The rows count hands, not cameras

The rows under a splay measure the bays, not the lean took one splayed strip apart and found the labour divided cleanly. The two straight sides fix the plane the strip lies on and nothing drawn between them can bend it. The rows say only how that plane is cut into bays, and what they say is lens-free: the same drawing is an exact picture of a lectern leaning 55.9° and of one leaning 39.5°, and the bays it depicts are the same in both readings.

A divergent picture is not one strip. A Byzantine interior has a floor, a footstool under the throne, the table top and the open book on it, each splayed by its own amount and each cut into bays by a hand that did the same thing on all of them. So the question this essay puts is the one that single strip could not answer: with several strips in front of it, what can a reader ask the rows that the sides have not already been asked?

The answer separates two counts that have been run together. The sides count cameras — how many viewpoints a picture needs — and four surfaces, and no one camera that draws them found that count leaning heavily on a focal length nobody can read off the page. The rows count hands, they count them as one number per strip, and that number does not move when the assumed lens does.

Four strips and one habit #

The picture is constructed rather than traced: four strips with the splays an icon uses — 1.12 for the floor, 1.20 for the footstool, 1.32 for the table and 1.38 for the book — each a band of equal width divided into bays, each drawn with its own near edge, far edge and two straight sides, and each with its interior rows placed by the same rule.

The rule needs a name and a number. Two placements are already known. A camera drawing bays of equal depth puts its rows where the reciprocal of the drawn width runs evenly from the near edge to the far one, because depth changes at a constant rate along a straight plane and the drawn width is the focal length times the true width divided by that depth. A hand spacing rows evenly down the page puts them where the drawn width itself runs evenly, because the strip’s sides are straight and so the width is an exactly linear function of the page row.

Those are two members of one family. A placement is a power: the rows go where some power of the drawn width runs evenly. Write the power q, and the hand’s even spacing is q = 1 while the camera’s equal bays are q = −1. Nothing else about the strip appears in the definition — not the lean, not the distance, not the lens — so q is a property of the drawing and not of the scene it is a drawing of.

The family is continuous and the two named habits are 2 apart in it, so the question of whether a picture was drawn by one hand becomes arithmetic rather than judgement: read q off each strip, and ask whether the four numbers are one number.

What a habit is, in the depicted world #

A number read off a drawing earns its keep by saying something about the thing drawn, and this one does, in a form that generalises a law already measured.

The rows under a splay measure the bays, not the lean found that a strip with evenly spaced rows depicts a near bay deeper than its far one by nearly the square of the splay — 1.74 at a splay of 1.32, approached from below as the strip is cut more finely. The square looked like a fact about even spacing. It is a fact about even spacing’s place in the family.

A habit of q depicts a near bay deeper than the far one by the splay raised to the power 1 + q. At q = −1 the power is zero and the bays are equal, which is the camera’s placement and has to come out that way. At q = 1 the power is two, which is the result the rows under a splay measure the bays, not the lean reached. Everything between is a painter dividing a leaning surface at a stated rate, and the drawing follows the law to 0.6 per cent across the whole range once the strip is cut into ninety-six bays.

The reason the law is a fine limit rather than an identity is the one that essay gave: the near and far bays of a coarsely cut strip are wide samples of their own ends, so they average the rate over a stretch rather than reporting it at a point. Cut into six, a splay of 1.32 at q = 1 gives 1.59 against the law’s 1.74; cut into ninety-six it gives 1.73. A painter’s strips are coarse, and a reader comparing a measured ratio with the law has to cut the law to the same number of bays before the comparison means anything.

So the habit is not only a description of where a hand put its lines. It is a statement about the surface the picture depicts — how fast a lectern’s boards, or a throne’s steps, shorten from the front of the strip to the back — and two painters with different habits have drawn two different pieces of furniture.

What the rows charge a camera: nothing at all #

Before the rows can count anything they have to be shown not to be counting something else, and the exactness established for a single strip makes that quick.

Every row of every strip is a horizontal segment of known true width, so its depth is that width times the focal length over its drawn width, and its height follows from its page row and that depth. Each row therefore reads back as a point in the vertical section through its strip, and the strip depicts a flat plane exactly when those points are collinear. Across all four strips of the picture above, drawn at any habit the family holds, the worst departure from collinearity is $4\times10^{-16}$ metres — the arithmetic floor, on a strip two metres long.

