A fold names the height

The marks name the place, not the height establishes the anamorph field’s sharpest ambiguity. Given the marks a design made on a flat floor, the reader’s position on the floor comes back exactly, with no assumption about the design at all — and their height comes back only in the product of height and the design’s aspect. Raise the assumed eye and the reconstructed picture is taller and exactly as rectangular, so every height explains the marks and nothing chooses between them.

The height a flat floor cannot give then shows what breaks it: six centimetres of corrugation, and the height comes back.

Between a plane and a corrugation there is a case neither essay covers, and it is the one most real floors are.

A crease is not a curvature #

A corrugated floor has curvature everywhere. That is what breaks the projective family: a wrong height needs the intended picture to have been curved, and it was not.

A crease has curvature nowhere, except on one line where it is infinite. Two flat half-planes meeting at an angle; every point of the surface is on a plane; and each half separately leaves the height exactly as free as a full plane does.

So nothing in the earlier argument settles it. The corrugation’s mechanism — a smoothly varying surface that no scaling of the picture can match — is absent, and the question is whether a single line’s worth of departure is enough.

It is, and by a lot #

Fold the far half of the floor up by two degrees — a slope of three centimetres in a metre, which is a floor a builder would call level — and a height wrong by ten centimetres leaves six tenths of a millimetre unexplained on a design 1.4 metres wide.

At ten degrees it is 2.35 millimetres. At thirty, 4.9. At ninety — a wall rather than a floor — 9.3.

And at zero it is $1.5\times10^{-16}$ metres, which is nothing: the flat case is exactly free, as it must be.

So one crease is enough, and it is not marginally enough. Two degrees of fold turns a completely undetermined parameter into one that a reader measuring marks to a millimetre can pin to about a fifth of a metre — and ten degrees, which is a wheelchair ramp, pins it to four centimetres.

The first degree buys most of it #

The shape of the curve is the second finding and it is the more useful one.

Going from flat to two degrees buys 0.58 millimetres. Going from two degrees to ninety buys another 8.7 — sixteen times as much, for forty-five times the angle. The curve is steep at the start and nearly flat by the end.

What that says is that the quantity doing the work is that the floor is not a plane, rather than how far from one it is. The first departure breaks the family; further departures only sharpen an answer that already exists.

That is a familiar shape in this collection and it is worth naming the resemblance. A second view buys a reconstruction that one view cannot give at all, and a third view buys much less than the second did. One conic calibrates a camera and a second refines it. The pattern is that identifiability is a threshold and conditioning is a slope, and this measurement separates them cleanly: the threshold is at zero degrees and everything after is slope.

Linear in the error it is asked to find #

The residual is exactly proportional to how wrong the height is: a two-centimetre error at ten degrees of fold leaves 0.47 millimetres, a ten-centimetre error leaves 2.35, and a forty-centimetre error leaves 9.40 — a factor of twenty for a factor of twenty.

That linearity is what makes the numbers above into a conditioning rather than a detection threshold. At ten degrees of fold the recovery leaves 0.235 millimetres per centimetre of height error, so a reader who can measure the marks to a fifth of a millimetre knows the eye’s height to about a centimetre; at ninety degrees it is 0.93 millimetres per centimetre and the same reader knows it to two millimetres.

Quoting a conditioning rather than a threshold is the honest form for a measurement of this kind, and this collection’s ladder of conditioning makes the general argument: a conditioning is what lets a reader compute their own answer from their own measurement precision, and a threshold is what somebody else’s precision produced.

Why the crease works, in one picture #

The mechanism is visible in section and does not need the algebra.

A mark on the near half of the floor is on the ground plane, and the family of eyes that explain it is the family the flat case has: raise the eye and the reconstructed design point rises in proportion, so the picture is taller and just as rectangular.

A mark on the far half is not on the ground plane — it is lifted off it by however far up the fold it landed — and the ray from a raised eye back to that mark reaches the picture plane at a height that is not simply proportional. The two families disagree, and the disagreement is the residual.

