Visual Plane
every picture states the point it is correct from
Where to stand

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.
18 min read 13 figures Determined up to somethingThe surface is a choiceThe round trip

Worth reading first: The marks name the place, not the height · The ceiling that is not a plane · The floor that is not a plane.

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.

The height a flat floor cannot tell youThe same design, the same eye, two floors. On the flat one every candidate height explains the marks to 1e-15 m, so the height is not in the marks at all. On a floor rippled by 6 cm the true height explains them exactly and the nearest wrong one leaves 2.0 mm.020406011.5022.503candidate height for the eye, in metreswhat the best rectangle leaves over, in millimetresa flat floor: every height fitsthe eye that made the markscurvature costs the collineationand buys the parameter back
Fig. 1 The same design, the same eye, two floors. On the flat one every candidate height explains the marks exactly, and the residual line lies along the bottom of the plot. On a floor rippled by six centimetres the residual has a sharp minimum at the height the marks were made from.

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. 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.

The eye, read back out of three numbersFour marks and the design's stated proportions, and the eye comes back 4.8e-12 mm from where it stood. Assume a different shape and the floor position is unchanged to 7.4e-11 mm while the height runs from 0.762 m to 2.592 m — the marks fix the product height × aspect, and it is 2.592.axis — the ground line, fixed pointwisecentreaxisthe ground linefixed pointwisecentre(0.150, 4.320)height + distanceratio-1.666667−distance / heightheight × aspect = 2.592, and neither aloneeye recovered to 2.2e-11 mmeye 1.62 m up, 2.70 m backthree numbers back to the eye: 5.0e-16 m
Fig. 2 The recovery on a plane, and what it does and does not return. The three numbers come out of the fitted homology; two of them give the eye’s position on the floor exactly, and the third gives a product rather than a height.
The height a flat floor cannot tell youThe same design, the same eye, two floors. On the flat one every candidate height explains the marks to 1e-15 m, so the height is not in the marks at all. On a floor rippled by 3 cm the true height explains them exactly and the nearest wrong one leaves 1.0 mm.010203011.5022.503candidate height for the eye, in metreswhat the best rectangle leaves over, in millimetresa flat floor: every height fitsthe eye that made the markscurvature costs the collineationand buys the parameter back
Fig. 3 Half the ripple. The residual at every wrong height is about half as large and the minimum is in the same place, which is what says the sharpness of the minimum is proportional to the departure from flatness.

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 the receiving surface costs a four-point fitThe same lamp and the same occluder, cast onto four surfaces. Four marks of the shadow are matched to four known points of the occluder, the map they determine exactly is built, and the other sixty-eight points are predicted by it. On the plane the prediction is right to 3e-13 mm; on the others it is not, and the fit is still exact at the four everywhere — 2e-13 mm — which is what makes the rest a prediction rather than a residual.a flat floor3e-13 mma homology — four points determine ita dished floor5.67 mmnot a homologya ridged floor9.07 mmnot a homologya floor with a step74.95 mmnot a homologyworst error of the predicted pointsexact at the four fitted, on every surface
Fig. 4 The neighbouring measurement, from the other direction. A shadow cast onto a floor that is not flat misses the best four-point fit, and the miss is what says the floor is not flat. Here the same departure is being used as information rather than reported as error.

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 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.

