The rows under a splay measure the bays, not the lean

An inverse perspective is a leaning plane asked a four-cornered divergent construction what it depicts, and got an answer nobody expected: a rectangle, with four right angles and its far edge equal to its near one, lying on a plane that leans toward the viewer. The splay — the far edge drawn wider than the near — does not encode a splayed object. It encodes a tilt. A lectern, an open book on a stand, a table top seen from a little below its far edge: all of them photograph divergently, and the recovery run on the photograph returns a square.

That essay worked with the four corners of one surface. The pictures it is about rarely stop at four corners. A Byzantine floor, the boards of a divergent table, the steps of a throne, the courses of a wall seen in inverse perspective — each is a splayed strip divided into bays, and the lines between the bays are drawn by hand, almost always evenly down the page. A camera does not draw them evenly. The lines between equal bays on a leaning plane fall closer together at one end of the strip than at the other.

So a drawn splay with evenly spaced rows cannot be the picture a camera would take of equal bays. The question this essay measures is what it is a picture of — whether the even rows bend the leaning surface, or unequal its bays, or both — and how large the difference from a camera’s rows is on the page, since a difference smaller than a brush is a difference no drawing can carry.

A strip of equal bays, photographed and drawn #

The strip is a band two metres wide and two metres long, divided into bays of equal depth, lying on a plane that leans toward a camera. The lean is chosen so that the far edge is drawn a stated number of times as wide as the near one — that ratio, the splay, is the painter’s number, and it is what every figure here is parameterised by. For a splay of 1.32, the value the earlier essay used, a strip two metres long seen from six metres leans 55.9° toward the eye.

Each line between two bays is a horizontal segment on the plane, and the camera draws it as a horizontal segment on the page. Its drawn width is its true width divided by its depth, so the two sides of the strip, which run through the ends of every one of those segments, are straight lines — straight for the camera and straight for a painter, since a strip’s sides on a plane are straight in space. Both drawings, then, share the near edge, the far edge and the two sides. They differ only in where the rows between fall.

A camera’s rows are not evenly spaced: they open out toward the far edge, because on a plane leaning toward the eye the far end of the strip is the nearer end, and nearer things are drawn larger in both directions. The evenly spaced rows sit between 0 and 6.8 pixels from them on the camera’s own scale, where the whole strip is drawn 99 pixels tall.

Why the sides are the camera’s and the rows are the hand’s #

The division of labour between sides and rows is not a modelling choice, and a real photograph shows why.

The square’s two sides come out of the camera as straight lines diverging toward the far edge, and a painter drawing a divergent table draws its sides the same way, with a straightedge from a meeting point below the near edge. The picture whose lines spread located that point: in front of the eye, below the picture, the vanishing point of a direction running down and away. Both hands, the camera’s and the painter’s, agree about the sides because a straight line in space has only one way to be drawn.

The rows are different. There is no straightedge construction for where the line halfway along a leaning surface falls that does not pass through the same apparatus dividing depth by eye measured for an ordinary receding floor — a diagonal, a measuring point, or a guess. The painter of an icon uses none of them. So a divergent picture carries the camera’s evidence in its sides and the painter’s habit in its rows, and it is the rows that have to be read for what the habit implies.

That is also why one camera means one horizon, not one point could test a divergent picture’s surfaces by their sides alone. Every surface’s sides are its own straight lines and every surface’s lean is set by them; a test for one camera is a test on the sides, and the rows could only add a second kind of test, which is where this essay ends.

The sides fix the plane; the rows cannot bend it #

The first thing to ask of the evenly spaced drawing is whether it still depicts a flat surface, and the answer is exact.

Every row is a horizontal segment whose true width is known — the strip’s width — so its depth can be read directly off its drawn width: depth is the true width times the focal length over the drawn width. Its height follows from its row on the page and that depth. Each row therefore reads back as a point in the vertical section through the strip, and the drawing depicts a flat surface exactly when those points lie on one straight line.

