Game Development Reference
In-Depth Information
A
r
m
O
Y
X
Figure 14.1. The angular velocity of a particle is a vector that is parallel to the axis of
rotation A and whose magnitude is equal to the rate of change of the angle formed in the
plane perpendicular to the axis.
The angular velocity is often written as a vector that is parallel to the axis of rota-
tion A and has the magnitude
()
()
ω t
. The vector angular velocity
ω is defined
as
()
()
()
ω
t ω t θ t
=
AA
=
.
(14.2)
The speed at which a rotating particle moves through space is calculated by
multiplying the particle's angular velocity by its distance from the axis of rota-
tion. For the particle shown in Figure 14.1, the speed
()
v t is given by
()
()
v t ω tr
=
.
(14.3)
However, this tells us nothing about what direction the particle is moving. Let the
vector function
()
r represent the position of the particle relative to a fixed origin
lying on the axis of rotation. As illustrated in Figure 14.2, the linear velocity vec-
tor
()
v
t
of the particle is given by
()
()
()
v ω r
t
=
t
×
t
(14.4)