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?

Bending Structure

I used mirrored segments of the cornu spiral to create these arches.
You can download the grasshopper file here:
http://omkrets.se/grasshopper/MN-002-bending-structure.zip

This script generates the cornu spiral.

Cornu spiral in Grasshopper

A test with construting a cornu spiral in Grasshopper.
Script written in VB.NET. Download .ghx-file below:

http://omkrets.se/grasshopper/MN-001-cornu-spiral.zip

Grasshopper

 I have used the Grasshopper plug-in for Rhino developed by David Rutten to do some more testing with creating the Cornu Spiral. First manually...

And later by using the Script function.

Generative components

 With GenerativeComponents (GC) I tried to mimic bending in 3d.

This shows an attempt to generate cornu spirals in GC. Not quite right - but beautiful!

Here I used scripting in GC to get a twisting strip.

Bending in 3D

 So far, most examples have been 2d-curves, but what happens when you bend in 3 dimensions.

I made a 3d loop with the saw blade and scanned it with a Microscribe 3d digitizer. Download 3dm-file here.

Cornu spirals in MicroStation

The CAD-program MicroStation has a command for clothoids (cornu spirals) that I used to draw this figure.

When I combined these curves with the ones I had traced by hand, the match was very good!

The traced curves from the saw blade were double (mirrored) clothoid segments.

Finding the Cornu Spiral

I involved a friend (Ola J.) who helped me searching. He found an article by B.K.P. Horn that was called “The Curve of Least Energy”, which contained some pages about a spiral called the “Cornu Spiral” or “Clothoid”.

This spiral is very useful when designing roads, rails and roller-coasters. Segments of the spiral are used as transitions between for example straight and curved parts and give the smoothest possible changes in curvature.

Arranging curves

I digitized these curves and used a CAD-program to arrange them in different ways in order to find some kind of clue to the geometry. The curves could roughly be represented as NURBS-curves with two control points in the application Rhinoceros (Rhino).

 Arranged in this way, the endpoints seemed to lie along two circles with the diameter 4/5 of the curves length. The displacement between the circles was 1/4 of the diameter.

I measured angles and distances in order to understand, but could not see any connections.

Tracing a saw blade

A saw blade, 90 centimetres (3 feet) long, was traced in different positions.

The result was 35 curves with equal length but different distance between the endpoints.

First steps

In order to answer these questions I started tracing strips of plywood. If the endpoints touched each other, there was almost a 90 degree angle and a drop-like shape.

 Here, a strip of paper was bent and traced several times. The anchor points are fixed, but the curve length vary.

I scanned the curves and investigated them in a CAD-program. They were obviously not arcs.

Trying to understand the geometry of bending

When you bend a thin strip of an elastic material you get a beautifully shaped curve. What geometry does this curve follow? Can the curve be calculated if you know the length of the material and the position of the end points? Is it possible to calculate more complex situations with several forces in different directions? Can you make similar calculations in 3d? Can this geometry be useful in design/production?