Game Development Reference

In-Depth Information

In each case
a
p
i
is the distance of particle
i
from the center of mass of the whole

structure, in the direction of axis
a
;
m
i
is the mass of particle
i
; and there are
n
particles

in the set.

A.2

C
ONTINUOUS
M
ASSES

We can do the same for a general rigid body by splitting it into infinitesimal masses.

This requires an integral over the whole body, which is considerably more difficult

than the rest of the mathematics in this topic. The formula is included here for com-

pleteness.

⎡

⎣

⎤

⎦

I
x

−

I
xy
−

I
xz

I

=

−

I
xy

I
y

−

I
yz

[A.2]

−

I
xz

−

I
yz

I
z

as before, where

a
p
i
dm

I
a
=

m

and

I
ab
=

a
p
i
b
p
i
dm

m

The components of both are as before. The integrals are definite integrals over the

whole mass of the rigid body.

A.3

C
OMMON
S
HAPES

This section gives some inertia tensors of common objects.

A.3.1

C
UBOID

This includes any rectangular six-sided object, where the object has constant density:

⎡

⎣

⎤

⎦

1

12
m(d
y
+

d
z
)

0

0

12
m(d
x
+

1

d
z
)

I

=

0

0

1

12
m(d
x
+

d
y
)

0

0

where
m
is the mass and
d
x
,
d
y
,and
d
z
are the extents of the cuboid along each axis.

A.3.2

S
PHERE

This inertia tensor corresponds to a sphere with constant density: