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)

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