For a camera’s rows they do, to the last digit of the arithmetic, and the bays between them are equal. For evenly spaced rows they do as well — to the same last digit. The reason is short enough to state. The drawn width is linear in the row, because the sides are straight; so the depth is one over a linear function of the row, and the height is a linear function of the row over the same linear function. Two quantities that are linear over one common linear denominator trace a straight line as the row varies. Any rows at all between two straight sides depict the same flat plane.

That is worth having as a statement rather than as a surprise, because it separates two things a divergent drawing says. The sides say how the surface leans, and they say it completely: nothing drawn between them can make the surface curve. The rows say only how the surface is cut. Even spacing does not depict a warped lectern; it depicts a flat lectern divided into bays of different sizes.

What even rows do to the bays #

Reading the depth of each bay along the plane off the evenly spaced drawing shows how they differ, and the difference has a pattern.

The bays shrink steadily from the near edge to the far one. Under a splay of 1.1 the near bay is 1.19 times as deep as the far; under 1.2, 1.40 times; under 1.32, 1.66 times. Each of those is close to the splay multiplied by itself — 1.21, 1.44, 1.74.

The square has a reason. A short stretch of a leaning plane at depth z is drawn smaller by one power of z across the page, which is the width, and by very nearly two powers of z along the recession, which is the spacing, because it is both further away and more steeply foreshortened. Where a camera draws equal bays, the spacing on the page therefore goes with the square of the width. A painter who holds the spacing constant while the width grows by the splay has made every far bay smaller than every near one by the square of how much wider it is drawn.

With three bays the ratio at a splay of 1.36 is 1.506; with six, 1.668; with twenty-four, 1.802, against the square’s 1.850. The square is the limit the end bays approach as the strip is divided more finely, since a finer division makes each end bay a better sample of its own end of the strip. At the coarse divisions a painter actually uses the ratio is a little smaller, and it is always on the same side.

This connects the divergent strip to a result from the convergent side of the subject, and to the rule that draws another room before it. The rule is exact for a floor that lengthens found that the taught constant-ratio rule for a convergent floor is an exact picture of a floor whose boards grow toward the back. Even spacing under a splay is the same kind of fact: an exact picture of a surface the painter did not mean, differing from the intended one only in how its divisions are sized. In both, the lines that run toward or away from the vanishing point are right, and the lines across are what carry the painter’s habit.

A number that depends on the splay alone #

The most useful property of these readings is one that was not expected, and it was found by trying to break it.

The strip was redrawn with the same splay at a different lean, a different distance and a different lens: a strip one and a half metres long seen from four metres through a lens of 900 pixels, which has to lean 47.4° rather than 55.9° to be drawn with a splay of 1.32. The row gap, as a share of the strip’s drawn height, came out the same to nine decimal places, and so did every bay ratio. A strip 2.6 m long at 6 m, leaning 39.5°, gave the same again.

The reason is projective. The near edge, the far edge and the two sides fix a quadrilateral on the page, and the rows a camera draws between them are fixed by that quadrilateral alone — they are where the images of equally spaced lines on the plane must fall, and equal spacing on a line is a statement a projective map of the page carries with it. So the whole comparison between a camera’s rows and a painter’s is a function of the splay and the number of bays, and a reader holding only the drawing has everything needed to make it.

That also marks exactly what the drawing does not hold. Four surfaces, and no one camera that draws them found that the tilt a divergent surface implies depends on the lens assumed as much as on the picture. The rows confirm it from the other side: the same drawing, rows and all, is exact for a lectern at 55.9° and for one at 39.5°. The rows add nothing that separates them.

How large a splay a brush can see #

Because the comparison depends only on the splay, its size on the page can be stated once for every drawing.

The three curves lie almost on one another. At a splay of 1.36 the worst row is out by 7.14 per cent of the strip’s height with three bays, 7.63 with six and 7.67 with twenty-four. Cutting the same strip into more bays does not make the difference easier to see, which answers directly the natural question of how many bays it takes before even rows and a camera’s part company. The number of bays is not what decides it. The splay is.

