This online article is a supplement to our manuscript on quartic solutions of the quadrupole gravitational lens.

Using the method of stationary phase, we were able to obtain a closed form solution of the quadrupole gravitational lens that is valid in most regions except for the immediate vicinity of the caustic boundary.

Key to this solution is the quartic equation

$$x^4-2\eta\sin\mu x^3+(\eta^2-1)x^2+\eta\sin\mu x+\tfrac{1}{4}\sin^2\mu=0,\tag{22}$$

where $x=\sin(\phi_\xi-\phi)$ is the sine of an angle, and the solution "lives" in the image plane (where the gravitational lens projects an image) characterized by polar coordinates $(\eta,\tfrac{1}{2}\mu)$. A corresponding equation for $y=\cos(\phi_\xi-\phi)$ is given by

$$y^4+2\eta\cos\mu y^3+(\eta^2-1)y^2-2\eta\cos\mu y+\tfrac{1}{4}\sin^2\mu-\eta^2=0.\tag{35}$$

These equations both have as many as four real roots, and these are not easy to visualize. To help with the visualization, we created an animation that shows the angle formed from these roots as a function of image plane coordinates $(\eta\cos\tfrac{1}{2}\mu,\eta\sin\tfrac{1}{2}\mu)$. The result is presented here.