More on Matrices - 3D Math Primer for Graphics and Game Development - page 188

to allow in only light that passes through a single point. However, in the
mathematical universe inside a computer, it works just fine.
As expected, moving the plane of projection in front of the center of
projection removes the annoying minus signs:

Projecting a point onto
the plane z = d

′

x ′ y ′ z ′

p = = dx/z dy/z d . (6.12)

6.5.2 Perspective Projection Matrices
Because the conversion from 4D to 3D space implies a division, we can
encode a perspective projection in a 4 × 4 matrix. The basic idea is to
come up with an equation for p

with a common denominator for x, y, and
z, and then set up a 4 × 4 matrix that will set w equal to this denominator.
We assume that the original points have w = 1.
First, we manipulate Equation (6.12) to have a common denominator:

′

x y z

′

p = dx/z dy/z d = dx/z dy/z dz/z = .

z/d

To divide by this denominator, we put the denominator into w, so the 4D
point will be of the form

x y z z/d .

So we need a 4 × 4 matrix that multiplies a homogeneous vector [x,y,z,1]
to produce [x,y,z,z/d]. The matrix that does this is

2
4 1

3

0 0 0

Projecting onto the
plane z = d using a 4 × 4
matrix

5

0 1 0 0

x y z 1 = x y z z/d .

0 0 1 1/d

0 0 0 0

Thus, we have derived a 4 × 4 projection matrix.
There are several important points to be made here:

• Multiplication by this matrix doesn't actually perform the perspective

transform, it just computes the proper denominator into w. Remem-
ber that the perspective division actually occurs when we convert
from 4D to 3D by dividing by w.

• There are many variations. For example, we can place the plane of

projection at z = 0, and the center of projection at [0,0,−d]. This
results in a slightly different equation.

• This seems overly complicated. It seems like it would be simpler

to just divide by z, rather than bothering with matrices. So why is