[jbay808]

A friend of mine broke a badminton racket during a match, and I was struck by how the sharply bent and twisted metal rim was transformed into a smooth, continuously double-curved surface by the racket weave. I looked closely at the balance of tension in the woven cord, thought of how it resembles Poisson's equation, and suddenly it all made sense.

Edit - it looked something like this:

https://thumbs.dreamstime.com/b/broken-badminton-racket-phot...

[codethief]

Hmmm, this looks more like a minimal surface, i.e. a solution to the minimal-surface equation[0], than a solution to Poisson's equation. Then again, both equations are of elliptic type.

Some links for people who've never heard of minimal surfaces:

https://en.wikipedia.org/wiki/Minimal_surface

https://minimalsurfaces.blog/ (lots of illustrations)

https://makmanx.github.io/math3435s18/talks/MSE.pdf (brief intro with historical remarks and illustrations)

[0]: More specifically, it's a solution to Plateau's problem: https://en.wikipedia.org/wiki/Plateau%27s_problem

[jbay808]

Isn't Poisson's equation basically describing a minimal surface for small z?

I'm not saying the badminton racket follows exactly a (discrete) 2D Poisson equation. But it's certainly related enough to be more than a surface similarly.

The cords are under high tension, which means that any curvature along x (that is, dz^2/dx^2) will result in a net z-axis tension force unless balanced by an oppositely curved cord running in the y direction. Since it's in static equilibrium, there can be no unbalanced forces and so that must be the case. Therefore at each intersection, (d^2/dx^2 + d^2/dy^2)z = 0, which is Poisson's equation in 2D for z height being the function. Approximately, assuming equal tension in x and y, small z, and so on.

[gpsx]

I think that is good intuition. The transverse force is the second derivative of the transverse position, to first order. But then there are higher order effects in it. I think that is where that math breaks down.

[codethief]

I didn't say you're completely wrong, either. :) (In fact, I stressed that both equations are of elliptic type, so solutions might look similar.)

One can make a similar heuristic argument, though, as to why the surface you observed should follow the minimal-surface equation.

EDIT, as you've updated your comment:

> Isn't Poisson's equation basically describing a minimal surface for small z?

Small changes in z, I think.

[gpsx]

If you want to get technical, this isn't a continuous surface but weave of strings. And something that makes this more complex is the location of the string intersections has a static/dynamic friction component that is difficult to model analytically. But I think we accept your approximation if you are looking for numeric results.