How many pieces?
Lay a grid over a curved surface and each cell becomes a candidate panel. Coarse grids give few, large, affordable panels that visibly miss the surface; fine grids hug it at the cost of seams, joints and money. The error shrinks fast — quarter the deviation for every doubling — so the game is finding the coarsest grid the eye forgives.
Slide the density and watch the deviation fall.
When is a quad flat?
Triangles are always flat — three points define a plane. A quad's four corners have no such guarantee: lift one and the panel warps, and warped means expensive: cold-bent glass, twisted frames, gaskets that leak. The standard measure is the gap between the two diagonals; zero means planar, and fabricators live by fractions of a percent.
Drag the corners in 3D and watch the warp gauge.
Surfaces that panel themselves
Here's the theorem that shapes real roofs: slide one curve along another — a translational surface, height = f(u) + g(v) — and every quad in its grid comes out exactly planar, automatically, forever. The cross-terms cancel; the algebra does the fabrication. Morph the same grid toward a dome, and the flatness quietly dies.
Slide between the vault and the dome; the colours are warp.
Pushing panels flat
For surfaces that don't panel themselves, you optimize: project each quad onto its own best-fit plane, average the answers, repeat — relaxation again, aimed at flatness. And here's the honest part: it takes the dome a good way down and then plateaus, and it barely dents a twisted grid at all. The optimizer polishes; the choice of surface and grid decides.
Run the planarizer on each surface and read the gauge.
The facade studio
Your roof, your call. Pick the surface — the friendly vault, the plateau-bound dome, the stubborn twist — set the panel density, and choose triangles or quads. Colour the panels by warp and every fabrication problem lights up; run the planarizer and watch how far it can and can't take you. Orbit with a drag. The readout keeps score: panel count against worst warp — the two numbers every facade meeting is really about.