A square plan is not a cube

The cube that is a box found the step every two-point cube construction leaves to the eye — then place the two far vertical edges — and measured what it costs. Put the two far edges at different fractions of the way to their vanishing points and the drawing depicts a box rather than a cube, eight points of difference giving a box 1.4 times shallower than it is wide. Put them at the same fraction and the depicted top is square, by symmetry, whatever the fraction.

It is tempting to read that second result as “a cube for free”. It is not, and the reason is worth a whole essay, because it is the half of the by-eye step nobody was tracking even after the first half had been found.

A cube is two statements, not one. Its top is square, and its height equals its side. The test the earlier measurement ran — back-project the drawn top face through the camera the vanishing points imply, and compare its two sides — checks only the first. A square slab and a square tower pass it as well as a cube does. The second number is decided by how far the far edges go, which symmetry does not touch.

So this essay does two things. It measures the height that a square-plan test cannot see, and finds the even-handed drawing is a cube at exactly one placement. And it draws the construction that supplies that placement — the measuring points — and prices every mark it asks a hand to make, since a construction replacing a judgement with marks has only moved the judgement unless the marks are more forgiving.

The height a square-plan test cannot see #

The height is read the same way the plan was. The two vanishing points and the principal point give a focal length, the drawn top face is back-projected onto its plane at a chosen scale, and the near vertical edge is reconstructed at the same scale: its top corner is the reference point the face was reconstructed through, and its bottom corner is where a vertical through the top corner meets the ray to the bottom corner. For the projection of a real cube both numbers come out 1 to the last digits of the arithmetic.

On the layout every figure here shares — vanishing points 1,190 pixels apart on a level horizon, a near edge 202 pixels tall standing just left of the principal point — the two far edges were placed the same fraction of the way toward their vanishing points, from 8 to 52 per cent.

At every placement the top is square to the arithmetic floor, which is the symmetry result, confirmed. The height is anything but constant. At 8 per cent the depicted solid is 2.76 times as tall as its side, a square tower. At 52 per cent it is 0.22 as tall, a tile. It passes through 1 at exactly one placement, 19.4 per cent of the way to each vanishing point, and nowhere else.

The drawing the cube that is a box used to illustrate the construction places its far edges at 42 per cent. That is a square slab whose height is 0.33 of its side. Every check the earlier essay ran on it came back clean — the receding edges meet their vanishing points, the corner angles reconstruct at 90 degrees, the top is square — and the solid it depicts is a third as tall as a cube.

Nothing on the page looks wrong. A reader shown the drawing sees a cube, because a reader does not reconstruct a height from a focal length; they see a box whose near edge and receding edges look plausible, and plausibility is satisfied by the whole curve above. That is exactly the shape of the error the earlier essay described — and that dividing depth by eye finds in a receding row of posts — found one level further in: a free parameter that nothing constrains and nothing displays, controlling the thing the drawing is nominally of.

Why the height depends on the placement is not mysterious once the reconstruction is written out. The near edge is fixed at 202 pixels on the page, so its reconstructed length is fixed. The further toward the vanishing point a far edge goes, the deeper the reconstructed corner lies, and the longer the reconstructed side. So the side grows with the placement while the height stays put, and their ratio falls through one exactly once. A square plan is a statement about the two sides against each other; a cube is a statement about them against the edge the drawing began with.

The construction that places the edges #

The classical fix is the one the measuring point sets out for a single receding row, applied twice — the two-vanishing-point cousin of the distance point that Alberti draws a pavement with. It needs every mark drawn, and the whole board, because the construction’s own inconvenience is that most of it lies off the sheet.

1. Drop the centre of vision onto the horizon, where the principal point is.
2. Draw a half-circle whose diameter joins the two vanishing points; the station point is where the vertical through the centre of vision meets it. Any point on that half-circle sees the two vanishing points at a right angle, which is what a cube’s two horizontal directions need.
3. With compasses centred on each vanishing point, swing the station point up onto the horizon. The two landing points are the measuring points.
4. Draw a level ground line through the near bottom corner, where the picture is at full size, and mark the cube’s side — the drawn height of the near edge — along it to the left and to the right.
5. Join each mark to the measuring point on the far side of the drawing. Where that line crosses the receding bottom edge is the far corner.

