The stretch decides the band

Every measurement of an anamorph in this collection starts by choosing how much design to cast, and the choice has been made geometrically: the band of design heights whose rays actually meet the object.

That is the right criterion for a question about the object and the wrong one for a question about paint, and the difference is a factor of twenty.

What the stretch is #

Take two design points a hair apart. Cast both. Measure how far apart their marks land, and divide by how far apart they were in the design.

That ratio is the local stretch, and it is the amount by which the picture is drawn out where it lands. A stretch of one means the mark is the same size as the design; a stretch of twenty means a centimetre of design covers twenty centimetres of floor.

It is not an error. The reader at the design eye receives the intended picture exactly, because the stretch is precisely what the projection undoes — that is the whole of what an anamorph is. The stretch is a fact about the painter’s job, not about the reader’s view.

Why a geometric band is the wrong band #

The band a ray test gives runs from just above the ground line to the highest ray that meets the object at all. Near the top of that band the rays are nearly parallel to the surface, and a ray nearly parallel to a surface lands a very long way from its neighbour.

On a bare floor, the worst stretch inside the geometric band is 19. On a corridor’s floor it is 18; on an ascending flight, where the last rays graze along a tread, it is 16.

So the outermost few per cent of a geometrically chosen design is being asked to be painted at twenty or thirty times its own scale — which for a design of any detail means a mark that was a millimetre wide in the drawing is half a metre on the floor, positioned to a tolerance the same map divides by twenty.

A designer who takes the geometric band and paints it discovers this at the far end of the job, which is the expensive place to discover it. The design that outruns the floor is the field’s earlier finding of the same shape from the other side: a design too tall for the floor available, found by measuring rather than by painting.

What a cap leaves #

Cap the stretch at four — a mark drawn at four times its design size, which is a reasonable limit for hand painting — and each object keeps a different share of its design.

A bare floor: 55 per cent. Nearly half the geometrically valid design is unpaintable.

A corner: 81 per cent. The wall takes over exactly where the floor’s rays start to graze, which is what a wall is for.

A cluster of blocks: 85 per cent. The block faces are nearly square-on to the rays, which is why occlusion rather than obliquity is the cluster’s limitation.

A seating rake: 84 per cent. An ascending flight: 87 per cent. A corridor: 82 per cent.

And the top of a descending flight: 100 per cent, the object that reaches least of itself and paints all of what it reaches — every mark inside a stretch of four, because that design’s whole band is short and near, and its worst stretch anywhere is 3.3.

The floor’s share, in closed form #

The bare floor is the one object in the set whose whole curve can be written down, and doing it says what a cap is really buying.

The stretch on a flat floor is $z/h(1-f)^{2}$ for a design reaching the fraction $f$ of the eye’s height, so capping it at $S$ keeps everything below

$$f_S \;=\; 1 - \sqrt{\frac{z}{h\,S}},$$

and the share of the geometric band that survives is $f_S/f_{S_{\max}}$. With the eye 1.62 m up and 2.40 m back, the geometric band runs to a stretch of 19 and so to $f = 0.721$; a cap of four gives $f = 0.391$; and the ratio is 0.543 against the 55 per cent measured. The median follows from the same expression at half the band — $f = 0.360$ gives a stretch of 3.62, against the 3.6 reported two sections below.

The shape of that expression is the part worth carrying. The reach grows as $1 - c/\sqrt{S}$, so the cap has to be quadrupled to halve what is left out. Reading it as a table:

A painter who can work at twice the stretch gains a quarter of the design; one who can work at three times gains a third. That is a much gentler return than the stretch column of the earlier table suggests, and it is gentle in the useful direction: the last few per cent of design cost enormously in stretch and the first eighty per cent cost almost nothing, so a cap set anywhere reasonable keeps most of the picture.

