Critical Curves for The SIE Lens

The SIE lens is defined by the convergence

\kappa(\xi_1,\xi_2)=\frac{\sqrt{f}\xi_0}{2\sqrt{\xi_1^2+f^2\xi_2^2}},

(1)

where $f$ is the ratio between the minor and major axes.

The magnification matrix for SIE may now be calculated Kormann 1994

and becomes $$ \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}

(2)

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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

\xi_{\mathrm{crit}} = \frac{\sqrt{f}\xi_0}{\sqrt{\cos^2\phi+f^2\sin^2\phi}}, $$