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.

According to Kormann 1994 (TODO Check specific paper) the magnification matrix for SIE now 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)

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}},

(3)