What an eye can paint

A stair does not use all its faces ends with a sentence it does not test:

Which faces exist and which faces can be painted are different questions.

It is demonstrated there on one object. A flight of nine steps is eighteen planes; from the top of a descending flight not one riser is reachable at any eye height; from the bottom of an ascending one the picture concentrates on the risers, which are a third of the surface and take more than half the design.

A statement about one object is an observation. This is the same measurement pointed at four more, so that it can be a law or fail to be one.

Five objects and a control #

The objects are chosen to fail differently rather than to be pretty, which is the same principle the four receivers were chosen on.

A flat floor is the control: one face, no folds, nothing hidden.

A corner — a floor and a wall — is the two-face case the anamorph field already understands, and it is the second control: both faces are used by every eye that can see the picture at all.

A cluster of blocks has twenty-one faces, of which most are pointing the wrong way and one is behind another.

A seating rake is a flight of steps at completely different proportions: 0.9 m of tread to 0.45 m of riser, against a stair’s 0.30 to 0.17. If the stair’s result is about flights, the change of shape should break it.

And a corridor with a doorway is the object the others cannot produce: some of the design goes through the opening and lands on a wall three metres further away.

What each of them offers #

From an eye 1.7 metres up and 2.4 metres in front of the intended picture, the five objects give:

The flat floor: one face of one, all of the surface, 98 per cent of the design landing.

The corner: two of two, all of the surface, all of the design.

The cluster of blocks: eight faces of twenty-one, 73 per cent of the surface by area, 88 per cent of the design landing — the rest goes past the blocks and off the far end of the floor.

The seating rake: seven of thirteen and 52 per cent of the surface, with nearly all of the design landing.

The corridor: six of seven and 99 per cent of the surface, with all of the design landing.

So the law holds, and it holds in a form worth stating carefully. Every object with more than one plane in it withholds part of itself from any given eye, and how much depends on the eye. The plane withholds nothing from anybody, which is what makes the statement about folds rather than about anamorphs.

The design band, and a bisection that had to go #

One piece of machinery had to be rebuilt to run this on five objects rather than one, and the rebuild is a finding rather than a chore.

A design has to be given a band of heights whose rays actually meet the object, because rays aimed too high sail over it into the room. The stair essay finds that band by bisection: a higher ray is shallower, so hitting is monotone in the design’s height, and a bisection on “does this ray hit” converges to the top of the band.

That is true of a flight and false of a corridor. There, the rays that pass through the doorway hit the far wall, the rays just above the doorway hit the end wall, and the rays above those hit nothing — three regimes interleaved, and a bisection on hitting returns a bound that is true of whichever sample it happened to test.

So the band here is sampled rather than bisected: a hundred and sixty rays across the range, and the band is the outermost that land. Slower, and correct on an object with a hole in it.

The general shape of that error is one this collection keeps meeting. A property that is monotone on the object a routine was written for becomes non-monotone on the next object, and the routine gives an answer rather than a refusal. It is the same species as the drag range computed from an option the generator never sees and as the plan reading that assumes the feet and the tips are coplanar.

Every face is still exactly a projective map #

Before any of the counting means anything, the piece the anamorph field has always relied on has to survive the generalisation.

On a single plane, an anamorph is a homology: four design points fitted to four marks predict every other point of that face exactly. That is what makes a corner anamorph two homologies and a flight eighteen, and it is the reason the construction can be drawn with a straightedge rather than computed.

Measured on all five objects — and on the corridor’s far wall, which is reached through a hole in another face — the worst per-face residual is $8\times10^{-15}$ metres. Arithmetic, on every object, including the ones with occlusion in them.

That is not a foregone conclusion. Occlusion could in principle break the map by handing a face a design region that is not simply connected; it does not, because a homology is a map between planes and does not care which parts of its domain were used.

Where the picture goes, against where the surface is #

The stair essay’s sharpest number is that 58 per cent of the picture lands on 36 per cent of the surface, and the reason it happens is worth generalising: the picture concentrates where the surface is nearly square-on to the rays, and that is not where the surface is.

