Rotation Transforms in Space - Rotation Transforms for Computer Graphics - page 122

Fig. 9.5 Four views of the unit cube using the rotation sequence R α,x R β,y R γ,z

Multiplying the three matrices in ( 9.4 ) together we obtain
⎡ ⎤

c β c γ − c β s γ s β

⎣ ⎦

s α s β c γ + c α s γ − s α s β s γ + c α c γ − s α c β (9.5)

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

or using the more familiar notation:
⎡ ⎤

cos β cos γ − cos β sin γ sin β

⎣ ⎦ .

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

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

Comparing ( 9.3 ) and ( 9.5 ) it can be seen that they are completely different.
Let's evaluate ( 9.5 ) by making α = β = γ = 90°:

⎡ ⎤

001
0

⎣ ⎦ .

10
100

−

(9.6)

The matrix ( 9.6 ) is equivalent to rotating a point 90° about the fixed z -axis, fol-
lowed by a rotation of 90° about the fixed y -axis, followed by a rotation of 90° about
the fixed x -axis. This rotation sequence is illustrated in Fig. 9.5 (a)-(d).
From Fig. 9.5 (d) we see that the cube's coordinates are as shown in Table 9.3 .
We can confirm that these coordinates are correct by multiplying the cube's original