A catenary curve is the shape taken by an idealized hanging chain or rope under the influence of gravity. It has the equation $y= a \cosh (x/a)$. My question is:
What is the shape taken by an idealized, thin two-dimensional sheet, pinned on a plane parallel to the ground, under the influence of gravity?
The answer surely depends on how it is pinned to the plane, the boundary conditions. Natural options are:
- A disk sheet fixed to a circle.
- A square sheet fixed to a square.
- A square sheet pinned at its four corners.
The middle option above would look something like this when inverted:
(Image by Tim Tyler at hexdome.com.)
I don't think any of these shapes is a catenoid, which is the surface of revolution formed by a catenary curve. Is there a simple analytic description of any of these surfaces, analogous to the $\cosh$ equation for the catenary curve? I have been unsuccessful in finding anything but simulations of solutions of the differential equations.
This question arose in imagining a higher-dimensional version of the property that an inverted catenary supports smooth rides of a square-wheeled bicycle (explored in this MO question). Thanks for pointers!