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.