More on Matrices - 3D Math Primer for Graphics and Game Development - page 189

homogeneous space interesting? First, 4×4 matrices provide a way to
express projection as a transformation that can be concatenated with
other transformations. Second, projection onto nonaxially aligned
planes is possible. Basically, we don't need homogeneous coordinates,
but 4×4 matrices provide a compact way to represent and manipulate
projection transformations.

• The projection matrix in a real graphics geometry pipeline (perhaps

more accurately known as the “clip matrix”) does more than just
copy z into w. It differs from the one we derived in two important
respects:

◦ Most graphics systems apply a normalizing scale factor such that

w = 1 at the far clip plane. This ensures that the values used
for depth buffering are distributed appropriately for the scene
being rendered, to maximize precision of depth buffering.

◦ The projection matrix in most graphics systems also scales the

x and y values according to the field of view of the camera.

We'll get into these details in Section 10.3.2, when we show what
a projection matrix looks like in practice, using both DirectX and
OpenGL as examples.

6.6 Exercises

(Answers on page 765.)
1. Compute the determinant of the following matrix:

3 −2
1

4

2. Compute the determinant, adjoint, and inverse of the following matrix:
2
4

3
5

3 −2 0

1 4 0

0 0 2

3. Is the following matrix orthogonal?
2
4 −0.1495 −0.1986 −0.9685

3
5

−0.8256 0.5640 0.0117

−0.5439 −0.8015 0.2484

4. Invert the matrix from the previous exercise.