## Scaling an inertia matrix ?

### Scaling an inertia matrix ?

Hi,

I'm looking for a matrix relation to scale an inertia matrix in the 3
directions by 3 different coefficients.
Does a such relation exist ? an someone knows it ?

sobi steff.

### Scaling an inertia matrix ?

Quote:> Hi,
> I'm looking for a matrix relation to scale an inertia matrix in the 3
> directions by 3 different coefficients.

It's not quite clear what "scaling an inertia matrix" is supposed to
mean.  You could be re-scaling the coordinate system, or the solid
body the inertia matrix belongs to.  You might even mean a re-scaling
of the mass of that body, not the spatial coordinates.  I'll assume a
scaling of the coordinate system.

Quote:> Does a such relation exist ?

Yes, it does.  The inertia tensor is an object of dimension
mass*length^2.  The individual elements I_{i,j} of this tensor are
essentially just sums or integrals of terms of the type

mass(x,y,z)*X_i*X_j

over the whole volume of the body (or complete R^3 space, with the
mass zero everywhere outside it) where X is the vector (x,y,z), and
X_i or X_j are components of that vector.  This is the moment of
inertia relative to the origin of the coordinate system (x,y,z) are
valid for.

Scaling the coordinates effects the X_i and X_j, and thus the elements
of the tensor, in a rather obvious way: they scale by a product of two
axis scale factors. I_xx scales by scale_x^2, I_xy by scale_x*scale_y,
and so on.

Note that for a solid body whose principal axes of inertia don't
happen to be aligned to the axes of the coordinate you're rescaling,
the scaling operation will affect not only the size, but also the
directions of those principal axes.  That's obvious once you realize
that such anisotropic scaling changes the *shape* of the body, not
just its size.
--

Even if all the snow were burnt, ashes would remain.

### Scaling an inertia matrix ?

Quote:> Hi,

> I'm looking for a matrix relation to scale an inertia matrix in the 3
> directions by 3 different coefficients.
> Does a such relation exist ? an someone knows it ?

> sobi steff.

Sobi:
The inertia matrix I has not the least meaning for mechanics,
if the distances are not measured in an Euclidian metric.
Therefore an arbitrary scaling of axes is useless.
Once the matrix I is found in x,y,z, it can be transformed
as J to main axes u,v,w by using the rotation matrix N, which
contains normalized eigenvectors (the directions of the main
axes in x,y,z).

J = N^T * I* N

J is diagonal: J = (Juu,Jvv,Jww)

Now Juu,Jvv,Jww may be scaled - which means of course a dif-
ferent body. The reverse transformation delivers a new I ,
without affecting the eigenvectors.
I can hardly see a purpose for this scaling.

Best regards  --Gernot Hoffmann

I'm writing a 3D-physics simulator, and have some questions:

1. How is the inertia tensor matrix interpreted?
2. Where can I find an algorithm to calculate it for
a given rigid body represented by points and triangles?

Thanks!