What shape does a piece of paper make when it is pushed from the edges?

differential-geometry

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

Solution 1:

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

Can some proof that $\det(A) \ne 0$ be checked faster than matrix multiplication?

Prove this determinant identity combinatorially

Probability of completing a self-avoiding chessboard tour

Prove $\sum\limits_{n = 1}^\infty \frac{( - 1)^n}{\ln n + \sin n} $ is convergent.

"Rectangularity" of integers

How would you evaluate $\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$

Which number fields can appear as subfields of a finite-dimensional division algebra over Q with center Q?

Haar Measure of a Topological Ring

Curious about an empirically found continued fraction for tanh

If two definite integrals are equal, does there exist a chain of substitutions and/or partial integrations which will get us from one to the other?

Why do we use big Oh in taylor series?