Game Development Reference

In-Depth Information

3

2

() ()

L

=

mrω

−

r

r

ω

.

(14.40)

i

k

k

i

k

i

k

j

j

k

j

=

1

ω
as

We can express the quantity

3

,

ωωδ

=

=

(14.41)

i

j

ij

j

1

where
i
δ
is the Kronecker delta defined by Equation (2.42). This substitution al-

lows us to write
L
as

3

2

()()

L

=

mr ωδ

−

rr

ω

i

k

k

j

ij

k

i

k

j

j

k

j

=

1

3

=

ω m δ r

2

−

()()

rr

.

(14.42)

j

k

ij

k

k

i

k

j

j

=

1

k

()

The sum over
k
can be interpreted as the

,
ij
entry of a 3

×

3

matrix
:

2

()()

=

m δ r

−

rr

.

(14.43)

ij

k

ij

k

k

i

k

j

k

This allows us to express
L
as

3

,

(14.44)

L

=

ω

i

j

ij

j

=

1

()

and thus the angular momentum

L

t

can be written as

()

()

t

=

.

t

(14.45)

L

The entity
is called the
inertia tensor
and relates the angular velocity of a

rigid body to its angular momentum. The inertia tensor also relates the torque

()

()

()

τ
acting on a rigid body to the body's angular acceleration

α ω
. Dif-

t

=

t

ferentiating both sides of Equation (14.45) gives us

()

()

()

L τ

t

==

t

.

α

t

(14.46)

Written as a 3

×

3

matrix, the inertia tensor is given by

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