Graphics Reference
In-Depth Information
Fig. 9.4
Four views of the unit cube before and during the three rotations
Ta b l e 9 . 2
Vertex coordinates
for the cube in Fig.
9.4
(d)
vertex
01234
5
6
7
x
0
1
0
1
0
1
0
1
y
0
0
1
1
0
0
1
1
z
0000
−
1
−
1
−
1
−
1
An observation we made with 2D rotations is that they are additive: i.e.
R
α
fol-
lowed by
R
β
is equivalent to
R
α
+
β
. But something equally important is that rota-
tions in 2D commute:
R
β
+
α
whereas, in general, 3D rotations are non-commutative. This is seen by considering
a composite rotation formed by a rotation
α
about the
x
-axis
R
α,x
,followedbya
rotation
β
about the
z
-axis
R
β,z
, and
R
α,x
R
β,z
=
R
α
R
β
=
R
β
R
α
=
R
α
+
β
=
R
β,z
R
α,x
.
As
an
illustration,
let's
reverse
the
composite
rotation
computed
above
to
R
α,x
R
β,y
R
γ,z
:
⎡
⎤
⎡
⎤
⎡
⎤
10
0
c
β
0
s
β
01 0
−
s
β
c
γ
−
s
γ
0
⎣
⎦
⎣
⎦
⎣
⎦
.
(9.4)
R
α,x
R
β,y
R
γ,z
=
0
c
α
−
s
α
s
γ
c
γ
0
0
s
α
c
α
0
c
β
0
0
1
