
320 Chapter 5 Controller-Based Animation
A slight extension of this construction shows that rotations and translations can
be baked into the child’s keyframes. The translational component is the same as
was shown previously. If the orientation keyframes are the quaternions q
0
and q
1
,
then q(t) is defined by Equation (5.3). If α
1
is the quaternion corresponding to the
parent’s world transformation, then the matrix component after application of the
world transformation is
α
1
q(t) = α
1
q
i
sin((1 − t)θ
i
) + q
i+1
sin(tθ
i
)
sin θ
i
=
(αq
i
) sin((1 − t)θ
i
) + (αq
i+1
) sin(tθ
i
)
sin θ
i
Using the algebraic properties of quaternions, the angle between αq
i
and αq
i+1
is θ.
Think