The height a flat floor cannot give

Hand the machinery the floor marks of an anamorph and ask for the eye back. It returns the spot on the floor to eleven decimal places, with nothing assumed about the design at all. It does not return the height.

The marks name the place, not the height establishes that, and reads it as a fact about anamorphs. It is a fact about planes, and the difference is worth an essay because the repair is available and cheap.

Why a plane cannot say #

The reason is short and it is the same one that makes the whole subject tractable.

The map from an upright design to a flat floor is a planar homology, and its three numbers are the eye. Raising the eye by a small amount changes one of them — the ratio — and the effect on the reconstruction is to multiply the intended picture’s height and leave its width alone.

So for any candidate height there is a design, of a different aspect ratio, that produces exactly these marks. Every member of that family is a perfectly good rectangle. Nothing in the marks prefers one.

What the marks fix is the eye’s position on the floor and the product of the eye’s height with the design’s aspect ratio. Two unknowns, one equation.

The test that says so #

That is an argument, and this collection prefers a measurement.

Take the marks — cast as the forward construction casts them. For each candidate height along the vertical through the true eye, run the rays backwards to the picture plane and see what design that eye would have had to intend. Then ask how far that design is from being a rectangular grid — allowing a rotation, a translation and two independent scales, so that any aspect ratio is free.

Two scales rather than one is the point of the test. Allowing a full affine map would forgive a shear, and a shear is one of the distortions under examination; allowing a single scale would forbid the aspect change that is the whole ambiguity. Two axis-aligned scales is exactly the freedom the family has.

On a flat floor the residual is one part in a thousand trillion of a metre at every candidate height — a flat line, which is what a perfect ambiguity looks like when it is drawn.

What six centimetres of ripple does #

Now put a corrugation in the floor: a sinusoidal ripple six centimetres in amplitude with a period of 1.6 metres, running across the design. A pavement with a camber, a floor of boards, a cobbled street.

The marks are now made on a surface that is not a plane, so the map is not a collineation, and the family that worked before does not.

The residual at the true height is still zero — it must be, since that eye made the marks. At every other height it is not. Five centimetres away it is 1.97 millimetres; fifteen centimetres away, 5.9; half a metre away, 25.6.

Two millimetres on a design 1.8 metres wide is one part in nine hundred, which is a large residual by the standards of anything else in this collection. And it is a residual against a fit that has been given every freedom the ambiguity ever used.

So six centimetres of ripple names the eye’s height to within a few centimetres, and the flat floor names it not at all.

What “explains the marks” is being asked of #

The test above deserves one more paragraph, because the phrase does a lot of work and it could mean several things.

It does not mean “reproduces the marks”. Every candidate eye reproduces the marks trivially, by definition of what a candidate is: given an eye and the marks, there is always some design that would have produced them, found by running the rays backwards. The question is never whether a design exists; it is whether that design is the kind of thing a designer would have drawn.

Here the criterion is “a rectangular grid of some aspect ratio”, which is a strong assumption and an honest one. It is strong because a real anamorph’s design is not a grid; it is a dragon or a waterfall. It is honest because any criterion at all has to be assumed — the marks alone cannot distinguish an eye that produced a grid from an eye that produced a wave, and a recovery that claimed otherwise would be smuggling a prior in.

For a real design the same test runs with the design supplied: cast the known design from each candidate eye and compare with the marks. That is a cleaner question and it is not the interesting one, because knowing the design is most of what a recovery is for.

The grid is therefore standing in for “a design with structure a viewer would recognise”, and the measurement should be read as saying that a rippled floor lets the eye’s height be pinned down given any such assumption, while a flat floor does not let it be pinned down given any assumption at all. The second half is the stronger claim and it is the one that does not depend on the choice of criterion.

The flat floor is a limit, not a case #

The two results — nothing on a plane, a few centimetres on a ripple — read as two regimes and they are one expression with the amplitude set to zero.

