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.
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.
(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: