For a free-standing arch (domes too hard!) we can make it have uniform thickness and uniform compressive (along the curve) stress. Result curve is cycloid. Alternatively we can make it have uniform thickness and uniform bending stress, in fact zero bending stress. Result curve is inverted catenary y=cosh(x) type curve. Cannot do both at same time. But if allow nonuniform curve-thickness, then can do both by controlling both thickness and shape functions. --------- Now returning to the dome, the paper Henry Brady found seems to enforce uniform thickness and zero tensional stress round latitudes. But: nonuniform compressive stress along meridians. And is there also a third kind of stress - bending? It is a famous rigidity theorem of Pogorelov that the surface-metric of a convex body in 3D determines its shape uniquely. Therefore, unlike an arch, which can bend without distorting any arc-lengths, a dome cannot bend without distorting its surface metric. So I think the moral of that is, if the tensional and compressional stresses are handled by the dome's material without stretching & shrinking its surface metric, then the dome shape will stay fixed even if the material is totally flimsy against bending (assuming base attached to ground by immovable hinges). [Similarly, a catenary-shaped arch (or hanging chain) will stay fixed shape even if totally flimsy against bending, provided it does not stretch/shrink.] If when manufacturing the dome, you were to build in those compressional & tensional stresses so dome was already in correct shape when stressed, then you'd be happy. However, to accomplish this seems to require building the dome from top downward jacking it up as we go (!) -- or if built in usual bottom-up way, then applying artificial stress using cables as it is built (the cables supply the forces representing the unbuilt missing part of the dome). Far as I know, nobody does either in practical construction. I guess the practical thing to do would be to build it in slightly intentionally wrong shape, so that under stress it comes out right shape. As far as I know nobody does that in practice either. There is also yet another issue: instabilities such as "buckling." Trying to make your arch or dome maximally stable against such, might be a whole different design challenge (probably a good deal more difficult too). So these things are quite subtle.