Game Development Reference
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u (1)
n (1)
v (1)
u (1)
e
z
y
z
n (1)
x
u (1)
y
v (1)
x
n (1)
v (1)
Figure 9.4. Attachment frames.
The quaternions that map vectors defined in these frames to the global frames
are then in world frame, so we have
re and sf ,
respectively. This changes the definition of the relative quaternion in Equation (9.9)
to
p = e r sf .
Following the steps in Equations (9.9) and (9.10), we get
p = e qf
= P ( f ) Q ( e ) T q .
Everything else follows. To define a hinge joint, for instance, we can either spec-
ify the quaternion transforms e and f directly or provide a hinge frame containing
at least the axis of rotation in world coordinates. If we have a full frame of ref-
erence for the hinge definition, it is possible to define the reference joint angle
also. Otherwise, the orthogonal complement of the axis must be computed and
the quaternions e , f extracted from the frame.
Once we have a full frame defining the hinge geometry in world coordinates
with three orthogonal axes, u , v , n , forming an orthonormal basis in which n is
the axis of rotation, we build the matrix
R = uvn

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