That is the null result, and it is worth stating in the form a reader would use. One camera redraws every strip of a divergent picture exactly, at every habit, with nothing left over. A picture whose strips are all drawn with evenly spaced rows is no less a one-camera picture than one whose rows a camera placed; the rows are simply not evidence about viewpoints. The question that essay left asked whether requiring one camera to agree with the sides and with a single assumption about the bays could charge it something. It cannot, because the assumption about the bays is free: any rows at all between two straight sides depict a flat plane cut into bays of whatever depths those rows imply, and a flat plane cut into unequal bays is exactly what a camera can photograph.

So the camera count stands where one camera means one horizon, not one point left it, resting on the sides and on the vanishing lines they determine. That essay narrowed the test from a shared meeting point to a shared vanishing line and left every earlier verdict standing; this narrows what the test is made of, and leaves them standing again. The rows add nothing to it and take nothing from it, and the reason to read them is that they are about something else.

The number a lens cannot move #

What the sides cannot do is report a lean without being handed a focal length. A drawn quadrilateral fixes the ratio of the far depth to the near one, because that ratio is the splay; turning the ratio into an angle needs to know how many pixels an angle is worth, which is the lens.

The same drawing, unchanged, is a picture of a lectern leaning 36.4° and of one leaning 86.0° — nearly edge-on — depending only on a number the picture does not contain. That is the weakness a camera count needs a tolerance worked around by reporting the count in pixels of redrawing rather than in degrees of tilt, and the weakness is real: the lean is not in the picture. The picture whose lines spread located the meeting point of the sides on the page, which is a page fact and needs no lens; turning that point into an angle is where the lens enters, and it enters every reading built on it.

The habit is. Every quantity its recovery reads is on the paper — where the rows fall, and where the two sides are — and no step of it divides by a focal length or assumes a distance. So across that whole twentyfold sweep the reported habit does not move in its ninth decimal place, and neither does the near-to-far bay ratio it implies, which stays at 1.587342. The drawing says one thing about the hand at every lens it could have been taken with.

This is the second lens-free reading this field has found, and the pair of them now describe the picture in a way the sides alone could not. The splay says how the surface leans, up to a lens. The rows say how the hand worked, full stop.

One hand gives one number; two give two #

A reading is only worth having if it can come back with something other than the answer it was looking for, so the picture was drawn twice.

Drawn by one hand — every strip at q = 1 — the four strips report 1.000000000 each, with a pooled departure of $2\times10^{-15}$ of a strip’s height. That is the control, and it is not trivial: the four splays differ by a quarter, the bay counts differ, and one number accounts for all of them.

Drawn by two — floor and footstool at 1, table and book at 0.4 — the strips report 1.000000000, 1.000000000, 0.400000000 and 0.400000000. The hands come back exactly, each from its own strip, with no fitting across the picture and no lens.

The instructive part is what the pooled test says about the same picture. Forced to describe all four strips with one habit, the best single number is 0.5057, and the worst row it leaves out of place is 1.13 per cent of its strip’s drawn height — 1.13 pixels on a strip drawn a hundred tall, which is under the two-pixel brush this field has been charging redrawing against since a camera count needs a tolerance. A reader who asked only “does one habit fit?” would answer yes and be wrong.

That is the opposite of how the camera count behaves, and the asymmetry is the useful part. A camera count is a residual test: the picture is redrawn under one viewpoint and the question is how many pixels that costs. A hand count is not, because each strip carries its own hand on its own rows and there is nothing to pool. Reading strip by strip recovers a split the pooled residual hides. The picture above is convicted by looking at its strips one at a time and acquitted by looking at them together, and the first reading is the right one.

The strip that convicts a hand is not the most splayed one #

A reading recovered from a drawing is worth what the drawing’s own imprecision allows, and the rows under a splay measure the bays, not the lean reported the difference between the two named habits as a percentage of the strip’s drawn height — a currency in which a stronger splay is always better, since the gap between a camera’s rows and a hand’s rises all the way to the strongest splay a strip can be drawn with.

A painter’s slip is not measured in percentages of anything. It is a brush wide, and a brush is a brush whatever it is drawn on.