On this layout the focal length is 595 pixels, so the station point stands 595 pixels below the horizon — 395 pixels below the bottom edge of a 400-pixel picture. The measuring points land at 591 and 99 pixels across, inside the picture’s width. The far corners land at 19.36 per cent of the way to each vanishing point, and with every mark exact the drawing depicts a square top and a height equal to its side, both to the last digits of the arithmetic.

Two cross-checks keep this from being a construction certifying itself. The recovery reading the finished drawing never sees the station point or the measuring points; it sees four vanishing-point-consistent edges and nothing else. And the far edges the construction produces coincide, to a millionth of a pixel, with the by-eye construction’s edges placed at the construction’s own fraction — so the one placement at which the even-handed curve passes through one is the placement the measuring points draw, found by two routes that share no step.

Every mark a hand must place #

A construction removes judgement only if the marks it substitutes are easier to get right than the judgement was. The measuring points replace two judged placements with five constructed steps, and three of those steps are pure straightedge — lines between marks already made. Two are not. The centre of vision is a mark on the horizon a hand has to put somewhere, and each measuring point is a compass swing whose landing a hand has to read. Those are the marks to price.

The price is stated the same way for every mark: how far, in pixels of the drawing, the mark may be out before the depicted solid is five per cent off a cube, in whichever of its two numbers — plan or height — goes first, and in whichever direction of slip is worse. The by-eye construction is priced the same way, with its two marks read as the difference between the far edges and their common placement.

The two by-eye slips are steep and nearly identical. Moving one far edge against the other reaches five per cent at 4.4 pixels; moving both together reaches it at 4.4 as well. Either way, a drawer who wants a cube within five per cent must place each far edge within about four pixels of a location nothing on the sheet shows.

The measuring-point marks are shallow. The centre of vision may be 29.0 pixels out before the solid is five per cent off, and a single measuring point 42.1. Four pixels on a drawing this size is about the width of a drawn line; forty is a mark that has visibly missed.

The construction drawn above shows it: at forty pixels out, the left measuring point has visibly left the arc it was swung along, and the drawing is 4.8 per cent off a cube. The mistake is large enough to see on the board, and its cost is still under the limit.

Why the construction forgives its marks #

The measuring point’s tolerance is a lever, and it can be read directly off the board.

The far corner lies on the measuring line, which runs from the ground mark to the measuring point. A slip of the measuring point swings that line about the ground mark, and the corner moves by the slip scaled down by how far along the line the corner sits. On this layout the ground mark on the right is at 532 pixels across, the measuring point at 99, and the far corner at 448 — close to the mark end. A one-pixel slip of the measuring point moves the far corner by 0.116 pixels. So the construction turns a forty-pixel slip into a corner about five pixels out, which is roughly what the by-eye drawer was being asked to hold directly.

The centre of vision works through the same lever with one more stage. Moving it along the horizon slides the station point round the half-circle, which moves both measuring points the same way by about 0.7 of a pixel each per pixel of slip; each measuring point then moves its corner by its lever. Both corners shifting the same way is why the centre of vision is less forgiving than a single measuring point: it breaks the construction’s mirror symmetry, and that is the one kind of slip — the earlier essay’s difference — to which a cube’s plan is sensitive.

That symmetry has a companion worth checking, because it tells the two errors apart. Move the two measuring points by the same amount inward, each toward the other, and the construction stays a mirror image of itself: the plan stays square to the arithmetic floor and only the height moves. Move them by the same amount the same way along the horizon and the height barely moves while the plan goes out of square. Mirror-symmetric slips cost height; asymmetric slips cost plan. That is the by-eye result again — common placement decides the height, difference decides the plan — reappearing inside the construction that was supposed to remove it.

The five marks in one table #

Set side by side, the tolerances say what finishing the construction buys, and they say it in the only unit a draughtsman can act on.

The measuring points slipping together the same way are the least forgiving of the constructed marks, at 20.5 pixels, and that is the slip a hand is least likely to make, since the two are swung separately from opposite ends of the horizon. A single measuring point tolerates 42.1 pixels, which is 9.7 times the slip an edge placed by eye tolerates.

