The room the eye may stand in

Everything written about anamorphosis says the picture is correct from one point. Nothing says how big the point is.

That is not a facetious question. A point has no size, and a person cannot stand at one; what a pavement painting actually offers is a region — the set of places an eye can be and still get a picture close enough to the intended one. Where the anamorph still works measured what a step away costs, which is the forward question. This is the reverse: fix what is acceptable and ask how far the eye may go.

The answer is a solid, and its shape is the useful part.

What is being measured #

The design here is a rectangular grid 1.8 metres wide and just over a metre tall, standing upright on a ground line, cast onto the floor from an eye 1.65 metres up and 3.2 metres back. That is a pavement painting of ordinary proportions seen by a person of ordinary height standing at an ordinary distance.

From a displaced eye the marks read back to a picture that is not the design. The measure of “wrong” used is the worst displacement of any point of the reconstructed picture from where it should be, in metres of the intended picture — so a ten-millimetre tolerance means no point of the design is more than a centimetre out of place, on a design 1.8 metres wide.

In each of ninety-six directions the eye is walked away from the design point until that tolerance is reached, and the distance is bisected to convergence. The ninety-six answers are the solid.

Two properties make that a legitimate way to get a solid rather than a sampling artefact, and both are checked rather than assumed. The error grows monotonically with displacement in every direction tested, so a single bisection per direction finds the whole boundary and there is no second crossing to miss. And the solid is star-shaped about the design eye, which follows from the monotonicity.

The numbers #

36 millimetres along the line of sight. 14 millimetres across it. Thirty-one cubic centimetres altogether.

Thirty-one cubic centimetres is about a large marble.

The long axis is the line of sight, to within the angular resolution of the sampling — which is what one would expect and is worth checking anyway. Moving toward or away from the design changes the reconstruction mostly by scaling it, and a scaled picture is nearly the same picture; moving across the line of sight shears it, and nothing forgives a shear.

The ratio is 2.56. That is a spindle rather than a ball and it is a mild spindle: the along-axis freedom is not enormously more than the across-axis freedom, which is worth saying because “the eye may move freely along the sight line” is a claim that gets made and it is only two and a half times true.

The reason it is only two and a half times true is worth spelling out, and it can be computed rather than described. The forward measurement gives both costs in closed form. A step $\delta$ across the sightline displaces a design point at height $y$ by $\delta\,y/e_y$, so its worst value is at the top of the design:

$$\delta\,\frac{y_{\max}}{e_y} \;=\; 0.64\,\delta$$

for an eye 1.65 m up and a design reaching 1.05.

A step along it does two things at once. It stretches the picture vertically by $(\delta/e_z)\,y(e_y-y)/e_y$ — a parabola, zero at the ground line and at eye level, largest halfway between — and it spreads the picture laterally by $|x|\,\delta\,y/e_y e_z$, which grows all the way to the design’s edge. Together,

$$\frac{\delta}{e_z}\cdot\frac{y}{e_y}\sqrt{(e_y-y)^{2} + x^{2}},$$

whose largest value on a design 1.8 m wide and 1.05 tall, cast from 3.2 m back, is at a far corner and comes to $0.215\,\delta$.

The predicted ratio is therefore $0.64/0.215 = 3.0$, against the 2.56 the search returns — the gap being the design’s exact extent and the resolution of the ninety-six directions, and the agreement close enough to say where the number comes from.

It comes from the lateral term. The vertical parabola alone would give a ratio near five, and it is the sideways spreading of the picture — the fact that walking toward a pavement painting fans its two sides outward as well as stretching it upward — that pulls the advantage back to three. So the received claim that the sightline is the free direction is right in its vertical half and wrong about the whole, and the correction is larger than the effect it corrects.

The design’s own proportions decide how large. The lateral term carries the design’s half-width and the vertical one does not, so a tall narrow design has an elongated region and a wide low one has a nearly round one. A pavement painting three metres wide and half a metre tall — which is what a great many of them are — has essentially no forgiving direction at all.

Ten times the tolerance is a thousand times the room #

The scaling is exact and it is the tidiest thing in the measurement.