So what a crease provides is a set of marks at a known height above the plane the rest of them are on, and one such set is enough. That is the same ingredient a corrugation provides continuously and a plane provides not at all — and it is why the essay’s title is about the fold rather than about the curvature.

The lift, and where the knee is #

The mechanism can be turned into the curve. A ray that would have landed at $q$ on the flat floor meets a plane hinged at $p$ and tilted by $\theta$ at horizontal distance $x$ beyond the fold, where the ray’s descent and the plane’s rise are equal:

$$x\,(\tan\theta + s) \;=\; (q-p)\,s, \qquad \text{lift} \;=\; x\tan\theta \;=\; (q-p)\,\frac{s\tan\theta}{s + \tan\theta},$$

with $s = e_y/(q + e_z)$ the slope at which the ray comes down — about 0.40 for a mark two metres out from an eye 1.62 m up and 2.4 m back.

Both of the essay’s findings are in that expression.

The knee is where the floor rises as fast as the ray descends. The lift is half its ceiling at $\tan\theta = s$, which here is 22°, and it approaches $(q-p)\,s$ and stops. So the sequence 0.58, 2.35, 4.9, 9.3 millimetres over 2°, 10°, 30° and 90° is a saturating curve rather than a slowing one: past about twenty degrees the marks are climbing the fold rather than being displaced along it, and further tilt moves them very little. A wall, at 90°, is worth only twice a thirty-degree ramp.

And the value is proportional to how much design lies beyond the crease. The lift carries $(q-p)$ as a factor, so moving the fold away from the reader reduces the residual linearly and takes it to exactly zero when the crease passes the last mark — which is the sweep’s 4.6, 3.3, 2.4 and 1.0 millimetres at 0.4, 0.8, 1.2 and 2.0 metres, extrapolating to nothing at about 2.4 m, where this design’s marks end.

Read together the two say where to look on a real floor. A shallow fold close to the reader is worth more than a steep one far away, because the angle saturates and the position does not. A doorway threshold under the near half of a pavement design is a better witness than a step at the far end of it, and a two-degree fall across the middle of the marks is worth more than a wall behind them.

That is an unusually cheerful result for a recovery. The ingredient it needs is the most common defect a floor has, it needs very little of it, and it needs it in the place a floor is most likely to have it.

Where the crease has to be #

A fold is only worth anything where the design lands, and the sweep that says so has an exact zero at one end of it.

Move the crease away from the reader while keeping the design and the eye where they are. At 0.4 metres the fold buys the most — a ten-degree fold leaves 4.6 millimetres for a ten-centimetre height error — and the number falls steadily: 3.3 at 0.8 metres, 2.4 at 1.2, 1.0 at 2.0.

At three metres the residual is exactly zero, at every fold angle tried, because the design’s marks all land on the near half and the crease is behind them. A surface that is folded somewhere the design never reaches is, for this purpose, a plane.

That gives the measurement its second control, and it is a better one than the flat floor because the surface is genuinely not planar and the answer is genuinely nothing. What determines the height is not the floor’s shape but the shape of the part the marks are on — which is the same statement the shadow field makes about what a floor can be read from: the marks are where the rays landed, so a surface is measured only along the curves a picture happened to touch.

The consequence for a real recovery is a checklist item rather than a caveat: before claiming a height from a fold, check that some of the marks are on the far side of it.

What the crease does not buy #

The fold fixes the height. It leaves everything else exactly where the flat case left it, and the list is worth being explicit about because a reader might reasonably expect more.

The place on the floor was never in doubt. The flat-floor recovery already returns the reader’s position on the ground exactly, with no assumption about the design at all, and the crease adds nothing to it.

The design’s own proportions are still a separate assumption. On a plane the recovery fixes the product of height and aspect, so stating either gives the other. With a fold the height comes back on its own — which means the aspect now comes back on its own as well, and that is a genuine gain the previous paragraphs understate: a folded floor determines what shape the intended picture was, and a flat one cannot.

And nothing about the picture’s content is recovered. The marks determine the eye and the design’s frame; what was drawn inside that frame is whatever was drawn.

Which floors are flat #

The practical consequence is a reversal of what a reader would expect, and it is worth stating plainly.

