Pottery meets Mathematics

Two of the things I really like in life come together. A type of construction I have used in the past is to make N symmetric, identical slab pieces (with N being 3 or 4 so far) and form them into a bowl. A recent example is show below (with N=4) and I think it turned out rather well.

    

What I thought would be fun is to figure out what the resulting form shape would be from a given piece shape and the "N". It was a fun bit of math shown below. There is another way you can derive it using rotations of curve tangents but this derivation is more my style.

I implemented a simple approximation in MATLAB and plotted out a few examples. The first is with N=4 and f(s) = sqrt(1+s) - 1 with s going from 1 to 4: 

and the second with f(s) = sqrt(1+2*s) -1:

You can see that the second has a flatter bottom -- completely flat at the centre since the derivative of f at s=0 is 1 (matching the N=4 case of 45 degrees). Note that you don't need to start the piece at s=0. You can start with s>0 and add a flat bottom (or anything else you want to do there). 

I think I need better visualization of the resulting surfaces to be able to use this as a tool to help pick aesthetically pleasing shapes. If I got really involved, I could generate the shapes for (small scale) 3D printing as a way to explore designs.