Graphics Reference
In-Depth Information
ˆ
Any quaternion
q
can be normalised to a unit quaternion
q
by dividing by its mag-
nitude:
q
q
ˆ
=
.
|
q
|
5.8 The Quaternion Conjugate
We have already discovered that the complex conjugate of a complex number
z
=
a
+
bi
is given by
z
∗
=
a
−
bi
and is very useful in computing the inverse of
z
.The
quaternion conjugate
plays a
similar role in computing the inverse of a quaternion. Therefore, given
q
=
s
+
v
=
s
+
x
i
+
y
j
+
z
k
its conjugate is defined as
q
∗
=
s
−
v
=
−
−
−
s
x
i
y
j
z
k
.
If we compute the product
qq
∗
we obtain
qq
∗
=
(s
+
v
)(s
−
v
)
s
2
s
2
=
+
v
·
v
+
s
v
−
s
v
+
v
×
(
−
v
)
=
+
v
·
v
s
2
x
2
y
2
z
2
=
+
+
+
which is a scalar and implies that
qq
∗
=|
2
q
|
or
qq
∗
.
|
q
|=
Similarly, we can show that
qq
∗
=
q
∗
q
.
Now let's show that
(
q
1
q
2
)
∗
=
q
2
q
1
. We start with quaternions
q
1
and
q
2
:
q
1
=
s
1
+
v
1
q
2
=
s
2
+
v
2
q
1
q
2
=
(s
1
+
v
1
)(s
2
+
v
2
)
=
s
1
s
2
−
v
1
·
v
2
+
s
1
v
2
+
s
2
v
1
+
v
1
×
v
2
(
q
1
q
2
)
∗
=
s
1
s
2
−
v
1
·
v
2
−
s
1
v
2
−
s
2
v
1
−
v
1
×
v
2
.
(5.3)
