In my Classical Mechanics course, we spent a week discussing the "calculus of variations", a field of differential calculus that focuses on finding functions that can be used to represent minimum and maximum (usually minimum) values. In the case of what my professor calls "the soap film problem" (though it's actually more commonly known as Plateau's problem), a soap film, when stretched out between two hoops, takes the shape of a catenoid in order to achieve the smallest possible surface area.
Using the formula for a catenoid, we may create a revolution plot to generate a 3D diagram of the catenoid. The only necessary starting values are the radii of the hoops, and the distance between them. For simplicity, I assume here that both hoops have the same radius. Here is the plot I made, with a hoop radius of 1.410, and a distance between the hoops of 1.170:
There are two surfaces shown: The outer, blue surface represents the catenoid's minimum surface area as previously described, that a soap film would exhibit in real life. The inner, orange surface represents the surface area that would theoretically exist if the catenoid's surface area was maximized. In real life, this solution couldn't possibly exist (the bubble would pop, if you tried to force it), but it illustrates something very fascinating; if you increase the distance between the hoops, then these "stable" and "unstable" surface areas will actually appear to become closer together, until we reach a point where the maximum and minimum surface areas perfectly overlap. Theoretically, this is the point at which the soap film will be guaranteed to have popped, as there can no longer exist a stable solution. Personally, I find this quite fascinating, as it means that this theory has some experimental basis! If I ever perform an experiment to predict the distance at which the film pops, I will definitely update this section.
If you'd like to examine the code I used to generate the plot, test different values for the hoop, or just play around with the plot itself (in Mathematica, you can click and drag to view the plot from different angles), then you can download the notebook here.