Graphics Reference
In-Depth Information
Fig. 11.9
The frame is rotated 180° about the vector
[
i
+
k
]
the matrix representing the triple
q
−
1
pq
,(
11.5
) we can show how quaternions can
be added to these techniques.
The triple
qpq
−
1
is used for rotating points about the vector associated with
the quaternion
q
, whereas the triple
q
−
1
pq
is used for rotating points about the
same vector, but in the opposite direction. But we have already reasoned that this
reverse rotation is equivalent to a change of frame of reference. To demonstrate this,
consider the problem of rotating the frame of reference 180° about
i
+
k
as shown
in Fig.
11.9
(a) and (b). The unit quaternion for such a rotation is
sin 90°
1
√
2
k
1
q
=
cos 90°
+
√
2
i
+
√
2
2
√
2
2
=
0
+
i
+
k
.
Consequently,
√
2
2
√
2
2
s
=
0
,x
=
,y
=
0
,z
=
.
Substituting these values in (
11.5
) we obtain
⎡
⎤
⎡
⎤
001
0
x
u
y
u
z
u
q
−
1
pq
⎣
⎦
⎣
⎦
=
10
100
−
which if used to process the coordinates of our unit cube produces
⎡
⎤
⎡
⎤
001
0
00001111
00110011
01010101
⎣
⎦
⎣
⎦
10
100
−
⎡
⎤
01010101
00
⎣
⎦
.
=
1
00001111
−
1
−
100
−
1
−
This scenario is shown in Fig.
11.9
(a) and (b).
