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?


More elastica curves...

Dr. Christopher D. Rahn at the Pennsylvania State University has developed a mine-hunting vehicle with whiskers (!) that uses the "elastica equations" to identify objects. Read more here.
Below is an image of experimentally predicted whisker shapes:

Elastica curves!

I went back to read "The Curve of Least Energy" by B.K.P. Horn more carefully. He writes: "Unfortunately, the Cornu spiral is not optimal either...".
In this web page, prof. Albert K. Harris explains the Elastica curve: "If you compress a long thin metal rod, when it eventually kinks its shape will approximate one of the elastica. This is said to optimize the spatial distribution of bending stress..."
And finally, in "Non-linear Beam Analysis" Japaneese aircraft structures engineer Toshimi Taki uses Elastica curves to generate something very interesting:

This is the first time I've seen anything similar to my own diagram of bending curves. I'm really excited!

Maybe the Cornu Spiral isn't the answer to 2d-bending after all?