Graphics Reference
In-Depth Information
is defined as
s
2
q
=
+
x
2
+
y
2
+
z
2
For example, given
=
+
−
+
q
1
2
i
3
j
4
k
then
1
2
√
30
3
2
q
=
+
2
2
+
−
+
4
2
=
If we represent the quaternion as
q
=
s
+
v
then
s
2
2
q
=
+
v
and from Eq. (7.20), we have
2
q
¯
q
=¯
qq
=
q
(7.21)
Developing Eq. (7.21), we can state that
2
q
=
q
q
¯
But as the single quaternion q can be a product of two quaternions
2
q
1
q
2
=
q
1
q
2
q
1
q
2
Applying Eq. (7.19) gives
2
q
1
q
2
=
q
1
q
2
¯
q
2
¯
q
1
Applying Eq. (7.21) gives
2
2
q
1
q
2
=
q
1
q
2
¯
q
1
Rearranging gives
2
2
q
1
q
2
=
q
1
¯
q
1
q
2
Applying Eq. (7.21) gives
2
2
2
q
1
q
2
=
q
1
q
2
Therefore,
q
1
q
2
=
q
1
q
2
which informs us that the norm of a quaternion product equals the product of the individual
norms.
As with unit vectors, there are unit quaternions, where
q
=
1, which are used extensively
when rotating vectors.