A ripple raises the floor by $h$ at a mark, so the ray that would have landed at depth $d$ lands short by $h/\tan\varphi = h\,(d + e_z)/e_y$, where $\varphi$ is the ray’s elevation. That displacement is a fact about the marks. A candidate eye $\delta$ higher reads it back with a slightly different magnification, so what survives the fit is

$$\text{residual} \;\sim\; \frac{A\,\delta\,(d + e_z)}{e_y^{2}},$$

linear in the ripple’s amplitude $A$ and linear in the height error, which is exactly the pair of behaviours the figures show: three times the offset gives three times the residual, and half the ripple gives half of it at every offset.

Setting $A = 0$ gives zero at every $\delta$. The flat floor’s flat line is not a different phenomenon; it is this expression at the one amplitude where it vanishes, which is why the ambiguity is exact rather than merely severe — there is no small residual to be teased out with a better fit.

Inverted, the expression gives the resolution. If marks can be located to $\tau$, the height is pinned to

$$\delta \;\sim\; \frac{\tau\,e_y^{2}}{A\,(d + e_z)},$$

which for two millimetres of marking accuracy on this arrangement is about five centimetres — the “few centimetres” the measurement reports, and inversely proportional to the ripple. Naming the height to a centimetre would need thirty centimetres of corrugation, or a design five times as long.

Two qualifications follow from the same expression, and both matter for anybody hoping to use a real floor.

A sloping floor is as useless as a level one. What the recovery needs is departure from any plane, not from the horizontal, because a tilted plane is still a plane and the homology argument is unaffected by which plane it is. A pavement with a camber running uniformly to a gutter supplies nothing; the same pavement with the gutter’s curve across the design supplies a great deal.

And the period has to be short enough. A ripple whose wavelength is long compared with the design’s own footprint is locally a plane over that footprint, so what counts is not the amplitude but the sagitta over the marks — the departure from the best-fitting plane across the region the design occupies, which for a long-wavelength ripple falls as the square of the ratio of footprint to period. That is the same quantity the developable surfaces measure as an obstruction and the tolerance on knowing the floor measures as an error, doing a third job here as a source of information.

Which gives the practical reading. A cobbled street, a floor of boards, a tiled surface with a fall across it, or a fold all supply sagitta at a scale comparable with the design and therefore name the eye. A concrete slab, a smooth ramp and a level pavement supply none and name nothing. The difference is not how rough the floor looks; it is how much it departs from a plane over the few metres the marks actually cover.

There is a design consequence too, and it runs against the grain of everything else in this field. A larger design is better for the recovery, because the footprint grows and with it the sagitta the marks span — where for every other question in the anamorph field a larger design costs floor, costs stretch and costs the reader’s standing room. So the arrangement that is hardest to paint is the one that best records who painted it, and the smooth pavement a painter would choose is the one that keeps the secret.

The same ambiguity in three other places #

The plane’s one-parameter blind spot is not peculiar to anamorphs, and the family is worth naming because the repair is the same each time.

A single photograph of a plane. Rectifying a façade from four points recovers the plane’s shape up to a similarity and gives no absolute size; a metre stick in the picture supplies the missing parameter, and nothing else does.

A pair of views. Two pictures give the scene’s shape and not its size — the whole scene and the whole baseline can be scaled together and nothing changes. One known length fixes it.

A camera and its own image. A photograph of a photograph is a photograph, so the chain of projections that produced a picture cannot be read off it.

In each case the free parameter is exactly the group of transformations the arrangement is invariant under, and in each case it is bought back by something that is not invariant under that group — a known length, a known angle, a surface with curvature. That is the general shape and it is a good deal more useful than the individual results.

What is worth adding here is that the cure is sometimes free. A metre stick has to be brought and placed. A ripple in the floor is already there, and costs nothing but the effort of measuring it.

Why curvature breaks the family #

The mechanism is worth stating because it generalises.

