Made by Denis Shiryaev, shir-man.com
Critical curve — where magnification diverges. Caustic — its image in the source plane. An isothermal lens shows four images from a source inside the astroid, two from one outside it, and only one once the source lies beyond θE altogether. Crossing the caustic is what makes an image pair appear or vanish.
Every pixel is solved backwards through the thin-lens equation
β = θ − α(θ) on the GPU, so multiple images, arcs, rings and
magnification are never drawn — they fall out of the mapping. Surface brightness
is conserved, as Liouville's theorem requires: the extra flux comes entirely from
the extra solid angle each image covers. Image positions in the panel are found by
Newton's method on the same equation, and magnifications from 1/|det A|.
Distances are angular-diameter distances in a flat ΛCDM universe
(H₀ = 70, Ωm = 0.3). The black hole uses the exact
Schwarzschild deflection integral, not its weak-field limit.
What is still approximate. The lens is thin and the sky is flat, which is standard and good to well under a percent at these angles. The isothermal profile is singular, so there is no faint central image. In the black hole view the bending is the full observer-at-infinity deflection applied at a static observer's finite radius — accurate to order rs/r, so a few percent at 30 rs — and there is no accretion disc, no frame dragging and no Doppler beaming, because the hole is Schwarzschild and the only light source is the background sky.