Game Development Reference

In-Depth Information

where
a
p
i
is the distance of particle
i
from the center of mass of the object, in the

direction of
a
.Weusethistocalculate
I
xy
,
I
xz
,and
I
yz
. In the case of
I
xy
,weget

n

I
xy
=

m
i
x
p
i
y
p
i

i

=

1

where
x
p
i
is the distance of the particle from the center of mass in the X axis direction;

similarly for
y
p
i
in the Y axis direction. Using the scalar products of vectors we get

n

I
xy
=

m
i
(
p
i
·

x
)(
p
i
·

y
)

i
=
1

Note that, unlike for the moment of inertia, each particle can contribute a negative

value to this sum. In the moment of inertia calculation, the distance was squared, so

its contribution is always positive. It is entirely possible to have a non-positive total

product of inertia. Zero values are particularly common for many different shaped

objects.

It is difficult to visualize what the product of inertia
means
either in geometrical

or mathematical terms. It represents the tendency of an object to rotate in a direction

different from the direction in which the torque is being applied. You may have seen

this in the behavior of a child's top. You start by spinning it in one direction, but it

jumps upside down and spins on its head almost immediately.

For a freely rotating object, if you apply a torque, you will not always get rotation

about the same axis to which you applied the torque. This is the effect that gyroscopes

are based on: they resist falling over because they transfer any gravity-induced falling

rotation back into the opposite direction to stand up straight once more. The prod-

ucts of inertia control this process: the transfer of rotation from one axis to another.

We place the products of inertia into our inertia tensor to give the final structure:

⎡

⎣

⎤

⎦

I
x

−

I
xy
−

I
xz

I

=

−

I
xy

I
y

−

I
yz

[10.2]

−

I
xz
−

I
yz

I
z

The mathematician Euler gave the rotational version of Newton's second law of mo-

tion in terms of this structure:

I
θ

τ

=

which gives us the angular acceleration in terms of the torque applied:

θ

I
−
1
τ

=

[10.3]

where
I
−
1

is the inverse of the inertia tensor, performed using a regular matrix inver-

sion.