----if you curl a piece of paper over (without creasing) so one edge is touching the opposite edge, what equation describes the resulting curve?
— James Tauber (@jtauber) March 1, 2013
---- Because you can view the sheet of paper as an extrusion of a curve, I'm happy just to know the equation of the 2-dimensional curve corresponding to the edge along the side (the one closest to the camera in the video, for example) ---- Continuing to think in 2D curve terms, it seems the parameters of the curve would include *at least* the two end points and the length (which must remain fixed). The *weight* of the paper may also come in to play. Clearly this isn't enough as you can form a circle (cylinder in the extruded case), teardrop or even something cardioid-like depending on the tangent vectors of the paper at the end points. ---- Potentially related: if you hang a sheet of paper over the edge of a ledge, what's the equation of the edge-curve in that case? ---- What relationship, if any, does all this have to the **Catenary**? ---- What relationship, if any, does all this have to the **Troposkein**? ---- Another related scenario regarding the shape of a sheet of paper: lie the sheet flat on a table then push the opposite edges towards one another. What's the curve created by the perpendicular edges? ---- Note that the answer would need to consider how much the fingers (or whatever's holding down the edges being pushed) are on the paper. ---- These are the sorts of curves I have in mind:  ---- Seems like the case in the top left is the same as a cantilever beam deflection. ---- The equation of deflection of a uniformly-loaded cantilever beam is:  according to [this PDF](http://ruina.tam.cornell.edu/Courses/ME4735/Rand4770Vibrations/BeamFormulas.pdf). ---- Found this paper which covers a number of the cases I'm interested in: [Analysis of the shape of a sheet of paper when two opposite edges are joined](http://sci-toys.com/bent_paper_problem.pdf). ---- Some different cases covered by the above paper: To be clear I'm talking about this sort of thing (particularly when edges first touch but also before and after) yfrog.us/15k37xtqwcjumq…
— James Tauber (@jtauber) March 1, 2013