Dammit ..... They may be simple geometrical functions, but I am sure it would have been a LONG time before they were seen had I not proposed them... And how many people would have guessed the analogy which led to them...? All of the forum folks must be exceedingly modest to not have proposed them. There are a few people here with math ability. But they did not use it.Maybe comparing many curves in a sort of statistical manner could give some clues? The archings may have been corrected some times, but nontheless it would give a collection of data to extract best fitted curves from, e.g. similar to Johns simple geometrical functions, or more complex ones like splines, which are used in 3D modeling for their conveniece and efficienecy for that.

Splines say nothing at all.** They are just best-fit polynomials. There is no unusual property of indefinite differentiation etc.

The simple functions all have the general properties desired. As a class, they are a matched set in this regard.

** I suppose you could specify a set of perhaps 10 points along each half-arch and see how those points fluctuate in the different examples. Certainly, one would need to scale the heights to be equal. Or at least the average heights to be equal. It is very likely that shape and height scaling are not related. I think that should be assumed if you are going to do this statistical exercise. Don't ask me how I know, it is just obvious to me.