Measured across the objects, the same disproportion appears everywhere there is a fold.

On the rake, the apron takes 32 per cent of the design and the first three risers take another 32 — so nearly two thirds of the picture is on four faces out of thirteen, and the treads above them, which are more than half the surface, take almost nothing.

On the descending flight, the first tread alone takes 68 per cent of the design. The remaining eight treads share the rest, and the risers take nothing at all.

The disproportion is a solid angle #

The pattern has a single expression behind it, and writing it down makes the five objects one measurement rather than five.

A design is a set of rays, so the share of the picture a face takes is the share of the eye’s solid angle it occupies — that is what casting from a point means. The share of the surface it occupies is its area. The ratio between the two is therefore the ratio of solid angle to area, which for a face at distance $r$ meeting the rays at incidence $\iota$ is

$$\frac{\text{design share}}{\text{area share}} \ \propto\ \frac{\cos\iota}{r^{2}},$$

normalised over the faces the eye reaches. Two terms, and both of them favour exactly the faces the measurements keep finding: near, and square-on.

For the horizontal faces of a flight the two terms collapse into one. A tread seen at depression $\delta$ from an eye $h$ above it has $\cos\iota = \sin\delta$ and $r = h/\sin\delta$, so its density is

$$\frac{\sin^{3}\delta}{h^{2}}$$

— a cube, which is why the first tread of a descending flight takes two thirds of the picture. At 55° below the horizon $\sin^{3}\delta$ is 0.55; at 10°, which is where the ninth tread sits, it is 0.005. A hundredfold spread across nine identical planes, from geometry alone.

The same expression gives the stair’s riser-to-tread ratio without any new work: at equal distance, $\cos\iota$ is $\cos\delta$ for a riser and $\sin\delta$ for a tread, so the density ratio is $\cot\delta$ — the reciprocal of the stretch ratio, as it must be, since design per unit area and area per unit design are the same statement inverted.

And it explains the two controls. A flat floor has one face, so there is nothing for the ratio to be a ratio of. A corner has two, and both are within a modest factor of each other in $\cos\iota/r^{2}$, so the disproportion is small and the object reads as evenly used. The disproportion is not a property of having many faces; it is a property of having faces at very different distances and orientations, and a stair is the ordinary object that maximises both at once.

That also says what an object would have to be for a design to land evenly on it: every face at the same distance from the eye and the same angle to its ray, which is a piece of a sphere centred on the reader. Nothing anybody walks on is one, which is the geometric reason a cast design always concentrates somewhere and the painter always has a part of the object doing most of the work.

On the corridor, the floor takes 56 per cent, the two side walls 14 each, and the far wall — reached through the doorway — takes 4.

The pattern is the same each time and it has a mechanism rather than a moral: a face nearly edge-on to the rays receives a long thin sliver of design stretched enormously, and a face square-on receives a compact patch at nearly unit scale. The picture therefore piles up on whatever faces the eye happens to face, and the count of faces used says nothing about how the picture is distributed among them.

Sweeping the eye #

The law says the subset depends on the eye, and the dependence can be watched rather than asserted.

Raise the eye above a cluster of blocks from 1.2 metres to 5, and the faces reached go from seven to twelve of the twenty-one, the surface reached goes from 71 per cent to 80, and the share of the design that lands at all goes from 91 per cent to 99. Everything improves, monotonically, and the improvement is entirely occlusion clearing: from higher up, the eye sees over the front block onto the one behind it.

The rake behaves the same way and more strongly, because it is a staircase and a staircase is a stack of things hiding behind each other: six faces reached at 1.2 metres and twelve at 5, with the surface going from 47 per cent to 90.

The corridor is the object where raising the eye is not free, and that is the finding of the sweep rather than of any single row. Its reached surface goes from 99 per cent to 100 — there is almost nothing to gain — while the share of the design that lands falls from 100 per cent to 92, because the rays that used to go through the doorway now hit the wall above it and the rays that used to hit the far wall go over it.

