Trying to reconcile the top layer with projected geometry with the bottom layer. This is negotiated here with triangles. Then to determine the relationship they have with each other. After working with it for a little while… need to have the rhomboids on top and the kites and darts on bottom. There are too many pieces below for the number above currently.
If the centerline of our vault is a normal catenary curve then we could have the top surface as kites and darts and a lower surface as rhomboids. Then we use evolute to optimize the top and bottom surface to match as closely as possible to the centerline as representative of a normal catenary vault. If it bow ties inward to the center line, we have the beginning of the shape we need for the interconnecting parts.
So we find the set of vertices for the top and bottom surfaces, then we can examine them in sections through the vault, and optimize each back to the catenary curve allowing the curve to vary as it extrudes through the surface. Nice as it gives us a much simpler algorithm to test against (catenary curve) definition, yes?