Game Development Reference

In-Depth Information

Expanding the triple vector product term yields:

H
cg
= ∫ {[(y
2
+ z
2
)ω
x
− xyω
y
- xzω
z
] i +

[ −yxω
x
+ (z
2
+ x
2
)ω
y
− yzω
z
]
j
+

[−zxω
x
− zyω
y
+ (x
2
+ y
2
)ωz]
k
} dm

To simplify this equation, let's replace a few terms by letting:

I
xx
= ∫ (y
2
+ z
2
) dm

I
yy
= ∫ (z
2
+ x
2
) dm

I
zz
= ∫ (x
2
+ y
2
) dm

I
xy
= I
yx
= ∫ (xy) dm

I
xz
= I
zx
= ∫ (xz) dm

I
yz
= I
zy
= ∫ (yz) dm

Substituting these
I
variables, some of which should look familiar to you, back into the

expanded equation yields:

H
cg
= [I
xx
ω
x
− I
xy
ω
y
− I
xz
ω
z
]
i
+

[−I
yx
ω
x
+ I
yy
ω
y
− I
yz
ω
z
]
j
+

[−I
zx
ω
x
− I
zy
ω
y
+ I
zz
ω
z
]
k

Simplifying this a step further by letting
I
be a matrix:

I
xx
−I
xy
−I
xz

I
= −I
yx
I
yy
−I
yz

−I
zx
−I
zy
I
zz

yields the following equation:

H
cg
=
I ω