So on two of the three objects a higher eye is better and on the third it is a trade. Which of the two quantities a designer is buying decides the answer, and the eye that reaches the most is the rung that prices that choice.

Why the rake behaves like a flight #

The rake is in this list to try to break the stair result, and it is worth saying explicitly that it failed to.

Its proportions are nothing like a staircase’s. A stair here is 0.30 of tread to 0.17 of riser — a ratio of 1.8 — and the rake is 0.9 to 0.45, a ratio of 2.0 at three times the size, which is a piece of seating rather than a way of getting upstairs.

Its behaviour is a staircase’s in every respect measured: faces withheld from a low eye and released by a high one, the picture concentrating on the near faces, half the surface unused at standing height, and every used face an exact homology.

That is the evidence that the stair essay’s finding is about folds rather than about stairs. An object made of alternating faces at two orientations hides some of them from any eye, whatever the proportions, and the count of what is hidden is a fact about the geometry rather than about the building trade.

The depth ratio, which the flight does not have #

One quantity separates the corridor from everything else, and it is the one a painter would care about most.

The ratio of the furthest mark’s distance to the nearest’s is 1.8 on a flight, 2.1 on a corner, 3.0 on the blocks and on a plain floor — and 3.2 on the corridor, with a discontinuity in the middle of it. The marks that go through the doorway land at 9.4 metres from the eye; the marks that hit the end wall beside the doorway land at 6.4; and the marks on the near floor land at 3.0.

So the corridor’s design is spread over two surfaces at very different depths with a jump between them, which is a different kind of object from a flight whose faces are all within a factor of two. A design that lands in two rooms is that case measured on its own.

What the law does not say #

Three limits, stated because the result is easy to over-read.

It is not about visibility. A face the design does not reach may be perfectly visible from the eye; the design is a bounded rectangle of rays, and a face outside that cone is missed rather than hidden. The two are separated in the next rung, where the difference turns out to be occlusion and is measured as an area.

It is not a claim that the unused faces are wasted. A design could be enlarged, or a second design added for a second eye — which is what a flight turns out to allow — and the measurement here is of one design from one eye.

And it is not about paint. Whether a face can carry a picture in practice depends on the stretch, which is a separate quantity with its own threshold and its own rung: a face can be reached by rays that arrive so obliquely that the picture on it is unpaintable at any scale.

What makes an object good for this #

The five objects sort themselves, and the ordering suggests what a designer looking for a surface should want.

Faces that turn toward the eye rather than away. A descending flight’s risers are unusable from the top and its treads are excellent; an ascending flight’s risers are excellent from the foot. The best objects are the ones whose faces mostly point at the reader, which is a fact about the reader’s position rather than about the object.

Faces at comparable depths. A corridor’s design is spread from three metres to nine and a half, so its scale varies by a factor of three across one picture; a flight’s faces are all within a factor of 1.8, so its picture is nearly uniform.

And few things in front of other things. Occlusion costs a cluster of blocks nearly a tenth of the surface that faces the eye, and it costs nothing at all on the objects with no depth ordering in them.

By those three, the best object in the set is the corner — two faces, both facing, no occlusion, a depth ratio of two — which is why it is the form every Renaissance quadratura and every modern pavement painting takes. The objects here that beat it on coverage do so by having more surface, and pay for it in every other column.

The short version #

An object with more than one plane in it offers part of itself to any given eye and withholds the rest, and the fraction is a measurement rather than a property: nine faces of twenty-one on a cluster of blocks, seven of thirteen on a rake, six of seven in a corridor, seven of eighteen from the top of a flight.

A plane offers all of itself to everybody, which is what makes this a statement about folds.

Every face that is reached carries an exact projective map, on every object, including one reached through a hole in another face. And where the picture goes among the reached faces is wildly uneven — one tread taking three-quarters of a design, three faces of thirteen taking half of another — because a picture piles up wherever the surface happens to face the eye.