Game Development Reference
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−1 , is defined as the conjugate
The inverse of a quaternion, denoted q
of a quaternion divided by its magnitude:
q .
q
Quaternion inverse
−1 =
q
(8.6)
The quaternion inverse has an interesting correspondence with the mul-
tiplicative inverse for real numbers (scalars). For real numbers, the mul-
tiplicative inverse a −1 is 1/a. In other words, a(a −1 ) = a −1 a = 1. The
same applies to quaternions. When we multiply a quaternion q by its in-
verse q
−1 , we get the identity quaternion [1, 0 ]. (We discuss quaternion
multiplication in Section 8.5.7.)
Equation (8.6) is the o cial definition of quaternion inverse. However,
if you are interested only in quaternions that represent pure rotations, like
we are in this topic, then all the quaternions are unit quaternions and so
the conjugate and inverse are equivalent.
The conjugate (inverse) is interesting because q and q
represent op-
posite angular displacements. It is easy to see why this is the case. By
negating v , we are negating the axis of rotation n . This doesn't change the
axis in the physical sense, since n and − n are parallel. However, it does
flip the direction that we consider to be positive rotation. Thus, q rotates
about an axis by an amount θ, and q
rotates in the opposite direction by
the same amount.
For our purposes, an alternative definition of quaternion conjugate could
have been to negate w, leaving v (and thus n ) unchanged. This would
negate the amount of rotation θ, rather than reversing what is consid-
ered positive rotation by flipping the axis of rotation. This would have
been equivalent to the definition given in Equation (8.5) (for our geometric
purposes, at least) and provided for a slightly more intuitive geometric in-
terpretation. However, the term conjugate has a special significance in the
context of complex numbers, so let's stick with the original definition.
8.5.7
Quaternion Multiplication
Quaternions can be multiplied. The result is similar to the cross product
for vectors, in that it yields another quaternion (not a scalar), and it is
not commutative. However, the notation is different: we denote quater-
nion multiplication simply by placing the two operands side-by-side. The
formula for quaternion multiplication can be easily derived based upon the
definition of quaternions as complex numbers (see Exercise 6), but we state
it here without development, using both quaternion notations:
 
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