The two curves disagree, and the disagreement is geometry rather than a choice of units. A strip two metres long at six metres has to lean 23.1° to be drawn with a splay of 1.14 and 73.3° to be drawn with a splay of 1.38, and leaning it further foreshortens it: the same strip is 160 pixels tall in the first drawing and 51 in the second. Above a splay of about 1.28 the strip is losing height faster than it is gaining signal, and the absolute displacement a change of habit produces starts to fall.

Put a hand’s own unsteadiness into the drawing and the same shape appears in the answer. Half a pixel of slip on every interior row leaves the recovered habit scattered by 0.079 at a splay of 1.26 and by three times that at either end of the range. Two hands are separable when they differ by more than about twice the scatter, so the picture’s own instrument for telling hands apart is its floor, at a splay near 1.26, and not its most dramatic surface. The book at 1.38 — the strip whose divergence a viewer notices first — is one of the picture’s worse instruments, and the footstool at 1.20 is one of its better ones. That inverts the intuition that percentage curve encourages, and it inverts it for a reason a reader can check with a ruler: measure how tall each strip is drawn.

It also puts a number on a question an art historian would actually ask. Two hands 0.6 apart in habit are told apart on a well-chosen strip at half a pixel of slip and are not told apart at a pixel and a half. The picture has to be drawn large, or the hands have to work very differently, and neither is a matter of opinion once the strip’s drawn height is measured.

What this reading does not settle #

It says nothing about which habit is older, or better, or taught. The habit is a description of where the rows are. Whether a workshop taught even spacing, whether a painter arrived at it by copying, and whether the exceptions are apprentices or later repairs are questions about people, and this measures paint. The convergent side of the subject has the same boundary: what a panel says about its maker found that a drawing names the class of error in it and not the recipe that produced it.

A hand need not have one habit. The family has one parameter because two placements were already named and the natural curve between them has one. A painter who spaced the first two bays by eye and stepped the rest, or who worked from the far edge on one strip and the near edge on another, is not in the family, and the fit will report the nearest member together with a residual that says it is not a member. The slip that leaves no trace is the standing warning here: a reading that always returns its best guess returns one for drawings it should refuse.

The strips are taken as drawn with straight sides and level rows. Everything above inherits that from the essay before it, and it is the assumption a real panel is least likely to honour exactly. A strip whose sides bow is a different object and its habit is not defined.

And the slip was modelled as independent on every row. A hand that drifts — spacing the near bays slightly wide and the far ones slightly narrow throughout a strip — is putting a systematic error into exactly the quantity the habit is read from, and it will be reported as a change of habit rather than as noise. That is the error this reading is least protected against, and the scatter figure above does not contain it.

What the two readings are each for #

The sides of a divergent picture and the rows between them answer different questions, and neither answers the other’s.

The sides carry the camera. They fix each surface’s vanishing line, they let a reader ask how many viewpoints a picture needs, and they report the lean only together with a focal length the picture does not hold — which is why that count is stated in pixels of redrawing.

The rows carry the hand. They charge the camera nothing at all, because any rows between two straight sides depict a flat plane. What they fix is one number per strip, recovered from the page with no lens anywhere in it, identical to nine decimal places across a twentyfold change in the assumed focal length. One hand across four splays gives one number; two hands give back exactly two, strip by strip, where a pooled fit would have called the picture consistent to within a brush.

And the instrument is not where it looks. A change of habit shows most strongly on a strip splayed about 1.28, not on the most splayed one in the picture, because a strip leaning far enough to splay hard is drawn short — 51 pixels against 160.

Still open: whether a hand’s drift reads as a change of habit #

The scatter above is independent slip: every row displaced on its own, nothing carried from one row to the next. That is the error a steady hand makes, and it is the error the habit is best protected against, because independent displacements average out of a fit with several rows in it.

The error a tiring hand makes is not independent. A painter working down a strip who lets the bays creep wider, or narrower, has put a monotone drift into the one quantity the habit is read from — and a monotone drift in the spacing is precisely what a change of habit is. The two are not obviously distinguishable, and if they are not, then a single hand working down a large panel would report a habit that changes from strip to strip, which is what two hands look like.

The measurement that settles it draws one strip at a fixed habit with a drift of stated size added along it, recovers the habit and the residual, and asks two things: how much habit a drift of a given size counterfeits, and whether the residual rises enough to give it away. If the residual stays flat while the reported habit moves, then every hand count in a picture drawn on a large panel needs a drift term before it can be believed — and the attribution readings this field has been building toward would be measuring fatigue.