Linear Algebra, 2020a

Topic: Computer Graphics 393
In this topic we will show how to use these ideas to perform some effects
from computer graphics. For that we will take the prior picture and redraw it
without the sphere, with a movie projector at the origin, and with plane z = 1
looking like a movie theater screen.
( x
p =
y)
projector, at (0, 0, 0) z = 1
This associates vectors in three-space on the grey line with p in the screen plane.
⎛
⎜
a
⎞
( ) ( )
⎟ x a/c
˜p = ⎝b⎠ ↦→ p = =
y b/c
c
We can adapt the things we have already seen about matrices to perform the
transformations. Rotation is an example. This matrix rotates in the plane z = 1
about the origin by the angle θ.
⎛
⎜
cos θ − sin θ 0
⎞ ⎛ ⎞ ⎛
x
⎟ ⎜ ⎟ ⎜
cosθ · x − sin θ · y
⎞
⎟
⎝sin θ cos θ 0⎠
⎝y⎠ = ⎝sin θ · x + cos θ · y⎠
0 0 1 1
1
Notice that it works on any homogeneous coordinate vector; if we apply the
matrix ⎛
⎜
cos θ − sin θ 0
⎞ ⎛
⎟ ⎜
a
⎞ ⎛
⎟ ⎜
cosθ · a − sin θ · b
⎞
⎟
⎝sin θ cos θ 0⎠
⎝b⎠ = ⎝sin θ · a + cos θ · b⎠
0 0 1 c
c
and then move to the z = 1 plane
⎛
⎜
cosθ · a − sin θ · b
⎞ ⎛
⎟ ⎜
(cosθ · a − sin θ · b)/c
⎞
⎟
⎝sin θ · a + cos θ · b⎠ ↦→ ⎝(sin θ · a + cos θ · b)/c⎠
c
1