Sebastian Gonzalez (https://www.researchgate.net/profile/Sebastian-Gonzalez-93#:~:text=Violin making is perhaps one,in a traditional master-apprentice format.) made a presentation for the March 6, 2026 Acoustics Workshop topic in which he discusses the effects of the violin's arch.
He modeled his arch shape with the equation y= (1-x^2)^2 where y is the arch height scaled to equal 1 at the maximum height and x is the width of half of the bout width scaled to equal 1 at the maximum distance from the center line.
I plotted his equation and got a nice smooth arch shape with a curve reversal inflection point but the plot (red line) seems to be a more slender cross arch than often seen. So I tried several different exponents n for the equation y = (1-X^2)^n:
n= 2, 1.75, 1.5, 1.25 and 1
This made the curves more bulgy and moved the inflection point nearer the plate's channel.
I also tried a different equation y = (1-x^3)^n also with different exponents n and found the curves were even more bulgy and some of them from either groups seemed very similar to the wide variety of real life violins. And with some futzing or fooling around any violin shape can probably be generated mathematically.
This might suggest the old violin makers simply carved their plates to whatever they thought was right and didn't waste time with this stuff.