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
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
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.
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.
Some different cases covered by the above paper:
