Graphics Reference
In-Depth Information
(
t
)
(
t
)
FIGURE 7.12
For a given number of rotations per unit of time, the angular velocity is the same whether the axis of rotation is near
or far away.
about an axis that is parallel to the previous axis but is ten miles away, again at the rate of two rev-
olutions per minute. The second point will have the same angular velocity as the first point. However,
in the case of the second point, the rotation will also induce an instantaneous linear velocity (which
constantly changes). But in both cases the object is still rotating at two revolutions per minute
(
Figure 7.12
)
.
Consider a point,
a
, whose position in space is defined relative to a point,
b ¼ x
(
t
);
a
's position
relative to
b
is defined by
r
(
t
). The point
a
is rotating and the axis of rotation passes through the point
b
(
Figure 7.13
). The change in
r
(
t
) is computed by taking the cross-product of
r
(
t
) and
o
(
t
)(
Eq. 7.22
)
.
Notice that the change in
r
(
t
) is perpendicular to the plane formed by
o
(
t
) and
r
(
t
) and that the mag-
nitude of the change is dependent on the perpendicular distance between
o
(
t
) and
r
(
t
), |
r
(
t
)|sin
y
, as well
as the magnitude of
o
(
t
).
r_ ðÞ¼o ðÞrðÞ
j r_ ðÞj¼jo ð ÞjjrðÞj
(7.22)
sin
Now consider an object that has an extent (distribution of mass) in space. The orientation of an
object, represented by a rotation matrix, can be viewed as a transformed version of the object's local
unit coordinate system. As such, its columns can be viewed as vectors defining relative positions in the
object. Thus, the change in the rotation matrix can be computed by taking the cross-product of
o
(
t
)with
a
b
r
(
t
)
(
t
)
r
(
t
)
b
x
(
t
)
FIGURE 7.13
A point rotating about an axis.
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