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I take an rectangular sheet of paper, of height $H$ and Young's Modulus $E$, and in the absence of gravity, I bend it into a "teardrop" shape so that the edges along the top and bottom touch only along a single line.

What analytic function describes this "teardrop" shape?

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Are you specifying the boundary condition that the tangent cones of the edges coincide, or that they form some specified dihedral angle? – S. Carnahan Jan 24 '12 at 23:45
You may find this question on MSE useful:… – Samuel Reid Jan 25 '12 at 0:02
up vote 7 down vote accepted

All in here (if using Euler Bernoulli beam theory)

(Added by Joseph O'Rourke). Here is Fig.1 from the paper by Antoni Colom, "Analysis of the shape of a sheet of paper when two opposite edges are joined," PDF link above.
         Tear Drop

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@Paul: I took the liberty of adding detail to your answer. – Joseph O'Rourke Jan 25 '12 at 1:34

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