Game Development Reference
InDepth Information
θ
ø
(2)
θ′
ø
(1)
Figure 9.3.
An illustration of the CV coupling.
circumference. Hold the pens 1 and 2 in your left and right hands, respectively,
and align the axes and the reference lines so that they face up. Now, rotate pen 2
by some angle
θ
about the vertical axis
z
away from you. Choosing
θ
45
◦
will
make things obvious. The two pens lie in the horizontal plane, with an angle
θ
between
n
(1)
and
n
(2)
. Now, realign the two pens and rotate them about their
common longitudinal axes by 90
◦
. Keep the reference lines aligned but make
them face you. Then rotate pen 2 by the same angle
θ
as before about the axis
z
. Clearly, the axis of rotation
r
is still perpendicular to
n
(1)
but is not the same
as before. If you had done this in small increments, you would have seen the CV
joint at work. You would probably scratch your head wondering how you would
actually construct something that worked like that. You can even change the angle
θ
as you move along, keeping perfect alignment between the reference lines. One
thing is constant though: relative rotation between pens 1 and 2, as seen from
pen 1, is about an axis
r
that is perpendicular to
n
(1)
. This axis
r
is not fixed,
however. This is what I've sketched in Figure 9.3.
Let's get rid of all the indices now. The conclusion from the experiment above
is that a rotation by any angle
θ
about any axis
r
such that
r
≈
z
=0always, will not
rotate the transformed
x

y
plane about the transformed axis
z
·
. Mathematically,
this implies that the relative quaternion
q
satisfies
q
s
=cos(
θ/
2)
q
v
=sin(
θ/
2)
r
,
·
z
=0
.
and
where
r
(9.5)
Therefore,
q
v
=
z
T
c
CV
=
z
·
P
q
=
P
CV
q
=0
,
(9.6)
where
P
CV
=
z
T
P
=[0
,
0
,
0
,
1]
.