have identical y coordinates, and the ratio of
their x coordinates is

x 1
x 2

x 1
( z 1 /k )+1 ( z 1 /k )+1 = x 1

x 2

=

x 2 .

(3.2)

Thus, when both x 1 and x 2 grow, the ratio x 1 /x 2 approaches 1, which implies that
the two projected points P 1 and P 2 get closer. Since P 1 and P 2 are any points with
the same y and z coordinates, this implies that all the points with the same y and z
coordinates produce projections that are very close. The object seems to have shrunk
in the x dimension (Figure 3.23a).
Thecasewhere k increases (i.e., the viewer moves away from the projection plane)
is different. Figure 3.23b shows how the projection of the object becomes bigger and
bigger in this case until, at the limit, when the viewer is at infinity, the projection
reaches the actual size of the object. The perspective projection is reduced in this limit
to a parallel projection. However, even though the projection itself gets bigger, the
viewer sees a small projected object because the projection plane and everything on it
look small to a distant viewer.
Condition 2 demands that a distant object feature less perspective. We already
know that an object can become distant in three ways each of which is individually
treated here.