Projective, affine, metric — what each stage buysThe photograph fixes the plane only up to a projectivity: the midpoint of a receding side lands 0.3970 of the way along. Supplying the plane's vanishing line buys the midpoint back exactly and nothing else. Supplying the image of one circle buys the last three numbers, at which point the right angle is 90.000° and two equal sides measure 1.000000. The cross-ratio is 1.333333 in all three, because it was never lost.projectiveaffinemetricmidpointtwo equal sidesa right anglecross-ratioprojective———1.333333333affine0.500000——1.333333333metric0.5000001.00000090.000°1.333333333— means the stage does not determine it at allcross-ratio 1.333333 throughout
Fig. 5 The ladder the whole subject is climbing. Each rung buys back one class of quantity, and each is paid for by something in the scene that is not invariant under the group being removed. A rippled floor is an unusually cheap payment.
One reconstruction, drawn at its own scale and at 2.0×The courtyard recovered from two pictures, seen from above. Every pairwise distance ratio matches the world's to 6.3e-14, so the shape is exact. The size is not determined at all: the right-hand plan is the same reconstruction 2.0 times larger and fits the same two pictures equally well. One measured length in the scene — here 0.800 m — fixes it, and nothing in the pictures can.as recovered0.308 across× 2.0 — same two pictures0.615 acrossworst ratio deviation 6.3e-14size fixed only by a supplied 0.80 m
Fig. 6 The best-known member of the family. Two pictures fix everything about a scene except how big it is, and the missing parameter is the scale — which is a symmetry of the whole arrangement rather than of one surface, and is the one thing a ripple does not buy.

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.

One photograph of one floor, undone three waysThe design is 1800 mm across. Knowing the surface returns it exactly — nothing is fitted, so there is no residual to report beyond arithmetic. Assuming the floor is flat is exact at the four marks the homography was given and 111 mm out elsewhere. And knowing the shape but getting its curvature 10% wrong costs 11.0 mm, which is the price of the parameter rather than of the shape.what the recovery was toldworst error in the recovered designthe surface, known1.1e-12 mmassumed flat, four marks110.97 mm6e-13 mm at the fourthe surface, curvature 10% out11.00 mma a ridged floor, k = 0.06, design 1800 mm wide1e-12 mm · 111 mm · 11.0 mm
Fig. 7 The forward version of the same fact. A design painted on a curved floor comes back exactly when the floor is supplied and is far out when it is not, because the map is not a collineation and no four marks determine it.

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.

Gaussian curvature decides whether a floor can be unrolled at allA surface can be laid flat without stretching exactly when its Gaussian curvature is zero everywhere — Gauss's theorem, which says that quantity survives any bending. Three of these four floors have none, including the ridge, which curves visibly. The dish has 0.0144 per square metre, and no cleverness in the flattening removes it.floorGaussian curvature, worst over the patchcan it be unrolled?a flat floor0yes, exactlya ridged floor0yes, exactlya floor with a step0yes, exactlya dished floor0.0144by nothing whateverfour floors, k = 0.06three at zero, one at 0.0144 m⁻²
Fig. 8 And the distinction that has to be kept alongside it. A ripple is developable — it can be flattened with no strain, so the design can still be printed and rolled on. Being flat enough to print and being flat enough for a collineation are different conditions, and only the second is what the ambiguity depends on.
One design, two planes, one joinThe rays from the eye through a design 1.2 m tall, meeting a floor and a wall standing 3.2 m away. 14 of the 22 sampled points land on the floor and 8 on the wall, the two maps agree on the join exactly, and the design's scale jumps by 3.88 as it crosses.wallthe intended picturethe joineyecontinuous across the cornerand 3.88× the scale on one side
Fig. 9 The other way to break the same family. A second plane removes the ambiguity as effectively as a ripple does and is a great deal easier to arrange, because most floors have a wall at the end of them.

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 height a flat floor cannot tell youThe same design, the same eye, two floors. On the flat one every candidate height explains the marks to 1e-15 m, so the height is not in the marks at all. On a floor rippled by 12 cm the true height explains them exactly and the nearest wrong one leaves 3.4 mm.025507510011.5022.503candidate height for the eye, in metreswhat the best rectangle leaves over, in millimetresa flat floor: every height fitsthe eye that made the markscurvature costs the collineationand buys the parameter back
Fig. 10 Twice the ripple, and the same sweep. The residual at every wrong height is about twice as large, and the minimum at the true height is unchanged — which is what says the curve is measuring the eye rather than the method.

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.

