# [Chimera-users] Measure inertia?

Tom Goddard goddard at sonic.net
Wed Jan 30 17:03:57 PST 2013

```Hi Yen,

The measure inertia command computes the moments of inertia of a set of atoms as computed in classical mechanics using the tensor

I_jk = sum_over_i ( mi * (djk*|xi|**2 - xi_j*xi_k) )

where mi and xi are the mass and position of atom i and djk = 1 for j=k and 0 for j!=k.  The principal axes are the eigenvectors of this 3 by 3 matrix and the moments about those axes are the eigenvalues.  Basically the moment is just a sum of mass times distance squared from the rotation axis.  Before we apply this formula we subtract the center of mass position from the atom coordinates.  So we measure the inertia about the center of mass.  For measuring the inertia of a surface we replace atoms by vertices of the triangulated surface where the "mass" of each vertex is 1/3 of the area of the attached triangles.  This is the inertia treating the surface as a thin shell.

The "inertia ellipsoid" shown by Chimera is not the same as the one defined in physics.  Instead for atoms we show the surface of a uniform density solid ellipsoid which would have the same principal axes and moments as the atoms.  For surfaces we show a ellipsoidal surface which as a thin shell has the same axes and moments as the measured surface.  So the Chimera definition  of "inertia ellipsoid" is the ellipsoid that has the same inertia as the measured object.  The physics definition turns out to be something different.

Elaine, perhaps you can put this information as a technical note on the "measure inertia" Chimera User's Guide page.

Tom

On Jan 24, 2013, at 4:17 PM, Yen-Ting Lai wrote:

> Thank you very much, Elaine.
>
> On Thu, Jan 24, 2013 at 3:53 PM, Elaine Meng wrote:
> Hi Yen,
> It is simply a calculation of principal axes where atoms are mass-weighted points, see:
> <http://www.cgl.ucsf.edu/chimera/docs/UsersGuide/midas/measure.html#inertia>
>
> I doubt someone would publish a paper about such a fundamental calculation, so although I did not read the J Mol Graph paper, it is likely to be a different method.
>
> The person who wrote the "measure inertia" code is away currently, but if there is anything else to say, he may send another reply later.
> Best,
> Elaine
> ----------
> Elaine C. Meng, Ph.D.
> UCSF Computer Graphics Lab (Chimera team) and Babbitt Lab
> Department of Pharmaceutical Chemistry
> University of California, San Francisco
>
> On Jan 24, 2013, at 1:07 PM, Yen-Ting Lai wrote:
>
> > Hi,
> > What's the formula that's used by "measure inertia"?
> > I found a paper:
> > An ellipsoidal approximation of protein shape. J. Mol Graph (1983)
> > Does the "measure inertia" use the same method?
> > Thank you!
> > Yen
>
>
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