The same square root says why the corner does so much better at 81 per cent. A wall removes the $(1-f)^{-2}$ entirely above the join, so its curve is not a square root at all but a step — everything below the join follows the floor’s law and everything above it is at the wall’s constant stretch. The corner is not a floor with a better constant; it is a different function.

The ordering is the opposite of the coverage ordering #

Reading the two measurements together is what makes the cap worth having.

What an eye can paint ranks the objects by how much of themselves they offer: the corridor and the corner offer nearly everything, the descending flight offers half.

By stretch the order reverses. The descending flight is perfect and the bare floor is the worst object in the set — and the bare floor is the one that offers a design its whole surface.

So the two questions genuinely conflict. A surface that offers a design a lot of itself is usually a surface the design meets at a shallow angle, because both are consequences of the same thing: a surface running away from the eye is large in the picture’s terms and grazing in the design’s.

That conflict is not resolvable by choosing a better object; it is a trade, and the cap is where a designer states which side of it they are on.

The median, which is the number a designer should ask for #

The worst stretch is a single point of the design and it is easy to over-weight. The median stretch is the number that describes the job.

A bare floor’s median is 3.6, so half its design is drawn out by more than three and a half times. A corner’s is 2.3, a corridor’s 2.6, a rake’s 2.2, a cluster of blocks’ 1.9, and an ascending flight’s 1.8 — the lowest in the set, because a flight presents a reader with a staircase of nearly square-on faces.

Set the medians beside the worst cases and the objects sort into two kinds. On a floor the worst case is five times the median, so the trouble is concentrated at the far end. On a flight the worst is nine times the median: almost all of the design is comfortable and a tiny sliver is impossible.

The second is a much better job to have, because the sliver can simply be dropped. The first cannot, because the badly stretched part is a large contiguous region.

The stretch is a smear, not a magnification #

One thing the number above hides, and it changes what a painter is actually being asked for.

The stretch measured so far is differenced along the design’s own vertical — two design points one above the other — because that is the direction an anamorph’s geometry varies in. Difference along the design’s horizontal instead and the answer is completely different at the same point.

At the worst point of a bare floor’s design, the along-design stretch is 18.7 and the across-design stretch at the same point is 3.6: a ratio of five. On an ascending flight the worst point stretches 15.8 one way and 1.8 the other, a ratio of nearly nine.

So a grazing mark is not a scaled copy of itself. It is a smear: drawn out along one direction and nearly unchanged across it, which is exactly what everybody has seen in a photograph of a pavement painting taken from the side.

That matters for the cap. A circle in the design lands as an ellipse five to nine times longer than it is wide, and whether that is paintable depends on the content rather than on the size: a horizontal line survives it, a face does not, and a letter is somewhere in between. The single number this essay caps is the larger of the two, so the cap is conservative for content that happens to run the right way.

The honest way to report it would be both singular values of the local map, or their ratio as an anisotropy alongside the scale. The median anisotropy is 2.3 on a floor, 1.6 on a corner and 1.0 on an ascending flight — a flight’s marks are nearly undistorted in shape even where they are enlarged, which is one more reason a staircase is a better surface than a floor.

What a real pavement painting does #

Pavement paintings are made on exactly the object this measurement finds worst, and painters have solved it in a way that the numbers explain.

The design is placed near — the marks are on floor a few metres from the eye rather than at the far end of the band — and the picture is composed so that the parts a reader looks at longest are the parts with the least stretch. The far end of the band, where the stretch runs away, carries background or nothing.

That is not a compromise forced by the medium; it is the right design. A picture whose important content is at a stretch of 2 and whose empty sky is at 15 has used the geometry rather than fought it.

It is the same reasoning the design band on a stair needs, and it also explains why the genre’s characteristic subject is a hole — a chasm, a staircase going down, a pool — since a depicted hole puts its most detailed content in the near part of the design where the stretch is low, and its featureless depths at the far end where nothing needs to be drawn precisely.