A shadow cast onto a ridged floorThe section shows what the ray diagram is: straight lines from the lamp, through the occluder's plane, down to whatever is there to receive them. Nothing about the light or the occluder changes between the four surfaces — only where the rays stop. Fitting the shadow's map from four marks and predicting the other sixty-eight leaves 9.07 mm of error on this one, against 6e-14 mm at the four fitted marks.the lampthe occluder's planea ridged floora vertical section — the rays are straightfour points fitted · worst prediction 9.07 mm
Fig. 11 The general form of that warning. Only a flat receiver makes the map a homology; every other shape produces marks that a four-point fit explains at the four points and nowhere else, and a recovery told the wrong shape spends that freedom on a plausible wrong answer.
The design, the eye, and where the rays landA 0.90 m design standing on the ground line, an eye 1.65 m up and 3.20 m back, and the marks the rays leave on the floor. Above: the section, with the ray through the top of the design reaching 3.84 m away. Below: the marks themselves, in plan.floorpicture planeeye level — no mark above thiseye · 1.65 m up, 3.20 m backthe ground line, seen from abovethe mark runs to 3.84 ma point 1.65 m up casts no mark at all
Fig. 12 The arrangement all of this is about, drawn in section and in plan. The marks on the floor are what the recovery is handed, and everything in this essay is a question about what they contain.

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.

Two scenes 137× apart, and the one picture they both makeEverything in the second plan — the room, the eye's distance, the eye's own height — is 137 times the first. Every projected vertex agrees to 1e-13 px. A single photograph has no scale, and this is what that means.a room 2.8 m across, eye 1.6 m up1 mthe same plan, 137× bigger137 midenticalpicturesthe picture — both scenes, drawn twice, one on top of the otherlargest disagreement 1e-13 px over 8 verticesone length has to come from outside the picture
Fig. 13 The parameter no surface helps with. A scene and a scaled copy of it, photographed from correspondingly scaled positions, produce identical pictures — so scale is not something a single view has ever had, and no receiving surface changes that.

Where this goes next

axis — the ground line, fixed pointwisecentreaxisthe ground linefixed pointwisecentre(0.150, 4.020)height + distanceratio-1.481481−distance / heightheight × aspect = 2.592, and neither aloneeye recovered to 4.8e-12 mmeye 1.62 m up, 2.40 m backthree numbers back to the eye: 2.2e-16 m The marks name the place, not the height Run the site's own round trip on an anamorph — hand the machinery the floor marks and ask for the eye back — and it returns the spot on the floor to eleven decimal places with nothing assumed at all. It does not return the height. What the marks fix is the product of the eye's height with the design's aspect ratio, and no amount of looking at the floor separates the two. a flat floor3e-13 mma homology — four points determine ita dished floor5.67 mmnot a homologya ridged floor9.07 mmnot a homologya floor with a step74.95 mmnot a homologyworst error of the predicted pointsexact at the four fitted, on every surface The floor that is not a plane A shadow on a flat floor is a homology, so four marks determine the whole map and the rest of the outline comes back exactly. Dish the floor and the same four marks mispredict the rest by 5.67 mm; ridge it and 9.07 mm; put a step in it — two planes, each of them exactly a homology — and 74.95 mm. The receiver's shape is what breaks the projective description, and it breaks it worst where the surface is flattest. barrel vaulteye · 1.62 mon the floor1e-12 mmon the vault529.4 mmworst miss of the best homography, log scalevault radius 4.0 m7.7% of the extent, against 4e-14% The ceiling that is not a plane Paint the same design for the same eye onto a floor and onto a barrel vault, then fit the best possible homography to each set of marks. On the floor it misses by femtometres, because the map is a collineation and four marks determine every other. On the vault it misses by half a metre, and no choice of four marks helps — which is where the projective machinery this site has used since its first phase stops applying.

What links here

Computed from the collection, not written here: the essays that point at this one.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

AmbiguityAnamorphosisCollineationDegenerate familyDevelopable surfaceGaussian curvatureGround planeHomologyleast squaresResidual