Game Development Reference
In-Depth Information
something that looks like L = Jω:
L
=
r
×
P
=
r
× (m
v
) = (m
r
) ×
v
= (m
r
) × (ω ×
r
)
2
4
3
5
2
4
3
5
2
4
3
5
r
x
r
y
r
z
ω
y
r
z
− ω
z
r
y
ω
z
r
x
− ω
x
r
z
ω
x
r
y
r
y
(ω
x
r
y
− ω
y
r
x
) − r
z
(ω
z
r
x
− ω
x
r
z
)
r
z
(ω
y
r
z
− ω
z
r
y
) − r
x
(ω
x
r
y
− ω
y
r
x
)
r
x
(ω
z
r
x
= m
×
= m
− ω
y
r
x
− ω
x
r
z
) − r
y
(ω
y
r
z
− ω
z
r
y
)
2
3
r
y
ω
x
r
y
− r
y
ω
y
r
x
− r
z
ω
z
r
x
+ r
z
ω
x
r
z
4
5
= m
r
z
ω
y
r
z
− r
z
ω
z
r
y
− r
x
ω
x
r
y
+ r
x
ω
y
r
x
r
x
ω
z
r
x
− r
x
ω
x
r
z
− r
y
ω
y
r
z
+ r
y
ω
z
r
y
2
3
(r
y
+ r
z
)ω
x
− r
z
r
x
ω
z
−r
x
r
y
ω
x
+ (r
z
+ r
x
)ω
y
− r
y
r
x
ω
y
4
5
= m
− r
z
r
y
ω
z
− r
y
r
z
ω
y
+ (r
x
+ r
y
)ω
z
−r
x
r
z
ω
x
0
2
3
1
2
3
r
y
+ r
z
−r
y
r
x
−r
z
r
x
ω
x
ω
y
ω
z
@
4
5
A
4
5
−r
x
r
y
r
z
+ r
x
=
m
−r
z
r
y
.
−r
y
r
z
r
x
+ r
y
−r
x
r
z
The key point is in the last line, which bears a striking resemblance to
L = Jω. In fact, the quantity in parenthesis is the three-dimensional
moment of inertia.
Inertia Tensor
The inertia tensor of a mass m with radial vector
r
is
2
3
r
y
+ r
z
−r
y
r
x
−r
z
r
x
4
5
−r
x
r
y
r
z
+ r
x
J
= m
−r
z
r
y
.
(12.30)
−r
y
r
z
r
x
+ r
y
−r
x
r
z
Notice that this quantity is a matrix, not a vector. In recognition of this,
in three dimensions we sometimes refer to the moment of inertia as the
inertia tensor. This mathematical artifact is a result of the physical fact
that an object's resistance to rotational acceleration is anisotropic: it can
be easier to rotate an object about one axis compared to another. For
example, compare the torque required to spin a piece of rebar around like
a helicopter versus rolling it about its lengthwise axis.
Notice that
J
is a symmetric matrix. One trivial (but thankful) conse-
quence of this is that we can be sloppy with row and column vectors, while
ordinarily we must be very careful in order to avoid transposed results.
19
19
Remember, our convention in this topic is to use row vectors, since the majority of
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