Transformations - Transformations and Projections in Computer Graphics - page 45

where I is the identity matrix. The transpose M T therefore equals the inverse, M − 1 ,
of M , which shows that a rotation matrix M is orthogonal.
Example . Consider a rotation about the z axis. The rotation axis is u =(0 , 0 , 1),
resulting in

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

000
000
001

0 10

100
000

− cos θ − sin θ 0

⎝ ⎠ and U = ⎝ ⎠ , and hence M = ⎝ ⎠ ,

uu T = sin θ cos θ 0

0 0 1

which is the familiar rotation matrix about the z axis. It is identical to the z -rotation
matrix of Equation (1.29), so we conclude that it rotates counterclockwise when viewed
from the direction of positive z .
The general rotation matrix of Equation (1.32) can also be constructed as the
product of five simple rotations about various coordinate axes. Given a unit vector
u =( u x ,u y ,u z ), consider the following rotations.
1. Rotate u about the z axis into the xz plane, so its y coordinate becomes zero.
This is done by a rotation matrix of the form

⎡ ⎤

cos ψ − sin ψ 0

⎣ ⎦ ,

A = sin ψ cos ψ 0

0 0 1

and the angle ψ of rotation can be computed from the requirement that the y component
of vector v = uA b e zero.

This com ponent is − u x sin ψ + u y cos ψ , which implies

cos ψ = u x / u x + u y and sin ψ = u y / u x + u y . Notice that rotating u does not affect
its magnitude, so v is also a unit vector. In addition, since the rotation is about the z
axis, the z component of u does not change, so v z = u z .
2. Rotate vector v about the y axis until it coincides with the z axis. This is
accomplished by the matrix

⎡ ⎤

cos φ 0 in φ

⎣ ⎦ .

B = 0 1 0

− sin φ 0 os φ

The angle φ of rotation i s comp uted from the dot product cos φ = v · (0 , 0 , 1) = v z = u z ,

implying that sin φ = 1 − u z .Since v is a unit vector, it is rotated by B to vector

(0 , 0 , 1).
3. Rotate (0 , 0 , 1) about the z axis through an angle θ . Thisisdonebymatrix

⎡ ⎤

cos θ − sin θ 0

⎣ ⎦ .

C = sin θ cos θ 0

0 0 1

This is a trivial rotation that does not change (0 , 0 , 1).
4. Rotate the result of step 3 by B − 1 (which equals B T ).