2.Calculating shape constraint of a cell – elongation term
The shape of single cell immersed in medium and not subject to too drastic surface or surface constraints will be spherical (circular in 2D). However in certain situation we may want to use cells which are elongated along one of their body axes. To facilitate this we can place constraint on principal lengths of cell. In 2D it is sufficient to constrain one of the principal lenghths of cell how ever in 3D we need to constrain 2 out of 3 principal lengths. Our first task is to diagonalize inertia tensor (i.e. find a coordinate frame transformation which brings inertia tensor to a giagonal form)
We will consider here more difficult 3D case. The 2D case is described in detail in M.Zajac, G.L.jones, J,A,Glazier "Simulating convergent extension by way of anisotropic differential adhesion" Journal of Theoretical Biology 222 (2003) 247–259.
In order to diagonalize inertia tensor we need to solve eigenvalue equation:
 or in full form
The eigenvalue equation will be in the form of 3rd order polynomial. The roots of it are guaranteed to be real. The polynomial itself can be found either by explicit derivation, using symbolic calculation or simply in Wikipedia ( http://en.wikipedia.org/wiki/Eigenvalue_algorithm )
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