The ambiguity on a plane exists because the family of eyes that explain the marks is a family of projective maps, and the design is being tested against a projective criterion — is it a rectangle. A collineation of the plane has enough freedom to turn one rectangle into another, so the test cannot choose.

A curved surface makes the map non-projective. Raising the eye now changes the marks in a way that is not a collineation of anything: the rays hit the ripple at different places, at different angles, and the pattern that comes back is not a rectangle at all — it is a rectangle with a wave in it, and no rotation, translation or pair of scales removes a wave.

So the curvature is not adding information about the eye directly. It is removing a symmetry, and what is left over is the height.

That is the same mechanism as the corner in the anamorph that crosses a corner, where a second plane breaks the same family. Curvature and a second plane are two ways of doing the same thing, and both work because the plane’s own family is exactly as large as its projective group.

The trade, stated in both directions #

This collection has recorded one half of this trade repeatedly and this is the first time it has had both.

Curvature costs the projective description. A collineation is available exactly while the receiving surface is flat. Curve it and four marks no longer determine the rest; the best homography fitted to the marks on a vault misses by half a metre, and no choice of four marks helps. That is the ceiling that is not a plane.

And curvature buys the parameter the projective description could not supply. The plane’s ambiguity is the plane’s projective group, and a surface with curvature does not have one.

Both sentences are about the same object and they are not in tension. What a plane offers is a description with four-mark determinacy and a one-parameter blind spot. What a curved surface offers is no compact description and no blind spot.

Which is worth having as a general shape, because it recurs. Anywhere a family of solutions is exactly a group of symmetries of the model, breaking the model breaks the family — and the thing that breaks it is usually something that was previously described as a nuisance.

How much curvature is enough #

The ripple used here is six centimetres over a period of 1.6 metres — a slope of a few degrees at its steepest, and something a person walking on it would barely notice.

The residual scales with the ripple: doubling the amplitude roughly doubles it, over the range tried. So the constraint on the eye’s height tightens in proportion to how uneven the floor is, and there is no threshold — any departure from flatness at all supplies some information, and a perfectly flat floor supplies none.

That is worth stating carefully, because it has a practical edge. A real pavement is never flat, so the height is in principle always recoverable from a real anamorph’s marks. What decides whether it is recoverable in practice is whether the floor’s shape is known well enough to be used, and a floor’s shape is exactly the thing nobody measures.

So the honest position is that the ambiguity is a property of the model rather than of the world. The world has no perfectly flat floors and therefore no perfect ambiguity; the model has a flat floor because a flat floor gives a collineation, and the collineation is what makes everything else in this subject computable.

The reverse direction, which is a warning #

There is a way to get this backwards that is worth naming, because the machinery here nearly did.

Given the marks and a wrongly assumed floor shape, the recovery still returns an answer. It returns the height that best explains the marks under the wrong assumption, and it returns it with a small residual, because the fit has freedom to spend.

So the height recovered from a rippled floor is only as good as the ripple is known. Assume a flat floor and the ambiguity comes back in full. Assume the wrong ripple and the answer is confidently wrong — and this collection has recorded exactly that failure elsewhere: an exact recovery has no residual and cannot report a wrong model, so marks must be held back or nothing can say the model was wrong.

The remedy here is the same one: fit the height on part of the design and check it on the rest. The check costs a few marks and it is the difference between a measurement and a number.

What is left ambiguous #

Even with the ripple, one thing is not recovered, and it is worth saying which.

The overall scale is still free, as it is everywhere in this collection: a design twice as large, an eye twice as far away and a floor twice as rippled produce the same marks scaled. That is the one thing a single view cannot give and no amount of curvature touches it, because scaling is not a symmetry of the plane in particular — it is a symmetry of the whole arrangement.

So the ledger for a rippled floor reads: position on the floor, exact; height, to a few centimetres; design aspect, determined; overall scale, free. For a flat floor it reads: position exact, height free, aspect free in the same one-parameter family, scale free.

One ripple converts two free parameters into one.