At a one-millimetre tolerance the solid is 3.7 mm long and 1.4 mm across, with a volume of 0.03 cubic centimetres. At ten millimetres it is 36.5 mm and 14.3 mm, at 30.6 cubic centimetres. Every linear dimension is ten times larger and the volume is a thousand times larger, to three figures.

That is because the error is linear in the displacement over this range, which is a consequence of the map being a homology whose parameters depend smoothly on the eye. So the whole family of solids is one shape scaled, and a single measurement at one tolerance gives every other.

It also means the question “how big is the region” has no answer without a tolerance attached, and that the answer is proportional to whatever tolerance is chosen. A reader who wants the picture right to a millimetre gets a marble the size of a pinhead. A reader who will accept five centimetres of error on a 1.8-metre design gets something the size of a fist — and standing in the wrong place is what every reader of every ordinary picture is doing at a tolerance nobody states.

Which is the honest way to answer “how big is one point”: it is not a size, it is a rate.

The solid, predicted rather than searched #

The two coefficients turn the whole measurement into arithmetic, and it is worth doing because it says which of the arrangement’s numbers the room depends on.

Dividing the tolerance by each coefficient gives the allowance directly. At ten millimetres of picture error the across-allowance is $10/0.64 = 16$ mm and the along-allowance is $10/0.215 = 47$ mm, against the 14.3 and 36.5 the bisection returns. The closed form runs about a fifth high because it evaluates the error at the design’s extreme corner as though the tolerance were reached there for every direction of step, and the search does not have to agree with that. The volume comes out near 50 cubic centimetres against 31, which is the same fifth cubed.

What the expressions add is the dependence. Every allowance is the tolerance over a coefficient, and each coefficient carries the design’s own extent: the across one is $y_{\max}/e_y$ and the along one is built from $y_{\max}$ and the half-width over $e_y e_z$. So

a design half as tall doubles the room in every direction and multiplies the volume by eight, and a design half as wide leaves the across-allowance alone while enlarging the along one. The reader’s freedom is set by how ambitious the picture is, not by how carefully it was painted — and the tolerance attaches to the picture’s own size, so a smaller picture at the same proportions buys nothing at all.

That last point is the one worth carrying out of the arithmetic. Scaling the whole arrangement — design, eye height, distance — leaves every ratio unchanged and every allowance proportional, so the room is the same fraction of the room it always was. There is no way to buy a bigger standing region except by accepting a larger error or by drawing a less ambitious picture, which is the same trade the floor’s own arithmetic imposes on height, arriving here as a constraint on the reader instead of on the pavement.

What a pavement painting actually does about this #

A photograph of a street painting is taken from a marked spot, and the marked spot is a stripe on the pavement about the size of a pair of feet — a few hundred millimetres across. The measurement above says the eye may move fourteen millimetres.

Both are true and they are not in conflict, because they are answering different questions.

The tolerance a street painting is drawn to is not ten millimetres on a 1.8-metre design. Nobody looking at a painted dragon is checking whether its snout is within a centimetre of where it should be; what is being judged is whether the illusion holds — whether the thing reads as standing up — and that survives errors of many centimetres.

So the useful reading of the measurement is the rate, not the value. On this design, one centimetre of eye movement across the line of sight buys about seven millimetres of error in the picture. A viewer who will tolerate a hundred millimetres of error may stand within about 140 millimetres of the mark, sideways, which is roughly the width of the painted stripe. The stripe is honest.

The eye’s own height is the tight direction #

One asymmetry inside the solid is worth pulling out, because it contradicts what people do.

The across-axis directions are not all alike. Sideways movement and up-and-down movement cost the same amount per millimetre — where the anamorph still works measures that they are exactly equal, which is a consequence of the homology’s structure and not an approximation. But a person’s eye height is far less under their control than their sideways position: standing on tiptoe, leaning, or being a different person changes it by tens of millimetres without anyone noticing.

So the direction the solid is tightest in is the direction people vary most in, and it is the one nobody thinks about. A pavement painting photographed by a person 1.75 metres tall from a spot marked by a person 1.65 metres tall is a hundred millimetres out along the tight axis, which on this design is seven times the tolerance.

