The cube that is a box

Every book on perspective teaches the two-point cube. Draw a horizon. Put a vanishing point on it to the left and one to the right. Draw the near vertical edge of the cube. Run the top and bottom edges from its ends to the two vanishing points. Then place the two far vertical edges, and complete the box.

The step before last is the one this essay is about. Then place the two far vertical edges — where? No printed method gives a construction, because doing it properly needs a measuring point that the method never introduces.

So the drawer places them by eye. The result is a perfectly good picture of a box. Which box is decided entirely by that step.

How the measurement is made #

The drawing is measured by asking what solid it is the projection of.

The two vanishing points and an assumed principal point give the focal length, from the relation that two orthogonal directions must satisfy: f² = −(v₁ − p)·(v₂ − p). With the focal length in hand, the drawn top face — a quadrilateral of four image points — can be back-projected onto the plane it came from, and the side lengths of the resulting rectangle read off.

For a cube the ratio of adjacent sides is 1. For the projection of a genuine cube, measured this way, it comes out 1.000000 with the corner angles at 90° to 2 × 10⁻¹³ degrees.

For the taught construction it comes out to whatever the by-eye step made it.

Why the corner angles prove nothing #

The first thing the measurement reports is that the depicted corner angles are exactly 90°, at every setting, in every variation.

That is not evidence the construction is right. It is a consequence of how the focal length was obtained: the relation used to get it assumes the two directions are orthogonal, so the reconstruction is guaranteed to produce right angles.

This is worth dwelling on, because it is the shape of the mistake that most often makes a check useless. A quantity computed under an assumption cannot test the assumption. The 90° is the assumption coming back out, and reporting it as a result would be the same error as the cross-ratio test that certified a wrong depth construction.

The side ratio is a genuine measurement because nothing in the derivation forced it to be 1. It is the shape of the reconstructed rectangle, and it can come out anything.

The result, which was a surprise #

The first version of this measurement moved both far edges together — one parameter, the fraction of the way toward the vanishing points — and reported a side ratio of exactly 1.000 at every setting.

That looked like a bug. It is a result.

Placing the two far edges at the same fraction keeps the construction symmetric about the near edge, and symmetry forces the depicted solid to be square in plan. A drawer who is even-handed gets a square plan for free, without knowing they were being tested on anything. A square plan is not yet a cube — the solid’s height against its side is a second number, decided by how far the two edges go rather than by their difference, and a square plan is not a cube measures it.

What decides the solid is the difference between the two placements. And that difference is a quantity nothing in the drawing displays, nothing in the method mentions, and nobody drawing a cube is tracking.

How much is eight points #

The difference is measured as a fraction of the distance from the near edge to the vanishing point, so eight points means one far edge placed 8% of that distance further along than the other.

On the figure, that is about nine pixels out of a 600-pixel span — a difference no one would notice and no one is trying to control. It produces a box whose depth is 1.4 times shallower than its width.

Sixteen points either way, still well inside what a hand would produce without intending anything, spans side ratios from 0.52 to 1.94: from a box about half as deep as it is wide, to one nearly twice as deep.

None of those drawings looks wrong. That is the point. Every one of them is a correct projection of some rectangular box, so every one satisfies every internal check a drawing can be given — the edges converge properly, the corner angles reconstruct at 90°, the cross-ratios are consistent. The drawing is not defective. It just depicts something nobody chose.

The ratio is exponential in the slip #

The four readings — 1.000 at no asymmetry, 0.719 at eight points, and 0.52 to 1.94 at sixteen points either way — are one law, and seeing it makes the tolerance computable rather than descriptive.

Take logarithms. $\log 0.719 = -0.330$ at $\delta = 0.08$ and $\log 0.52 = -0.654$ at $\delta = 0.16$: twice the slip, twice the logarithm. So the depicted side ratio is

$$r = e^{-k\delta}, \qquad k \approx 4.1$$

for this layout, and the two ends of the sixteen-point span are exact reciprocals of each other — $0.52 \times 1.94 = 1.01$, which is 1 to the precision the figures are quoted at.

Three things follow.

Slips compose by multiplying. Two independent misplacements do not add their errors; they multiply their ratios, because each contributes additively to the logarithm. A drawing built from several by-eye steps therefore accumulates its shape error the way a chain of scales does rather than the way a sum of angles does.

