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