Analyzing the Drop shape

The characteristic Drop shape [that appears when the ends of the saw blade touch each other] turned out to look rather different in the elastica version and the cornu spiral version. The latter is closer to the saw blade. With the help of grasshopper I plotted the curvature of the three curves above. The curvature of the elastica curve and the cornu spiral curve look as expected, but the curvature of the saw blade is a bit ambiguous and noisy. I probably need to redo the measurements.


Cornu Spiral curves scripted in Grasshopper

I also tried scripting the Cornu Spiral curves (described earlier) in Grasshopper VB:

When superimposed with the saw blade curves it shows a much better fit, even though the overlap is not 100%:

(I remember that the last drop shape was a bit hard to orient correctly when drawing it, because it had both supports in one point.)
Please also note that my scripting skills are very basic, so I may very well have made errors when producing the elastica curves and cornu spiral curves.


Elastica curves scripted in Grasshopper

I tried some Grasshopper scripting to produce elastica curves.
(The script is really basic, it uses Loop to copy and rotate a line a bunch of times. The Sin values between 0 and 1 Radians are used to vary the amount of rotation. To get many curves at the same time a range of values are fed into the VB script component.)

Below are the curves that I drew manually along the saw blade:

To my surprise it's quite a bad fit as can be seen in the image below, showing both sets of curves superimposed:

The typical "drop shape" (that appears when the ends of the curve meet) seems too wide and low. Also curves seem to go in the wrong direction (see at left arrow).