There is no safe average slip. Equal and opposite slips give reciprocal boxes, so the median outcome of an even-handed hand is a square plan exactly — which is the symmetry result above — while the mean ratio is $\tfrac{1}{2}(r + 1/r) \geq 1$ and is therefore biased deep. Averaging many drawings by one draughtsman does not recover the cube; it recovers something deeper than a cube, and the discrepancy grows as the square of their typical slip.

And the tolerance is a number. Holding the side ratio inside five per cent needs $|\delta| < 0.05/4.1 = 0.012$ — a bit over one per cent of the distance from the near edge to the vanishing point, which on the figure’s 580-pixel span is about seven pixels of difference between the two edges. A careful hand can hold that — but only a hand that knows it is being asked to, and nothing in the method says so, which is the whole finding restated as a specification: the step the method leaves to judgement has a tolerance and never states it.

What the drawing also decides silently #

The same construction fixes a second quantity without mentioning it: the focal length.

With the two vanishing points placed 1,190 px apart on a 690 px canvas and the principal point at the centre, the implied focal length is 580 px, which is a field of view of 59°. The picture is therefore a correct projection only from about 14 cm when shown at a normal size.

Bringing the vanishing points closer together — which is what a drawer does when they want both to fit on the sheet — widens the implied lens further and brings the correct viewing point closer still. That is the mechanism behind the exaggerated look of student perspective work: nothing in the method says how far apart to put the vanishing points, so they get put where the paper allows, and the paper is not a camera.

The fix, which is not new #

None of this is an argument against the construction. It is an argument for finishing it.

The classical method has the missing step. The measuring point places depths exactly, its position is determined by the focal length, and using it removes both free parameters at once — the far edges are no longer placed, they are constructed, and the drawing depicts the solid that was specified.

Checked against the projection, the measuring-point construction lands on the correct positions to 6 × 10⁻¹⁴ px over six divisions. The construction is exact; it is only ever left out.

Why it gets left out is not mysterious. The measuring point sits at the distance from the vanishing point to the station point — usually far off the sheet — and using it forces the drawer to commit to a viewing distance before drawing anything. Placing the far edges by eye avoids both inconveniences, and the price is a solid nobody selected and a lens nobody chose.

What this generalises to #

The pattern is worth naming because it is not confined to cubes.

A construction has a free parameter. Nothing in the construction constrains it. Nothing in the resulting drawing displays it. The drawing is internally consistent whatever value it takes, so no examination of the drawing reveals that a choice was made. And the parameter controls something the drawing is nominally about.

Under those conditions the method will be taught, used and believed for centuries without anyone noticing, because there is no way to notice from inside. Dividing depth by eye is the same shape, and so is placing the vanishing points wherever the paper allows.

What breaks the pattern is having a second, independent way of producing the same picture. Here it is the projection: the construction can be laid over a computed projection and the difference read off. Without that second route there is nothing to compare against, and the only available check is another construction of the same kind.

That is the argument for computing rather than constructing, and it is not that hand construction is inaccurate. It is that hand construction cannot be audited, and this is what an audit finds when one becomes possible.

What a real camera does instead #

It is worth setting the taught construction beside what a projection produces, because the difference is not that the projection is more accurate — it is that the projection has no free parameter at all.

Given an eye, a target, a field of view and a cube, every one of the box’s eight vertices has a position, and there is nothing left to decide. The three vanishing points follow, the depicted side ratio comes out 1.000000, and the corner angles come out 90° to 2 × 10⁻¹³ degrees.

The construction has the same inputs available — a horizon, two vanishing points and an implied focal length — and does not use them to place the far edges. It could: the vanishing points and the focal length determine the two measuring points, and the measuring points determine the depths exactly. The information is present in the drawing and the method does not consult it.

That is the precise sense in which the construction is incomplete rather than wrong. It is a correct procedure with a step missing, and the missing step is the one that would connect the drawing to the dimensions it is meant to depict.

Why the error survives inspection #

Someone checking a two-point cube drawing has a small number of things they can check, and the drawing passes all of them.

Do the top and bottom edges run to the vanishing points? Yes, by construction.

Are the verticals vertical? Yes.

Do the receding edges of the top face meet at the vanishing points? Yes — that is how the back corner was located.

