Developable surface — where it appears
Named by 12 essays across 7 fields — each of them below, with the objects they name alongside it.
The wall under the paint
An anamorph's design is a set of rays from a known eye, fixed before any paint is applied — so the marks are a calibration target whose rays are known exactly and whose shape is not. One ordinary photograph supplies the second ray for each mark and the wall comes back at the arithmetic floor, on a dished floor, a ridged one, and a floor with a step in it. What it says between the marks is nothing, by the sagitta.
The pond with its trees laid flat
An Egyptian garden pond is drawn in plan with its trees rotated outward about the bank they stand on. A rotation is an isometry, so every length in that drawing is exactly the length it is in the garden — zero error, not a small one. What is spent is the angle between any two faces, which reads 180° across every hinge and is 90° in the garden, and the four walls admit sixteen assemblies, so a reader supplies four bits to fold it back up.
The height a flat floor cannot give
The marks of a floor anamorph name the eye's position on the floor exactly and say nothing about how high it was — every candidate height explains them perfectly, to one part in a thousand trillion. That is a fact about planes rather than about anamorphs. Ripple the floor by six centimetres and the family collapses: the true height explains the marks exactly and the nearest wrong one, five centimetres away, leaves two millimetres on a design 1.8 metres wide.
A stair does not use all its faces
A corner anamorph is two homologies and a flight of nine steps is eighteen, which is arithmetic and is the least of it. What a flight has that a corner does not is that which faces exist and which faces can be painted are different questions. From the top of a descending flight not one riser is reachable at any eye height, so half the planes are unpaintable by construction — and from the bottom of an ascending one, 58 per cent of the picture lands on risers that are 36 per cent of the surface.
The residual has a shape
A flat-floor map mispredicts a shadow by millimetres on any floor that is not flat, and the number everybody quotes is the worst one. Tune a dish, a ridge and a step until all three mispredict by exactly 25.0 millimetres and the scalar can no longer tell them apart — by construction. The signed residual around the ring still can. The second harmonic of it reads 0.09%, 2.79% and 19.37%, a factor of two hundred across three floors the headline number calls identical.
No view draws a curved plate true
The auxiliary view is descriptive geometry's answer to a foreshortened plane — turn until the plane is parallel to the paper and it draws at true shape. A bent plate has no such direction and a dished one has none twice over: the best view of the first is out by 1 − cos w and the best of the second by 1 − cos²w, worse by exactly 1 + cos w, because its normals need two parameters rather than one.
The floor is a choice of coordinates
Four rungs of this field have measured what a curved floor costs a shadow reading, in millimetres. It costs nothing. A shadow mark sits on the ray from the lamp through the occluder's edge, and where along that ray the floor caught it is a fact about the floor alone — so un-casting in rays returns the occluder to three parts in ten thousand of a millionth of a millimetre on a plane, a dish, a ridge and a step alike, while the same shadow read in the plan is wrong by up to 216 millimetres.
The drawing and the development
A bent plate gets two flat pictures on the same sheet and each is exact in what the other loses. The parallel drawing keeps the generators at one scale and stretches the arc over a factor; the development keeps every length on the surface and keeps nothing of the shape in space. Neither is the plate and the pair of them is.
The corners a floor cannot add
A wire with no corner anywhere on it casts a shadow with one, wherever its tangent runs along the ray. That condition contains the lamp and the wire and no surface at all — so the same helix over a plane, a dish and a ridge draws shadows that differ by metres and each has exactly one corner. A floor with a crease draws five more, and the two kinds are separable by a hundredfold: a real corner is where the shadow stops dead, and a crease is where it turns at full speed.
A picture that can be printed
Two screens are fed the surface that is exactly right for each, so nothing about the viewer is left in the answer. What remains is whether the picture can be made flat before it goes up — and a cylinder unrolls while a sphere does not, so the dome's picture is stretched by eighteen per cent between its middle and its rim before anybody sits down.
A floor is read along curves
Whatever a shadow says about the floor it landed on, it says only where the shadow is — and a shadow is a curve while a floor is a surface. Shadow curve length grows exactly linearly in the number of lamps, by a fitted exponent of 0.999, and the fraction of floor within two centimetres of one grows more slowly at 0.94, because the curves begin to overlap. At thirty-two lamps, seventy-one per cent of a nine square metre patch has still never had a shadow on it.
A curved surface made of straight lines
A hyperboloid of one sheet carries two families of exactly straight lines, so a photograph of a cooling tower is full of straight lines bounding nothing flat — 28 of them here, drawn to 2.3e-13 px of straightness against a parallel of the same surface that bows 120 pixels. Both families' directions satisfy one asymptotic equation, so their vanishing points lie on one conic in the picture and not on two, to 1.6e-12 px with no fitting anywhere.
Named alongside it
The objects these essays reach for when they reach for this one.
Gaussian curvatureReceiving surfaceDemonstrationHomologyPlan viewReconstructionResidualShadow projectionAnamorphosisConditioningDevelopmentOrthographic