Explaining the five cases of elastic bending

At the moment, this is how I would explain the geometry of the 5 cases of elastic bending (see previous post):

Case #1 This one follows the elastica curve, which means the curvature varies with the sin of distance along the curve (explanation here). The curve equals half a cycle of Sin (180 degrees) which means the curvature will be zero at start point and endpoint.

[EDIT 2010-06-13] This case probably involves the Cornu spiral (clothoid), see here and here.

Case #2 This is probably* a part of a clothoid curve (Cornu spiral). Curvature is maximum at the clamped end and zero at the loose end. There is a linear change in curvature in between. (A loose end cannot store any bending energy and the curvature there must be zero).
*(This could possibly be simply half an elastica curve, but I find that less likely).

Case #3 This is a circle (cylinder). Curvature is constant along the curve.

Case #4 This is a helix. Curvature is constant along the curve and there is also a constant twist. This could also be called a cylindrical plank line, which means it has the shape of a thin (straight) strip that has been wrapped around a cylinder.

Case #5 This is a conic plank line, which means it has the shape of a thin (straight) strip that has been wrapped around a cone.

Five cases of elastic bending

I have so far identified five cases of elastic bending in a thin strip:

Case #1
Two loose ends pushed together (2d).

Case #2

One clamped end, the loose end is pushed (2d).

 Case #3
Both ends clamped together to form a loop (2d).

Case #4Both ends clamped to form a loop, but with a distance sideways between endpoints (3d).

Case #5Both ends clamped and twisted to form a loop (3d).

Rubber band

I found this rubber band on the table looking just like this. The shape of the bending and twisting is similar to that of the saw blade. (Christmas tree in the background...)

Analysis of helix angle in loop surface

Grasshopper helped me plot the difference in angles between curve direction and curvature direction along the loop surface. Seems to start around 30° and reach 0° at surface mid point. It flips over to 180° and then decreases another 30° to 150° during the second half. Interesting! This would indicate a linear decrease/increase of the twist amount (measured as the helix angle). I think this pretty much solved the puzzle...

More helix testing

Some more helix testing in Grasshopper (no scripting yet as you may notice from the messy layout). The question is now: How to make a smooth transition between helixes with different angle?
EDIT: Download ghx-file.

Developable helix surface

The measurement for bending is obviously curvature, but how to measure the amount of twisting? The image above shows how the Curve Direction differs from the Curvature Direction in a developable helix surface. Perhaps this "Helix Angle" can be a useful measurent?