Game Development Reference
In-Depth Information
8.5.5 Quaternion Magnitude
We can compute the magnitude of a quaternion, just as we can for vectors
and complex numbers. The notation and formula shown in Equation (8.4)
are similar to those used for vectors:
x y z
q
=
w
=
w
2
+ x
2
+ y
2
+ z
2
Quaternion magnitude
(8.4)
w
2
+
v
2
.
=
w
v
=
Let's see what this means geometrically for a rotation quaternion:
w
2
+
v
2
Rotation quaternions
have unit magnitude
q
=
w
v
=
cos
2
(θ/2) + (sin(θ/2)
n
)
2
=
(substituting using θ and
n
)
cos
2
(θ/2) + sin
2
(θ/2)
n
2
=
cos
2
(θ/2) + sin
2
(θ/2)(1)
=
(
n
is a unit vector)
√
(sin
2
x + cos
2
x = 1)
=
1
= 1.
This is an important observation.
For our purposes of using quaternions to represent orientation, all quater-
nions are so-called unit quaternions, which have a magnitude equal to unity.
For information concerning nonnormalized quaternions, see the techni-
cal report by Dam et al. [11].
8.5.6 Quaternion Conjugate and Inverse
The conjugate of a quaternion, denoted
q
∗
, is obtained by negating the
vector portion of the quaternion:
∗
∗
q
=
w
v
=
w
−
v
(8.5)
Quaternion conjugate
−x
∗
=
w
x y z
=
w
−y
−z
.
The term “conjugate” is inherited from the interpretation of a quaternion
as a complex number. We look at this interpretation in more detail in
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