Perspective Projection - Transformations and Projections in Computer Graphics - page 84

y

z

z

y

x

x

(a) (b)

Figure 3.20: (a) Standard and (b) Nonstandard Positions.

object and the viewer looking at the object from different directions, examining various
projections of it on the plane. It is pointless to move the viewer around the object while
the projection plane stays at the same location (Figure 3.21b) because such a viewer
will generally not even be looking at the plane. Thus, the projection plane must move
with the viewer and must remain perpendicular to the line of sight of the viewer and at
adistanceof k units from him (although k maybevariedbytheuser).

k k

(a) (b)

Figure 3.21: Moving the Viewer and the Projection Plane.

Step 2 . The two similar triangles of Figure 3.22 yield the simple relations

x ∗
k

y ∗
k

x
z + k

y
z + k ,

= and =

from which we obtain

x
( z/k )+1

y
( z/k )+1 .

x ∗ = y ∗ =

and (3.1)

(Some authors assign the x coordinate a negative sign. This is a result of the difference
between left-handed and right-handed coordinate systems as discussed in Section 1.3.
See also Exercise 3.27.) The +1 in the denominator of Equation (3.1) is important. It
guarantees that the denominator will never be zero. The denominator can be zero only
if z/k =

− 1, but k is positive and z is nonnegative.