Graphics Reference
In-Depth Information
Using the trace operation, we can write
Tr
(
R
90
°
,x
R
90
°
,y
R
90
°
,z
)
=−
1
therefore,
β
=
arccos
((
−
1
−
1
)/
2
)
=
180°
.
As promised, let's explore another way of identifying the fixed axis of rotation,
which is an eigenvector. Consider the following argument where
A
is a simple rota-
tion transform:
If
v
is a fixed axis of rotation and
A
a rotation transform, then
v
suffers no rota-
tion:
Av
=
v
(9.9)
similarly,
A
T
v
=
v
.
(9.10)
Subtracting (
9.10
) from (
9.9
), we have
A
T
v
Av
−
=
0
(9.11)
A
A
T
v
−
=
0
(9.12)
where
0
is a null vector.
In Chap. 4 we defined an antisymmetric matrix
Q
as
2
A
A
T
1
Q
=
−
(9.13)
therefore,
A
A
T
=
−
2
Q
.
(9.14)
Substituting (
9.14
)in(
9.12
)wehave
2
Qv
=
0
Qv
=
0
which permits us to write
⎡
⎤
⎡
⎤
⎡
⎤
0
q
3
−
q
2
v
1
v
2
v
3
0
0
0
⎣
⎦
⎣
⎦
=
⎣
⎦
−
q
3
0
q
1
(9.15)
q
2
−
q
1
0
where
q
1
=
a
32
q
2
=
a
31
−
a
13
q
3
=
a
23
−
a
12
−
a
21
.
Expanding (
9.15
)wehave