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Deformation Lab
The generator bends space with a sum of four 3-D sinusoids. Slide the bound B up: the rainbow grid stretches; past B = 1 folds become possible and the label rule can break. Drag to orbit.
B = 0.75: builder's default ceiling · B = 1: proven fold-free · B = 2.5: pool of record (cycle-checked)
rainbow grid = material coordinate of the undeformed surfacered = det(I + ∂D/∂p) ≤ 0, the map folds here and a deformed point no longer has one material point
The field of record: D(p) = Σᵢ aᵢ sin(⟨fᵢ, p⟩ + φᵢ), i = 1..4, and V(p) = p + D(p). The slider is not a distance: it is the analytic Jacobian bound B = Σᵢ |aᵢ||fᵢ|, and B < 1 provably forbids folding. Rendered pools use B = 2.5 (43–70 px of object motion at 512 px) and certify every pair with an A→B→A round trip; the paper's sweep reports a median round-trip error of exactly 0.00 px at B = 0.75, 2.5, 4 and 6. Here the surface is displayed as p + D(p) with the same coefficient draw as the generator (fᵢ ~ 0.8–1.6, so the warp bends the whole object rather than rippling it).