Game Development Reference

In-Depth Information

You already know that
I
represents the moment of inertia, and the terms that should

look familiar to you already are the moment of inertia terms about the three coordinate

axes,
I
xx
,
I
yy
, and
I
zz
. The other terms are called
products of inertia
(see
Figure 1-9
):

I
xy
= I
yx
= ∫ (xy) dm

I
xz
= I
zx
= ∫ (xz) dm

I
yz
= I
zy
= ∫ (yz) dm

Figure 1-9. Products of inertia

Just like the parallel axis theorem, there's a similar transfer of axis formula that applies

to products of inertia:

I
xy
= I
o(xy)
+ m d
x
d
y

I
xz
= I
o(xz)
+ m d
x
d
z

I
yz
= I
o(yz)
+ m d
y
d
z

where the
I
o
terms represent the local products of inertia (that is, the products of inertia

of the object about axes that pass through its own center of gravity),
m
is the object's

mass, and the
d
terms are the distances between the coordinate axes that pass through

the object's center of gravity and a parallel set of axes some distance away (see

Figure 1-10
).

You'll notice that we did not give you any product of inertia formulas for the simple

shapes shown earlier in
Figure 1-3
through
Figure 1-7
. The reason is that the given

moments of inertia were about the
principal axes
for these shapes. For any body, there

exists a set of axes—called the principal axes—oriented such that the product of inertia

terms in the inertia tensor are all zero.