That is why street paintings are photographed rather than looked at, and why the mark on the pavement is often a mark for a camera on a tripod rather than for a pair of feet.

Where the tolerance should be measured #

There is a decision buried in the measurement that changes the answer by a factor of two or three, and it is worth putting on the table rather than leaving in the machinery.

“How wrong is the picture” was taken above to mean the worst displacement of any point of the reconstructed design. That is a strict reading. Three others are defensible:

The average displacement rather than the worst. It is smaller by a factor of about two here, and it forgives an error concentrated at one corner — which is exactly the error a wrong eye produces, since the departure grows with height above the ground line and is zero at the bottom.

The displacement after a best-fitting similarity has been taken out. A viewer cannot tell that a picture is slightly larger or slightly rotated, so allowing a similarity before measuring is closer to what a person perceives. It roughly doubles the room, and it is the right measure for asking whether an illusion holds.

An angular measure rather than a metric one, since what reaches the eye is angles. That is the most defensible of all and it is also the least comparable with anything else in this collection, which quotes millimetres on the design throughout.

The strict reading is used here for one reason: it is the one that cannot be argued with. A number produced under the most demanding reasonable definition is a lower bound on every other, and a reader who prefers a looser definition can multiply. Quoting the loosest would have required defending the choice every time the number was used.

That is a general habit worth naming, because this collection has been bitten by the opposite. A measurement whose definition is generous is a measurement that will be quoted in contexts where the generosity is not appropriate, and nothing downstream carries the caveat.

The one direction that is genuinely free #

There is a fourth direction, and it is not in the solid at all, because it is not a translation.

Rotating the head — turning to look at a different part of the design — does not move the eye. The anamorph depends on where the eye is, not on which way it is pointing, so a viewer may look wherever they like, and the picture they receive at each moment is the correct one for that part of the design.

That sounds obvious and it is worth stating because it is the reason a large pavement painting works at all. A design ten metres long cannot be taken in at once; the viewer scans it. If the anamorph depended on gaze direction the scan would break it. It does not, and the whole of the design is simultaneously correct for one point in the air.

That in turn is why the machinery in this collection treats an anamorph as a map from the design to the floor rather than as a picture: a picture has a frame and a direction, and this object has neither. It has a centre.

Against an ordinary photograph #

The comparison that makes the number mean something is with a picture that is not an anamorph.

An ordinary photograph shown at the wrong distance is wrong too. Standing in the wrong place measures it: read from twice the correct distance, a picture depicts a scene twice as deep. So the same question can be asked of a photograph — how far may the eye move before the depicted scene is wrong by some amount — and the answer is in centimetres rather than millimetres.

The difference between an anamorph and a photograph is therefore one of degree and not of kind. Both have a station point; both degrade away from it; the anamorph degrades faster because its homology is more extreme. Describing an anamorph as “correct from one point” and a photograph as merely “best from one point” suggests a difference in kind that the measurement does not support.

What is different in kind is the visibility. A photograph read from the wrong distance looks like a photograph — the error is a change in the depicted scene, and nothing on the paper says so. An anamorph read from the wrong place looks like a smear. The error is the same sort of thing; only one of them is legible.

What the measurement needed to be trustworthy #

Three things had to be checked, and each of them is the kind of thing that would have made the answer confidently wrong.

Monotonicity. A bisection finds a boundary. If the error dipped below the tolerance again further out, the bisection would return the first crossing and the solid would be reported smaller than it is. Checked over every sampled direction and every step of the outward bracket.

That the tolerance decides the answer. A boundary-finder always returns something. Running the whole measurement at a tenth of the tolerance and getting a solid a thousand times smaller in volume is what says the boundary belongs to the tolerance rather than to the method — a routine that returned roughly the same solid at any tolerance would have been reporting its own step size.

That the long axis is the sight line and not the sampling. Ninety-six directions on a sphere from a spiral construction have no preferred axis, so the fact that the longest reach comes out along the line of sight is a result rather than an artefact.

None of the three is interesting in itself. All three are the difference between a number and a number worth quoting, and the pattern of running a measurement twice at different settings to see whether the setting decides the answer is the one this collection returns to most often.