The ruling direction of the strip of paper coincide with the ruling direction of the cone. That is pretty obvious looking at the pic Tom posted. If you wrapped the strip around a cylinder only then would the ruling direction be perpendicular to the edges of the strip.
Yes that should also be obvious.
Itâs not obviously the case that the ruling directions of the true elastic solution exactly match those of the cone.
The issue with the radius is that when a longer segment of the ruling line lies on the surface, you effectively have a longer hinge which is more resistant to bending than a short one.
Itâs also true that a conical strip has a larger curvature radius further from the tip.
Whatâs not clear is how these two factors interact.
Then thereâs the issue of the overlapping square, which has double the bending resistance of the rest of the strip.
There have been a number of interesting attempts to develop 1D models of such elastic ribbons (instead of simulating them as plates or shells). Itâs trickier than it might first seem though.
Well that issue might come into play if you donât have a cone to wrap the strip around. The resistance to bending is irrelevant if you wrap it around a cone. Its going to bend only as much the cone allows.
I didnât think the stiffness of the material was specified? Is it going to be stiff glossy stock or tissue paper? If you use tissue paper it doesnât look like it wants to hold any particular shape. Gravity influences the shape more than the resistance to bending.
In theory, if we are talking about a hypothetical ideal material with no thickness then there is no resistance to bending no matter how long the ruling lines are.
