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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

Did
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Hugh
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  • 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... – Steven Stadnicki Aug 19 '13 at 05:00
  • Done (hopefully) – Hugh Aug 19 '13 at 05:09
  • Good question (+1). The first thing that came to mind was cubic splines, but their physical basis is bending, not compression. – bubba Aug 19 '13 at 06:09
  • Reminds me of a gauss curve http://en.wikipedia.org/wiki/Normal_distribution#Standard_deviation_and_tolerance_intervals – Plom Aug 19 '13 at 07:41
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    This doesn't have a whole lot to do with differential geometry since the problem is basically one-dimensional. Maybe try physics SE instead? – Scaramouche Aug 19 '13 at 08:11
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    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". –  Aug 19 '13 at 08:19
  • The aspect of differential geometry that I'm reaching for is a theorem which shows that it is easier to square a quire of paper if you bend it into a parabola than if the paper is flat. (If you pick up some sheets of paper and tap it on the table to square them, you'll naturally bend the paper.) It was a while since I did diff geo but I remember this result! – Hugh Aug 19 '13 at 11:17
  • This looks like an inverse catenary to me, but I could be wrong. –  Aug 31 '13 at 21:36
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    Buckling of sheets - the shape is similar to a standing wave and depends a lot on how the ends are constrained while compression. – Macavity Sep 01 '13 at 18:13
  • If you push the edges of a crisp or short paper it takes the shape of upside down parabola or catenary or a sort of oval with a sharp corner if you bring the edges together. If you do this with flimsy or long paper, or if you press the edges down, you get that bell-shaped curve. It looks like it depends on the physics of the situation quite a bit. There could be multiple families depending on boundary conditions and the material properties. – Maesumi Sep 05 '13 at 19:21

1 Answers1

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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: $\text{curvature} = -k\, y$

Narasimham
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  • 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. – Daniel R Sep 20 '13 at 10:40