CompuCell3d manual and Tutorial Version 2
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so in our case the eigenvalue equation takes the form: 
This equation can be solved analytically, again we may use Wikipedia ( http://en.wikipedia.org/wiki/Cubic_function ) Now, the eigenvalues found that way are principal moments of inertia of a cell. That is they are components of inertia tensor in a coordinate frame rotated in such a way that off-diagonal elements of inertia tensor are 0: 
In our cell shape constraint we will want to obtain ellipsoidal cells. Therefore the target tensor of inertia for the cell should be tensor if inertia for ellipsoid: 
where a,b,c are parameters describing the surface of an ellipsoid: 
In other words a,b,c are half lengths of principal axes (they are analogues of circle's radius) Now we can determine semi axes lengths in terms of principal moments of inertia by inverting the following set of equations:  
Once we have calculated semiaxes lengths in terms of moments of inertia we can plug –in actual numbers for moment of inertia (the ones for actual cell) and obtain lengths of semiexes. Next we apply quadratic constraint on largest (semimajor) and smallest (seminimor axes). This is what elongation plugin does. Directory: BinDoc -> cc3d binaries -> Manuals Manuals -> 111Equation Chapter 1 Section 1 CompuCell3d reference Manual Version 4 Download 6.67 Mb. Share with your friends:
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