Transformations - Transformations and Projections in Computer Graphics - page 39

We now select (Figure 1.22b) the plane x + y + z − 1=0andthepoint P =(1 , 1 , 1).

Equation (1.28) becomes

2(1+1+1 − 1) 1

3 (1 , 1 , 1) .

P ∗ =(1 , 1 , 1)

− (1 , 1 , 1) = −

1+1+1

Similarly, point P =(0 , 0 , 0) is reflected to

2(0+0+0 − 1)

(1 , 1 , 1) = 2 P ∗ =(0 , 0 , 0)

−

3 (1 , 1 , 1) . 1+1+1

The special case of a reflection about one of the coordinate planes is also obtained
from Equation (1.28). The equation of the xy plane, for example, is z =0,where

Equation (1.28) yields

2(0+0+ z +0)
0 2 +0 2 +1 2

P ∗ =( x, y, z ) − (0 , 0 , 1) = ( x, y, − z ) .

1.4.2 Rotation

Rotation in three dimensions is di cult to visualize and is often confusing. One ap-
proach is to write three rotation matrices that rotate about the three coordinate axes:

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

cos θ − sin θ 00 cos θ 0 − sin θ 0 10 0 0

sin θ cos θ 00 0 1 0 0 0 os θ − sin θ 0

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

, , .

0 0 1 0 sin θ 0 θ 0 0 in θ cos θ 0

0 0 0 1 0 0 0 1 00 0 1

(1.29)
Let's look at the first of these matrices. Its third row and third column are (0 , 0 , 1 , 0),
which is why multiplying a point ( x, y, z, 1) by this matrix leaves its z coordinate un-
changed. The sines and cosines in the first two rows and two columns mix up the x and
y coordinates in a way similar to a two-dimensional rotation [Equation (1.4)]. Thus,
this transformation matrix causes a rotation about the z axis. The two other matrices
rotate about the y and x axes.

(0,1,0)

y y (0,1,0)

y

(1,0,0)

x x x

z

z z

(a) (b) (c)

Figure 1.23: Rotating About the Coordinate Axes.