Four surfaces, and no one camera that draws them

An inverse perspective is a leaning plane asked one divergent construction what it depicts and got a definite answer: a rectangle, on a plane leaning toward the viewer, with the lean set by where the construction’s sides meet. It then quantified the catch. The lean is the arctangent of the focal length over the distance of that meeting point from the centre, the focal length is an assumption, and a drawing supplies no evidence about it — so the recovered tilt ran from 20° to 71° for one drawing as the assumed lens went from 260 px to 2,084 px, while the rectangle and the sign of its lean stayed put.

That essay asked its question of a single quadrilateral, and it said so: a single quadrilateral is exactly the input this site’s oldest recovery takes. A picture is not one quadrilateral. An icon has a floor, a footstool, a table and a book, each drawn as its own one-point surface, each splayed at its own rate — and a question arises that one surface cannot raise. Could one camera have drawn all of them?

Four surfaces, one assumed camera #

The picture drawn here is constructed, not traced. Four one-point surfaces are laid out as a floor, a footstool, a table top and a book, each with its near edge and far edge horizontal on the page and its sides splayed outward at its own rate: 1.12 for the floor, 1.40 for the footstool, 1.32 for the table and 1.25 for the book. Nothing about any real painting is measured, and nothing about one is claimed. The construction is the smallest picture in which the question has an answer that can be computed.

Each surface’s sides meet somewhere below the picture: the floor’s 697 px below the centre row, the footstool’s 241 px, the table’s 375 px and the book’s 109 px. Read under the camera the rung below found its tilts consistent with — 520 px of focal length, centred on the picture — each meeting point gives its surface a tilt: floor 36.7°, footstool 65.1°, table 54.2°, book 78.2°.

A floor, a footstool top, a table top and a book lying on it are, in the ordinary world, parallel surfaces. The picture reads them as four surfaces at four different tilts, spread over 41.4°.

The control: one camera, four parallel surfaces #

What one camera does with four parallel surfaces is not a matter of opinion, and the control measures it. Four rectangles lie on parallel planes leaning 35° toward a pinhole, at different heights and different depths, and the pinhole photographs them.

Every one of the four photographs is divergent, because every plane leans toward the camera. And every one of their sides meets at the same point — 957.6 px below the top of the picture, to thirteen decimal places — so every one implies the same tilt, 35.0°, and the spread between them is 2 × 10⁻¹³ degrees.

That is the whole mechanism of the question. One camera and parallel planes means one direction for every surface’s receding sides, one vanishing point for that direction, and so one meeting point for every surface. A picture whose surfaces meet at four different points cannot be one camera’s picture of parallel surfaces, and the constructed stack’s surfaces meet at four.

Agreement tests the camera; the tilt tests the lens #

The control has a second lesson, and it is easy to miss because the photographed stack was read with the camera that took it.

Read the photographed stack under the wrong lens — 260 px, or 5,000 — and its four surfaces still imply one tilt. The tilt is wrong: it is no longer 35.0°. But it is the same wrong tilt for all four, because the four meeting points are one point, and every lens maps one point to one angle.

So a reading produces two numbers that answer two different questions. Whether the surfaces agree answers whether one camera drew them, and it needs no lens. What tilt they agree on answers which lens, together with the drawing, would produce that tilt — and the single-surface reading showed that a drawing supplies no evidence about the lens. Recovering the camera from a photograph needs more than one surface’s worth of structure for the same reason: a single vanishing point fixes a direction and not a focal length.

The constructed stack fails the first question before the second can be asked, and that is why the rest of this essay measures its distance from one camera rather than its tilts.

The spread belongs to the lens as much as to the picture #

It would be natural to report 41.4° as the measure of how far the constructed picture is from one camera. It is not a measure of the picture, and the reason is the catch from the single-surface reading arriving in a new form.

Read the same drawing under a different assumed lens and the spread changes, and changes a lot. At 260 px the four tilts spread 46.8°; at 520 px, 41.4°; at 1,040 px, 27.8°; at 5,000 px, 6.7°. A reader who assumes a very long lens concludes that the four surfaces are nearly parallel, and a reader who assumes a short one concludes that they diverge by nearly fifty degrees, and the drawing supports both equally.

