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?