Graphics Reference
In-Depth Information
using (
9.19
)wehave
v
=
(
0
0
)
=[
T
−
0
)(
−
1
−
1
)
0
−
0
−
20
]
which is the
y
-axis.
⎡
⎤
0
10
100
001
−
⎣
⎦
R
90
°
,z
=
using (
9.19
)wehave
v
=
(
0
1
)
=[
T
−
0
)
0
−
0
)(
−
1
−
00
−
2
]
which is the
z
-axis.
However, if we attempt to extract the axis of rotation from
⎡
⎤
001
0
⎣
⎦
R
90
°
,x
R
90
°
,y
R
90
°
,z
=
10
100
−
A
T
we have a problem, because
q
1
=
q
2
=
q
3
=
0. This is because
A
=
and the
A
T
.
So let's consider another approach based upon the fact that a rotation matrix
always has a real eigenvalue
λ
technique relies upon
A
=
=
1, which permits us to write
Av
=
λ
v
Av
=
λ
Iv
=
Iv
(
A
−
I
)
v
=
0
therefore,
⎡
⎤
⎡
⎤
⎡
⎤
(a
11
−
1
)
a
12
a
13
v
1
v
2
v
3
0
0
0
⎣
⎦
⎣
⎦
=
⎣
⎦
.
a
21
(a
22
−
1
)
a
23
(9.20)
a
31
a
32
(a
33
−
1
)
Expanding (
9.20
)wehave
(a
11
−
1
)v
1
+
a
12
v
2
+
a
13
v
3
=
0
a
21
v
1
+
(a
22
−
1
)v
2
+
a
23
v
3
=
0
a
31
v
1
+
a
32
v
2
+
(a
33
−
1
)v
3
=
0
.
Once more, there exists a trivial solution where
v
1
=
0, but to discover
something more useful we can relax any one of the
v
terms which gives us three
equations in two unknowns. Let's make
v
1
=
v
2
=
v
3
=
0:
a
12
v
2
+
a
13
v
3
=−
(a
11
−
1
)
(9.21)
(a
22
−
1
) v
2
+
a
23
v
3
=−
a
21
(9.22)
a
31
.
(9.23)
We are now faced with choosing a pair of equations to isolate
v
2
and
v
3
.Infact,we
have to consider all three pairings because it is possible that a future rotation matrix
a
32
v
2
+
(a
33
−
1
) v
3
=−
