Rotation Transforms in Space - Rotation Transforms for Computer Graphics - page 120

We can create a composite, moving-point, fixed-frame rotation by placing
R α,x , R β,y and R γ,z in any sequence. As an example, let's choose the sequence
R γ,z R β,y R α,x

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

c γ − s γ 0 c β 0 s β

01 0

10 0

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ . (9.1)

R γ,z R β,y R α,x = s γ c γ 0 0 c α − s α

0 0 1 − s β 0 c β 0 s α c α

Multiplying the three matrices in ( 9.1 ) together we obtain
⎡ ⎤

c γ c β c γ s β s α − s γ c α c γ s β c α + s γ s α

⎣ ⎦

s γ c β s γ s β s α + c γ c α s γ s β c α − c γ s α (9.2)

− s β c β s α c β c α

or using the more familiar notation:
⎡ ⎤

cos γ cos β cos γ sin β sin α − sin γ cos α cos γ sin β cos α + sin γ sin α

⎣ ⎦ .

sin γ cos β sin γ sin β sin α + cos γ cos α sin γ sin β cos α − cos γ sin α

− sin β cos β sin α cos β cos α

Let's evaluate ( 9.2 ) by making α = β = γ = 90°:

⎡ ⎤

001
010
−

⎣ ⎦ .

(9.3)

100

The matrix ( 9.3 ) is equivalent to rotating a point 90° about the fixed x -axis, followed
by a rotation of 90° about the fixed y -axis, followed by a rotation of 90° about the
fixed z -axis. This rotation sequence is illustrated in Fig. 9.4 (a)-(d).
Figure 9.4 (a) shows the starting position of the cube; (b) shows its orientation
after a 90° rotation about the x -axis; (c) shows its orientation after a further rotation
of 90° about the y -axis; and (d) the cube's resting position after a rotation of 90°
about the z -axis.
From Fig. 9.4 (d) we see that the cube's coordinates are as shown in Table 9.2 .We
can confirm that these coordinates are correct by multiplying the cube's original co-
ordinates shown in Table 9.1 by the matrix ( 9.3 ). Although it is not mathematically
correct, we will show the matrix multiplying an array of coordinates as follows
⎡

⎤ ⎡ ⎤

001
010

00001111
00110011
01010101

⎣ ⎦ ⎣ ⎦

− 100

⎡ ⎤

01010101
00110011
0000

⎣ ⎦

=

− 1 − 1 − 1 − 1

which agree with the coordinates in Table 9.2 .
Naturally, any three angles can be chosen to rotate a point about the fixed axes,
but it does become difficult to visualise without an interactive cgi system.
Note that the determinant of ( 9.3 ) is 1, which is as expected.