Graphics Reference
In-Depth Information
q
1
q
2
=
cos 15°
sin 15°
j
cos 30°
sin 30°
j
+
+
=
cos 15° cos 30°
−
sin 15° sin 30°
+
cos 15° sin 30°
j
+
cos 30° sin 15°
j
√
2
2
+
√
2
2
=
j
which is a quaternion that rotates 90° about the
y
-axis. Using the matrix (
11.4
)we
have
⎡
⎤
001
010
⎣
⎦
−
100
which rotates points about the
y
-axis by 90°.
11.5 Eigenvalue and Eigenvector
Although there is no doubt that (
11.3
) is a rotation matrix, we can secure further
evidence by calculating its eigenvalue and eigenvector. The eigenvalue should be
θ
,
where
Tr
qpq
−
1
=
1
+
2 cos
θ.
The trace of (
11.3
)is
Tr
qpq
−
1
=
2
s
2
+
x
2
−
2
s
2
+
y
2
−
2
s
2
+
z
2
−
1
+
1
+
1
2
s
2
z
2
−
4
s
2
x
2
y
2
=
+
+
+
+
3
4
s
2
=
−
1
4 cos
2
(θ/
2
)
=
−
1
4sin
2
(θ/
2
)
=
+
−
4 cos
θ
1
=
4 cos
θ
+
2
−
2 cos
θ
−
1
=
1
+
2 cos
θ
and
2
Tr
qpq
−
1
−
1
.
1
cos
θ
=
To compute the eigenvector of (
11.3
) we use the three equations derived in Chap. 9:
v
1
=
(a
22
−
1
)(a
33
−
1
)
−
a
23
a
32
v
2
=
(a
33
−
1
)(a
11
−
1
)
−
a
31
a
13
v
3
=
(a
11
−
1
)(a
22
−
1
)
−
a
12
a
21
.