Where the worst point is #

The worst stretch is not where a reader would guess, and finding it is worth a paragraph because the guess is so natural.

The obvious candidate is the mark furthest from the eye, since everything in perspective gets worse with distance. It is not: on a bare floor the worst point is at the top of the design — the highest ray, which is the shallowest — and its mark is not the furthest one; on an ascending flight the worst point is on the sixth tread, halfway up, where a ray happens to run nearly along the tread’s own plane.

The quantity that decides it is the angle between the ray and the surface, and nothing else. A ray meeting a face square-on lands its neighbours at nearly the design’s own spacing however far away the face is; a ray grazing a face lands them far apart however close it is.

So the stretch map on a faceted object is not smooth in any useful sense: it has a local maximum near the far edge of each face and drops back at the start of the next one, and the global worst case is on whichever face the rays happen to graze hardest. That is why the ascending flight’s worst point is in the middle of the object and its median is the best in the set — the flight has many faces, and a ray grazing one of them is square-on to the next.

What the cap is not #

Two things the stretch does not decide, stated because the cap looks more general than it is.

It is not the eye’s tolerance. How far a reader may move from the design eye is a separate measurement with a separate answer — a spindle of a few tens of cubic centimetres — and it is unaffected by how stretched the marks are. A badly stretched design is exactly as forgiving of a wrong eye as a well-behaved one.

And it is not resolution. A stretch of twenty does not blur anything: the marks are exactly where the design says, and a reader at the eye sees them exactly where they were drawn. What it costs is the painter’s precision, since a positioning error at the surface is divided by twenty by the time it reaches the reader — which is, if anything, a help, and is the one direction in which a large stretch is a gift.

That last point is worth dwelling on, because it is the opposite of what the word suggests. A stretched region is easy to paint accurately and hard to paint at all, since it is large; a compressed one is small and demands precision. The cap above is about the size of the job rather than about its accuracy.

Two faces of one object, measured separately #

A last number, because it is the one that connects this rung to the one about two pictures on one object.

On an ascending flight seen from its foot, the design lands on risers and treads, and the two kinds behave completely differently. The risers take 68 per cent of the design at a median stretch of 1.41 and a worst case of 2.1 — nearly undistorted everywhere. The treads take the other 32 per cent at a median of 2.59 and a worst case of 12.

So the flight’s whole stretch problem is on its treads, and every mark the cap rejects is a tread mark. That is a much more actionable statement than a number about the object as a whole: a designer told “87 per cent of the design survives a cap of four” learns less than one told “the risers are perfect and the treads are the difficulty”.

It is also the geometric reason every stair mural is painted on the risers, stated in the units a painter works in rather than as a share of the picture — and it is what makes a flight able to carry two pictures at once practical rather than merely possible.

The cap is a decision, and it should be stated #

One last point about the number four, which has been used throughout without justification.

Four is a plausible limit for hand painting and it is not derived from anything. A projector laying out a design tolerates far more, because it does not care how large a mark is; a mosaic tolerates far less, because a tessera has a size and a stretched design cannot be made from smaller ones. A cap is a statement about the medium and the geometry has no opinion about it.

That is why the figures here plot the whole curve rather than a single number, and why the cap is a slider rather than a constant. A reader with a different medium reads a different column of the same table, and the ordering of the objects is what survives the choice: the floor is worst and the descending flight is best at every cap from 1.5 to 20.

The short version #

An anamorph’s design should be given the band its paint can carry rather than the band its rays can reach, and the two differ by a factor of twenty at the far end.

Capped at four times the design’s own scale, a bare floor keeps 55 per cent of a geometrically valid design, a corner 80, a flight 85, and the top of a descending flight all of it. The ordering is the reverse of the ordering by coverage, because a surface that offers a design a lot of itself is a surface the design grazes along.

The number to design against is the median rather than the worst — 1.6 on a flight, 3.6 on a floor — and the shape of the tail says whether the unpaintable part can simply be dropped.