The curve has a peak, and the peak has a closed form. Each tilt is the arctangent of f over a meeting point’s distance from the centre row; with the nearest and furthest distances a and b, the spread is $\arctan(f/a) - \arctan(f/b)$, which is largest where $f^{2} = ab$ and is there equal to $2\arctan\sqrt{b/a} - 90°$. For the stack that is 46.85° at 276 px. As the assumed lens gets very long or very short the spread falls to zero from both sides — not because the surfaces become parallel, but because every meeting point implies nearly the same tilt when the lens is so long that every tilt is near 90°, or so short that every tilt is near 0°. At both ends the reading has stopped distinguishing anything.

The photographed control is flat at zero across the whole sweep. Its four meeting points are one point, so under any lens at all they imply one tilt.

That flat line is the part of the figure that does not depend on the lens, and it is the part worth keeping. The spread is zero under every lens exactly when the meeting points coincide, and it is not zero under any lens when they do not. Whether a picture is one camera’s is a fact about its meeting points. How far from one camera it is, in degrees, is a fact about the meeting points and an assumed lens together.

Why the peak sits at the geometric mean #

The closed form for the peak is short enough to derive, and the derivation explains the shape of the whole curve.

Take the two surfaces whose meeting points are nearest to and furthest from the centre row, at distances a and b. Under a lens of focal length f they imply tilts of arctan(f/a) and arctan(f/b), and the spread is the difference. Its derivative with respect to f is a/(a² + f²) − b/(b² + f²), which vanishes where a(b² + f²) = b(a² + f²). Collecting terms gives ab(b − a) = f²(b − a), so the spread peaks at f² = ab — the geometric mean of the two distances — and nowhere else.

For the constructed stack the book’s sides meet 109 px below the centre row and the floor’s 697 px below it. Their geometric mean is 275.6 px, which is where the sweep finds its peak, and the peak value is $2\arctan\sqrt{b/a} - 90°$, which for these two distances is 46.85°.

Two things follow. The peak depends only on the ratio b/a. A picture whose surfaces’ meeting points spread over a factor of 6.4 in distance from the centre row can never be read with more than 46.85° between its surfaces, under any lens whatever; one whose meeting points spread over a factor of two can never exceed 19.5°. And the lens that maximises the spread is set by the drawing, not by anything a reader believes about the painter’s instrument. A reader who argues that a picture’s surfaces are as far from parallel as they can be read is, knowingly or not, choosing the lens at the geometric mean of the drawing’s own meeting distances.

A distance that owes nothing to a lens #

A measurement of how far that belongs to the picture alone has to be taken on the page, in the page’s own units, and it can be.

Keep each surface’s near edge exactly where it is drawn, and keep the height of its far edge. Move its two far corners sideways, symmetrically, until its sides pass through a common point at some height h — the same point for every surface — and choose h so that the corners move as little as possible, in the root-mean-square sense. The result is the nearest drawing, of the same kind, that one camera could have made of parallel surfaces. No focal length enters and neither does the centre row: the construction asks only for a common meeting point, and one camera’s parallel surfaces are exactly the drawings with one.

For the constructed stack the nearest one-camera drawing moves the far corners 17.6 px, root-mean-square: 13.1 px for the floor, 25.2 px for the footstool, 18.1 px for the table and 10.4 px for the book. The common meeting point it settles on is 772 px down the page, between the floor’s own and the table’s.

The same construction applied to the photographed stack moves nothing: 3 × 10⁻¹⁴ px. It is already one camera’s drawing, and the measurement says so without having been told that a camera was involved.

A number in pixels is less evocative than a number in degrees, and that is the point of it. Seventeen and a half pixels is how much a copyist would have to redraw to make this picture consistent with a single camera, on this page, at this size. It can be compared across pictures drawn at the same scale without anyone agreeing on a lens, and it cannot be moved by assuming a different one.

What the redraw holds fixed, and why #

The redraw distance depends on what it is allowed to move, and the choice made here is worth stating as a choice.

Each surface keeps its near edge exactly where it was drawn and keeps the height of its far edge. Only the two far corners move, and only sideways. That holds fixed everything a draughtsman sets when placing a surface in depth — how far up the page its near and far edges sit — and leaves free only the splay, which is the quantity in question. A redraw that also moved the near corners or the far edge’s height would find a smaller distance by spending the correction on properties nobody doubted. It is the same kind of correction the cube that is a box measured for a two-point construction whose far edges were placed by eye: the far corners are where the judgement went, so the far corners are what a correction moves.

