Suppose we have computed the beam moment matrix $\Sigma_{ij}(s)$ around the ring.
Consider the normalized spatial distribution in $(x,y)$:
If we define the matrix
(2)the exponential could be written as
(3)with $\vec X = \pmatrix{x\cr y}$. Now, we have (see here for a reference)
(4)with
(5)We can thus derive that
(6)We have a family of ellipses, now, and we'd like to choose one to represent the electron beam.
Perhaps we may use
where the particle density has dropped by a factor of $\frac{1}{e}$
This is an ellipse with a rotation angle $\theta$ satisfying
(8)The equation for the ellipse may also be written in terms of the moments as
(9)First we find the positions we want to plot at:
dr278=findcells(ring_err,'FamName','DR_278….')
Then we can use the function
plotContours(ring,dr278(1:2:end))