When a cable is supported at its ends and droops due to its own weight, the resulting curve is called a catenary.
However, is there a three-dimensional analogue of this shape?
For example, let's say I take the unit square $([0,0,0], [1,0,0], [0,1,0], [1,1,0])$, support it at these points, and fill it with a uniform material. It will droop due to its own weight. I have a few questions about the resulting surface:
- Do the edges turn into local catenaries?
- Is there a parametric function that describes this shape?
- Is there a name for this shape?
Any information would be much appreciated!
Not exactly: A flexible (but un-stretchable) wire or thread can bend isometrically into a catenary.
By contrast, two-dimensional materials have internal rigidity coming from intrinsic curvature. Consequently, an ideal flexible (but un-stretchable) sheet (think of paper or a wire screen) suspended at its corners can only bend upward at the corners with four-fold symmetry.