The distances are also worth reading against the size of what moves. The floor’s far corners move 13.1 px on a near edge 560 px wide, about two per cent of the surface’s width. The footstool’s move 25.2 px on a near edge 170 px wide, about fifteen per cent. The smallest surface carries the largest share of the redraw, because the common meeting point is placed where the large surfaces want it and the small ones have to follow.

That is a useful property in a measure of this kind, and an honest caution. It is useful because a copyist’s eye would find the footstool’s correction the conspicuous one, and the measure agrees. It is a caution because a root-mean-square over corners weights every surface equally, whatever its size, and a measure weighted by area would report a different number. The figure reports the unweighted one, and the essay says so rather than implying the number is the only reading.

When no lens reconciles anything #

A second construction takes the question somewhere the first cannot. It is the same kind of stack, with one surface — the book — splayed hard enough that its sides meet above the picture’s centre rather than below it.

Now the meeting points lie on both sides of the centre row, and no lens brings the tilts together. Three surfaces read as leaning one way and the fourth as leaning the other under every focal length — the sign, as the single-surface reading found, does not depend on the lens — and the spread has no peak. It runs from 80.0° at 260 px down to 10.5° at 5,000 px, and toward 180° as the lens shortens further.

The lens-free distance registers the difference as well. The nearest one-camera drawing of the mixed stack moves the far corners 23.8 px root-mean-square, against 17.6 px for the stack whose meeting points all lay below. The book costs 24.6 px on its own, and the footstool and table almost as much, because a common meeting point placed to suit three surfaces splayed one way must drag the fourth all the way across the centre.

What the geometry establishes and what it does not #

The results are exact for the constructions, and the constructions are not paintings. It is worth stating the reach of the argument carefully, because the subject invites more.

It establishes a test. A picture of surfaces a reader takes to be parallel is one camera’s picture of them exactly when their receding sides share a meeting point. The test uses no lens, it can be run on any one-point drawing, and the photographed control shows it passing where it should.

It establishes a lens-free size. When the surfaces disagree, the redraw distance says by how many pixels, on the page as drawn.

It does not establish that any icon fails the test. No painting was measured. Whether the divergent constructions of a particular tradition share meeting points is an empirical question about particular pictures, and nothing here answers it.

And it depends on which surfaces are taken to be parallel. A floor and a table top are parallel in most rooms. A lectern is not, and the single-surface essay noted that the objects shown divergent in icons include lecterns and books presented on stands, which genuinely lean toward a reader. Four meeting points are evidence against one camera only if the four surfaces are parallel in the world depicted, and that is a fact about the depicted world, which a picture does not supply. One picture cannot supply it for exactly the reason no single-view recovery can supply a size.

So the honest form is conditional, like the single-surface result. If these surfaces are parallel, no single camera drew them, and the nearest one that could have moves the picture by a stated number of pixels. Whether they are parallel is outside what the drawing contains.

Why this is not the same question as the table’s #

It is worth placing this next to what each system answers, because there are two separate kinds of question about divergent pictures and they are easy to merge.

The table asks what a drawing system preserves, and every row of it is a convention applied uniformly: every surface of a cabinet drawing is drawn by the same rule. A centre and a measure are exclusive is a statement about rows, and a yes in the table is a price priced them on a scene drawn by one rule throughout. A divergent construction applied uniformly, with every surface splayed at the rate its lean requires under one camera, is not a separate system at all — it is an ordinary perspective picture of an unusual object, and it does not need a row.

What this essay measures is a picture in which the rule is not applied uniformly: each surface splayed on its own terms. That is a property of a picture rather than of a system, and it is the property the picture whose lines spread first drew attention to. The table cannot represent it, because a row is one rule, and the redraw distance is the measure a table does not have: how far a picture is from being the output of any single rule of the pinhole kind.

Still open: how few cameras can draw a divergent picture #

If one camera cannot draw a picture’s surfaces, the natural next question is how few can. The redraw distance forced every surface to one meeting point; the open question relaxes that to several. With a tolerance stated in pixels — say, the width of a brush line — the surfaces of a divergent picture can be partitioned into groups whose receding sides share a meeting point within that tolerance, and the smallest number of groups is the number of cameras the picture needs. A measurement of it would build constructed pictures with two and three groups hidden in them, test whether the partition recovers the groups it was built from as the tolerance narrows and widens, and find the tolerance below which a picture drawn by one careful hand splits into more cameras than it was drawn with — the point at which the count stops describing the construction and starts describing the brush.