Against a brush, the gap crosses two pixels on a strip drawn a hundred pixels tall at a splay of 1.083. Below that, no drawing of that size can carry the difference between evenly spaced rows and a camera’s; above it, the difference is a brush’s width or more, and grows almost linearly with the splay. The splays that make a divergent picture look divergent — the earlier essay’s 1.32, most icons’ tables and floors — are far above it: at 1.32 a hundred-pixel strip has rows six to seven pixels from where a camera would put them.

So in any divergent picture strong enough to be recognised as one, evenly spaced rows are measurably not a camera’s rows, and what they measure is the painter’s bays.

What a reader can take from a divergent strip #

Three readings, in the order a reader with a ruler would make them.

The sides give the lean, and only up to the lens. The splay fixes the plane’s inclination given a focal length and nothing without one, as the earlier essays found. No row placement can change that and no row placement can remove it.

The rows give the bays, independently of the lens. Measure where the rows are as fractions of the strip’s drawn height and compare them with where a camera puts equal bays under the same splay; the difference is a statement about the depicted bays that holds for every lens and every lean the sides allow. Even rows under a splay of 1.32 depict a near bay about 1.6 times as deep as the far one.

A camera count can use the rows as well as the sides. A camera count needs a tolerance found that how many cameras a divergent picture needs depends on the width of a brush. The rows give that count a second kind of evidence: a picture whose strips are all drawn with even rows is consistent with one camera only if every strip’s bays are allowed to be unequal in the same way, and the splay squared says how unequal. Whether that makes a camera count sharper or only longer is not measured here.

What this reading does not decide #

A painter may have meant unequal bays. A throne’s steps can genuinely shorten toward the top, and a tiled floor drawn in inverse perspective may be a floor of tiles that really do differ. The reading says what bays the drawing depicts. It cannot say the painter was wrong about them, only that a painter who meant equal bays drew them unequal by a stated factor.

The strip’s sides are taken as drawn straight and its rows as drawn level. A painter who curves the sides of a divergent table, or tilts its boards, has drawn a different object, and the exact flatness above belongs only to straight sides.

The brush threshold is a stated width on a stated size. Two pixels on a hundred-pixel strip is one choice. Because the gap is a share of the drawn height, any other brush and size scales it directly: a strip drawn five hundred pixels tall carries the difference at a splay about a fifth as far above one.

And one strip at a time. Every measurement here reads a single strip. A picture of a divergent table with a divergent floor below it has two strips whose bays may be drawn by one habit and whose leans are fixed by two different splays, and whether the habit is consistent across them is the kind of question the camera count asks and this reading does not.

The plane, the bays and the splay #

Under two straight diverging sides, rows spaced evenly down the page depict the same flat plane a camera’s rows do; the sides fix the plane exactly and nothing between them can bend it. What the rows decide is the bays. Even rows make the near bay deeper than the far by nearly the square of the splay — 1.66 times with twelve bays under a splay of 1.32, approaching 1.74 as the strip is divided more finely.

The gap between even rows and a camera’s is a function of the splay alone, identical to nine decimal places for the same splay at different leans, distances and lenses. It passes two pixels on a hundred-pixel strip at a splay of 1.083, and at the splay of a recognisably divergent picture it is six or seven per cent of the strip’s height, however many bays the strip is cut into.

Still open: whether one habit draws every strip in a picture #

The rows of one strip say what bays that strip depicts, and the reading is lens-free. A divergent picture holds several strips — floor, table, footstool, book — each with its own splay, and a painter who spaces rows evenly does so on all of them.

The measurement that follows takes a constructed picture with several splayed strips, each drawn with evenly spaced rows, and asks whether one camera, now required to agree with the sides and with a single assumption about the bays, can draw them all within a brush: whether “every strip’s bays follow its own splay squared” is a consistent reading of the whole picture, or whether strips at different splays demand different habits. If the whole picture reads as one habit under one camera, the rows are no evidence against a single viewpoint. If they demand several, the rows are a second, lens-free route to the camera count the sides could only give up to a focal length.