Do the corner angles reconstruct at 90°? Yes, and as the essay above shows this is forced and proves nothing.

Does it look like a cube? Yes, for every value of the free parameter across the whole range measured.

There is no check available from inside the drawing that fails. That is what makes this a good example of the general problem: the drawing is internally consistent, and internal consistency is the only thing a drawing can be tested for without a second, independent route to the same picture.

The same construction with the step supplied #

For anyone who wants the corrected procedure rather than the diagnosis, it is short.

Choose the viewing distance first — it is the one decision that fixes everything else, and avoiding it is what leaves the parameters free. Mark the station point at that distance on the plan.

Place the two vanishing points so that the angle they subtend at the station point is the angle between the two horizontal directions of the box, which for a cube is 90°.

Mark each measuring point on the horizon at the distance from its vanishing point to the station point.

Draw the near vertical edge and mark the cube’s edge length along the ground line with a ruler. Run those marks to the measuring points; the crossings on the two receding base edges are where the far verticals go.

Nothing in that is judged, and the resulting drawing depicts a cube because the depths were transferred rather than guessed. It also takes about twice as long as the taught version, which is the honest reason it is not the taught version.

The same audit on other taught constructions #

Once the method exists — draw the construction, recover the camera, ask what the drawing depicts — it can be run on anything, and it is worth listing what else it finds.

Depth division by eye fails, and by more than this: the best of three common methods misplaces a post by 3.5 m in a row spaced 1.4 m.

The horizon-crossing rule for figures passes, exactly, provided the picture plane is vertical — and fails measurably as soon as the camera is tilted, which is a condition the rule is almost never stated with.

The measuring-point construction passes to 6 × 10⁻¹⁴ px, which is arithmetic noise. It is exact, and it is the part of the classical method that gets left out.

The diagonal method for doubling a rectangle passes exactly, for the same reason: it is built from incidences only.

The pattern in those results is not that hand construction is unreliable. Two of the four are exact. It is that the exact ones are the ones built entirely from incidences, and the failing ones are the ones with a step that a person supplies by judgement — which is a criterion that can be applied to a method before testing it.

Running the audit on a photograph #

The measurement in this essay is applied to a construction, and nothing about it is specific to one. Given any picture containing something rectangular, the same three steps run: find the vanishing points from the drawn edges, get the focal length from the orthogonality relation, back-project a face and read its proportions.

Pointed at a photograph, the audit answers a different question — not is this a cube but is this picture consistent. A genuine photograph of a rectangular object returns three focal-length estimates that agree, because the object really did have three mutually perpendicular edge directions and one lens really did photograph them.

A composite does not. Two objects photographed with different lenses and pasted together give two different focal lengths from the same frame, and the disagreement is the measurement. That is one of the standard tests for image manipulation, and it is the same arithmetic this essay applies to a drawing.

The distinction worth keeping is between the two failure modes. A drawing made by the taught method fails on the proportions while remaining a perfectly consistent projection — there is a camera and a box that would produce it, just not the box intended. A composite fails on consistency: there is no single camera that produces the picture at all. Both are found by the same computation, and telling them apart is a matter of which number came out wrong.

The same audit, on a plane instead of a solid #

The measurement in this essay asks what solid a drawn quadrilateral depicts, and it needs intrinsics to answer: a focal length and a principal point, either known or recovered from a third bundle of edges.

The expansion phase’s metrology field runs the same question in the case where the intrinsics are not available, and the difference between the two is worth having beside each other.

There, four corners of a plane are given, along with the aspect ratio of the rectangle they are assumed to be the image of, and a homography rectifies the plane. Lengths on it come back exactly. What cannot come back is the aspect ratio itself: four image corners are consistent with a rectangle of every proportion, one for each homography taking four points to four points, so the shape has to be supplied.

That is precisely the free parameter this essay is about, relocated. Here, the two by-eye placements of the far verticals are free and the drawing does not say what solid resulted. There, the reference rectangle’s proportion is free and the rectification does not say what plane resulted. In both cases a correct construction has an unnamed input, in both cases the output looks entirely plausible whatever the input was, and in both cases the only way to find out is to ask the drawing what it depicts and compare.

The same question, on a construction that diverges #

Asking a drawing what solid it depicts is this field’s method, and it has an answer for constructions whose sides spread with depth as well as for ones whose sides converge.