So the measuring points do not make a hand exact. Nothing does. What they do is move the hand’s inevitable slip from a mark where it is expensive to marks where it is cheap, by interposing a lever between the hand and the corner. That is a better reason for the construction than the one usually given, which is that it is correct. The by-eye drawing is correct as well, for some cube, at some placement; the construction is the version whose slips a hand can afford.

What the tolerance does not buy #

Two prices sit outside the table, and both are the construction’s own.

The station point has to exist. On this layout it lies 395 pixels below the picture, which on a sheet sized to the drawing means taping on more paper or working on a board — and the vanishing points, at −250 and 940 pixels across, already lie off a 690-pixel sheet. That is why the cube that is a box found the step left out rather than taught wrongly. The distance point is the viewing distance makes the same observation about the one-point version: the only mark on the whole apparatus that encodes where the eye is sits where nobody wants to draw.

The vanishing points are taken as given. Every tolerance above holds the two vanishing points fixed and asks about the marks drawn from them. Where the vanishing points go is itself a choice that decides the focal length, and recovering the camera reads the focal length back out of them. A drawer who places them where the paper allows chooses a lens without meaning to; the construction then draws a correct cube for that lens, seen correctly only from its station point, which the point to stand at turns into a distance in front of the printed page.

And the construction does not check itself. A drawer whose measuring point lands forty pixels out has no way to know it from the finished drawing, which is as internally consistent as the correct one. What the construction offers is not auditability from the page but a slip budget ten times larger; the audit still needs a second route, which here is the recovery.

Where the even-handed curve comes from #

The curve in the first figure is not an accident of this layout, and the one cube on it has a meaning that can be stated without the reconstruction.

At the near corner the picture is at full size: a length along the ground line through that corner is drawn at its true length, which is why the construction can mark the cube’s side off with a ruler. A corner further back is drawn smaller in proportion to its distance from the eye, and how much further back a given placement puts it depends on how fast the drawn edge converges on its vanishing point. The placement that makes the far corner exactly one side deep is therefore fixed by the focal length, the angle of the cube and the size of the near edge — and the measuring point is the device that finds it with a compass instead of a calculation.

So the 19.4 per cent is not a fact about cubes. On a longer lens, with the vanishing points further apart, it would move, and on a larger near edge it would move. Turned so one face is nearly square to the picture, the two fractions would no longer be equal, and symmetry itself would stop being the right target. None of those layouts is measured here; the structure is what carries over. What does not move is the structure: symmetry fixes the plan for free, the height needs one number, and a hand that does not know the number draws a slab or a tower without being told.

What a finished cube asks of a hand #

A drawn cube is right when its plan is square and its height equals its side, and the taught construction’s by-eye step decides both — the plan through the difference between the two far edges, the height through their common placement. Symmetry settles only the first. On the layout measured, an even-handed drawer gets a cube at 19.4 per cent of the way to each vanishing point and a square slab a third as tall at the taught drawing’s 42 per cent, and no test run on the plan can tell them apart.

The measuring points place both far edges exactly. Every mark they ask a hand for may slip much further than a by-eye edge may: a single measuring point 42.1 pixels before the solid is five per cent off a cube, the centre of vision 29.0, against 4.4 for either by-eye slip — because each constructed mark reaches the corner through a lever of about a ninth.

Still open: whether the lever survives a cube turned toward the picture #

Every tolerance here is measured on one symmetric layout, with the cube’s two faces turned equally from the picture plane. That is the case in which the two measuring points sit at equal distances from their vanishing points, the two levers are equal, and a slip on one side costs what a slip on the other does.

Turn the cube so one face is nearly square to the picture and the layout stops being symmetric: one vanishing point runs far off the sheet, its measuring point runs with it, and its lever lengthens while the other shortens. The question that leaves is whether the construction’s advantage over the eye holds there or collapses on one side — whether a measuring point hundreds of pixels off the sheet still forgives its slip by a factor of ten, or whether the long side of a nearly frontal cube is where the taught shortcut and the finished construction come to cost the same. The measurement sweeps the cube’s turn, prices each mark at each turn, and asks at what turn, if any, the ratio of 9.7 falls to one.