CosmoSim is a simulator for gravitational lensing.
The SIE lens is defined by the convergence \begin{equation} \kappa(\xi_1,\xi_2)=\frac{\sqrt{f}\xi_0}{2\sqrt{\xi_1^2+f^2\xi_2^2}}, \end{equation} where $f$ is the ratio between the minor and major axes.
According to Kormann 1994
(TODO Check specific paper)
the magnification matrix for SIE now becomes
\begin{equation}
\mathcal{A}(\boldsymbol{\theta})
=
\begin{bmatrix}
1 - 2\kappa\sin^2\phi &
\kappa\sin(2\phi)
\\\
\kappa\sin(2\phi) &
1 - 2\kappa\cos^2\phi &
\end{bmatrix}
\end{equation}
where $(\xi,\phi)$ are the polar coordinates of the normalised screen space point
$\mathbf{x}$.
Now the critical curve is given as the points $(\xi_{\mathrm{crit}},\phi)$ given as
\begin{equation}
\xi_{\mathrm{crit}} =
\frac{\sqrt{f}\xi_0}{\sqrt{\cos^2\phi+f^2\sin^2\phi}},
\end{equation}