Graphics Reference
In-Depth Information
7.6 The complex conjugate of a quaternion
Now let's define the quaternion
q. The complex conjugate of an ordinary complex number is
¯
defined as
z
¯
=
a
−
b
i
where
z
=
a
+
b
i
Similarly, given a quaternion
q
=
s
+
v
its complex conjugate is defined as
q
¯
=
s
−
v
which we innocently used in demonstrating the rotational properties of quaternions.
In Section 7.3.3 we proved that for two complex numbers
z
1
z
2
=¯
z
1
¯
z
2
Could this relationship hold for two quaternions? Let's see.
Given two quaternions
q
1
=
s
1
+
v
1
and
q
2
=
s
2
+
v
2
then
q
1
q
2
=
s
1
+
v
1
s
2
+
v
2
which must be expanded using the rules of the quaternion product:
q
1
q
2
=
s
1
s
2
−
v
1
·
v
2
+
s
1
v
2
+
s
2
v
1
+
v
1
×
v
2
Therefore,
q
1
q
2
=
s
1
s
2
−
v
1
·
v
2
−
s
1
v
2
−
s
2
v
1
−
v
1
×
v
2
(7.17)
Now let's expand q
1
q
2
:
q
1
q
2
=
s
1
−
v
1
s
2
−
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
(7.18)
