Therefore we have found an expression for moment of inertia passing through the center of mass of the cell with additional pixel. Note that this expression involves moment of inertia but for the old cell (i.e. the original cell, not the one with extra pixel). When we add new pixel we know its coordinates and we can also easily calculate dnew .Thus when we need to calculate the moment of intertia for new cell instead of performing summation as given in the definition of the inertia tensor we can use much simpler expression.
This was diagonal term of the inertia tensor. What about off-diagonal terms? Let us write explicitely expression for Ixy :
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