When I push a piece of (A4) paper oriented landscape to me from the shorter edges, it makes a pretty shape, resembling a bell-curve. I seem to remember these sort of situations being a motivation for or concrete instance of some theorems in differential geometry, but apart from that I have no idea how to determine what the true shape of the paper in this situation.

Not a great example (as I'm pushing with one hand to take the photo) but similar to what I'm after. (Hey, if you can generalize to a one-sided push I might just award the checkmark!)

Example of a one-sided push

  • $\begingroup$ All I'd ask is an illustration of the specific sort of shape you're thinking about - the details of the pushing can make a huge difference in the answer... $\endgroup$ – Steven Stadnicki Aug 19 '13 at 5:00
  • $\begingroup$ Done (hopefully) $\endgroup$ – Hugh Aug 19 '13 at 5:09
  • 1
    $\begingroup$ This doesn't have a whole lot to do with differential geometry since the problem is basically one-dimensional. Maybe try physics SE instead? $\endgroup$ – Scaramouche Aug 19 '13 at 8:11
  • 6
    $\begingroup$ The solution is the curve minimizing the bending energy $\int \kappa(s)^2\,\mathrm ds$ for prescribed length, endpoints, and tangents, a.k.a. the elastica. See e.g. Sec. 9 onwards (and the bottom of Fig. 11) of Raph Levien's "The elastica: a mathematical history". $\endgroup$ – Rahul Aug 19 '13 at 8:19
  • 1
    $\begingroup$ Buckling of sheets - the shape is similar to a standing wave and depends a lot on how the ends are constrained while compression. $\endgroup$ – Macavity Sep 1 '13 at 18:13

It is the Elastica. For small wave heights, the equation is like $\sin^2 k x$ , where $k$ depends on bending rigidity $EI$, applied force $P$. More accurately described in terms of Elliptic functions.

The differential equation is simply: $\text{curvature} = -k\, y$

  • $\begingroup$ Welcome to math.SE: I have tried to improve the readability of your answer by introducing Tex. It is possible that I unintentionally changed the meaning of your answer. Please proofread the answer to ensure this has not happened. $\endgroup$ – Daniel R Sep 20 '13 at 10:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.