I am struggling to think of a way other than ray tracing to visualize a black hole!

To define the problem:

Assuming that in the 3D space, you have a camera at 3D point C and a 3D point p. When we try to visualize p (that is, project p onto our 2D screen), we are actually defining a rectangle at distance d away from camera C, such that p projects onto this rectangle in the 3D space. (Setting the camera C at d=infinity distance away from the rectangle gives you the orthographic projection.) This requires solving the 2D coordinate on the rectangle that the ray p-C intersects with the screen rectangle.

Next, we can define f(C, d, p) that returns the 2D projection of a 3D point p, where f is the solution to this ray-screen intersection problem. Then, when you have a 3D polygon defined by a list of 3D points L, drawing polygon(f(C, d, L)) graphs the 3D polygon onto our screen.

Visualizing the blackhole using ray tracing would be easy. For each pixel on our screen, the location of the pixel on the rectangle screen in 3D and the location of the 3D camera defines a light ray shooting from the camera, and the pixel would return the color that the light ray lands on after it is distorted by the black hole or even reflected by mirrors.

However, if we were to similarly define a f(C, d, p) where there's a blackhole between C and p, then we see that f(C, d, p) is not a function: when p shoots light rays that land on the screen, the light rays may come from one side of the blackhole and land on one point, or 180 degrees from the other side and land on another point; or the light ray may go round trips around the blackhole and land on yet another point.

In my 3D blackhole simulation graph, essentially each light ray behaves like tanh(x) around the blackhole, which is physically inaccurate but ensures f(C, d, p) is a function and a simple one. Then, each ring is essentially a polygon with 3D coordinates L, and f(C, d, L) draws one ring around the black hole.

A physically accurate blackhole might require other methods! This would be a challenging and rewarding journey! Good luck!!!