 Solution

The conditions are singularities of the image-plane Jacobian matrix, where the formal lensing magnification of a point-like astronomical source diverges. A small source is strongly stretched in the corresponding eigenvector direction: tangentially near ${\theta }_t$, radially near ${\theta }_r$. Their mapped loci are source-plane gravitational-lensing caustics, which determine where image counts change.

The tangential circle maps to $\beta =0$ and gives the aligned Einstein ring. To find the radial caustic, put $R={\theta }_E/{\theta }_c>1$. At ${\theta }_r$, $\sqrt{{\theta }_r^2+{\theta }_c^2}={\theta }_c{R}^{1/3}$ and ${\theta }_r={\theta }_c\sqrt{R^{2/3}-1}$. Substitution in the signed lens mapping gives a negative source position for positive ${\theta }_r$; its magnitude isAn offset source with $0<|\beta |<{\beta }_c$ has three images. For $\beta >0$, one outer image is on the positive side and two are on the negative side, as the sketch shows. The central and outer images have positive parity; the intermediate image has negative parity. On the radial caustic the inner pair merges, with a separate outer image remaining; outside it only the outer image survives. At exact alignment a central image coexists with the ring. Finite source extent smooths the formal divergences. The singular zero-core limit removes the central image, so its image-count rule differs from the finite-core case.