The flat floor is the special case. A pavement is flat; a floor with a kerb, a threshold, a step, a ramp, a change of level between two rooms, or a slight fall for drainage is not, and any one of those is a crease.

Since two degrees is enough — and a fall for drainage is one in eighty, which is 0.7 degrees, still leaving two tenths of a millimetre at a ten-centimetre error — most real anamorphs are made on surfaces that determine their own design eye completely. The ambiguity the field has been carefully reporting for three rungs is a property of an idealised surface that real buildings mostly do not have.

That is a good outcome for anybody trying to recover an eye from a photograph of a mural, and a slightly awkward one for the field’s own worked examples, which have all been on ideal planes.

What the measurement had to be careful about #

Two pieces of care were needed and both were forced by getting it wrong first.

The fold is written by its normal rather than by its slope. A slope needs a tangent, and the measurement runs to a right-angled fold where the tangent is $1.6\times10^{16}$ — at which point every ray that should meet the far half misses it, no marks land there, and the recovery reports that a right angle says nothing about the height. Which is the opposite of the truth, and it reads exactly like a finding.

And the residual is measured against the design that made the marks, scaled by whatever the candidate height implies, rather than against a rectangle fitted to the reconstruction. Fitting a rectangle would absorb part of the disagreement into the fit — the same confound the four-and-predict rule exists to avoid — and would report a smaller residual for a reason that has nothing to do with the floor.

Both are the kind of error that produces a well-formed number, which is why this collection’s habit is to require a control that must fail. Here the control is the flat floor, which has to come back at exactly zero and does.

The same shape, three fields over #

The result has a form this collection keeps arriving at, and naming the form is worth more than the result.

A parameter that is free under one arrangement and determined under a slightly different one, with the difference costing almost nothing. A flat floor leaves the height free; two degrees of fold fixes it. A single view of a plane leaves its metric structure free; one conic in the scene fixes the camera. A pinhole leaves every size free; a ruler in the room fixes them all.

In each case the free parameter is not nearly determined by the ideal arrangement and then better determined by the improved one. It is exactly free, and then it is determined — a discontinuity in what is knowable, produced by a continuous change in the scene.

That is the practical reason to measure identifiability separately from precision, and it is why this rung’s figures put an exact zero on the same axes as a millimetre. The zero is not a small number; it is a different kind of statement, and a plot that cannot show the difference between “nothing” and “not much” is hiding the only interesting thing in the sweep.

What to measure on a real mural #

The result is usable on a photograph of an existing anamorph, and the steps are short enough to list.

Find the crease in the picture. A change of level, a threshold, a kerb: the line where the surface’s slope changes. This is the one ingredient the flat-floor recovery does not need and this one cannot do without.

Measure the fold. Not precisely — the conditioning curve above is flat past about twenty degrees, so an estimate to five degrees is enough for anything past that, and the shallow end is where precision pays.

Check that marks land on both sides of it. A crease behind the design’s last mark contributes exactly nothing, which the hinge sweep above measures as a hard zero rather than as a small number.

Then recover. The eye’s place on the floor comes from the near half alone, exactly as it always has; the height comes from the disagreement between the two halves; and the design’s own proportions come out with the height rather than being assumed.

The one thing to be careful about is the same one every measurement in this field warns of: the recovery is only as good as the assumption that the marks were made from a single eye. A mural painted by eye rather than projected — and most historical ones were — has marks that no single eye explains exactly, and the residual then contains the painter’s own errors along with everything else.

The short version #

A flat floor leaves an anamorph’s eye height entirely free, and it is the only surface that does.

One crease is enough to fix it: two degrees of fold turns a ten-centimetre error in the height into six tenths of a millimetre of unexplained mark, and ninety degrees turns it into nine millimetres. The curve is steep at the start and flat at the end, because what does the work is that the floor is not a plane rather than how far from one it is.

The residual is exactly linear in the error, so the result is a conditioning a reader can apply to their own measurements: 0.235 millimetres per centimetre at ten degrees of fold, 0.93 at ninety.