Graphics Reference
In-Depth Information
⎡
⎣
⎤
⎦
s
−
x
−
y
−
z
xs
−
z
y
L(
q
)
=
.
yz
s
−
x
z
−
yxs
R(
q
−
1
)
is also easy, but requires converting
q
2
in the original definition into
q
−
1
2
which is effected by reversing the signs of the vector components:
⎡
⎤
s
xyz
R
q
−
1
=
⎣
⎦
−
xs
−
zy
.
−
yz
s
−
x
−
z
−
yx s
So now we can write
qpq
−
1
=
R
q
−
1
L(
q
)
p
⎡
⎤
⎡
⎤
⎡
⎤
s
xyz
s
−
x
−
y
−
z
0
x
u
y
u
z
u
⎣
⎦
⎣
⎦
⎣
⎦
−
xs
−
z
y
xs
−
z
y
=
−
yz
s
−
x
yz
s
−
x
−
z
−
yx s
z
−
yx s
⎡
⎤
⎡
⎤
1
0
0
0
0
x
u
y
u
z
u
⎣
⎦
⎣
⎦
2
(y
2
+
z
2
)
01
−
2
(xy
−
sz)
2
(xz
+
sy)
=
.
2
(x
2
z
2
)
+
−
+
−
0
(xy
sz)
1
2
(yz
sx)
2
(x
2
y
2
)
0
(xz
−
sy)
2
(yz
+
sx)
1
−
+
If we remove the first row and column and treat
p
as a vector, rather than a quater-
nion, we have
⎡
⎤
⎡
⎤
2
(y
2
+
z
2
)
1
−
2
(xy
−
sz)
2
(xz
+
sy)
x
u
y
u
z
u
⎣
⎦
⎣
⎦
2
(x
2
z
2
)
=
2
(xy
+
sz)
1
−
+
2
(yz
−
sx)
2
(x
2
y
2
)
2
(xz
−
sy)
2
(yz
+
sx)
1
−
+
which is identical to (
11.4
)!
11.3.2 Geometric Verification
Let's illustrate the action of (
11.3
) by rotating the point
(
0
,
1
,
1
)
, 90° about the
y
-
axis, as shown in Fig.
11.6
. The quaternion must take the form
q
=
cos
(θ/
2
)
+
sin
(θ/
2
)
v
ˆ
which means that
θ
=
90° and
v
ˆ
=
j
, therefore,
sin 45°
j
.
q
=
cos 45°
+
Consequently,
√
2
2
√
2
2
s
=
,x
=
0
,y
=
,z
=
0
.
