Dividing to a point off the board

A wall stands on the ground at an angle to the direction the camera is looking. Its base and its cap are parallel courses of the same masonry, so in the picture they converge, and every course of brick between them converges on the same point. That point is where the drawing has to aim, and on a wall turned forty degrees it is 450 pixels past the right-hand edge of a 690-pixel sheet — nearly two thirds of a canvas width of nothing.

Turn the wall further and it gets worse rather than better. At fifty-five degrees the meeting is 1009 pixels out, one and a half canvas widths, and the hero figure above is the whole of the difficulty in one picture. A longer sheet does not solve it; a longer sheet moves it. The vanishing point of a horizontal direction sits at $f\tan\psi$ from the principal point, and $\tan\psi$ has no bound, so for every sheet there is a wall whose vertex is off it.

The construction the drawing offices used does not go to the point at all. Divide each end of the wall into the same number of equal parts, join like-numbered marks, and the joins are the courses — a pencil through a vertex nobody drew, correct to the arithmetic floor. The measurement below is that the construction really produces that pencil, and the three things it quietly assumes, one of which fails on any picture taken with the camera tipped.

The homothety that does the work #

The reason is one sentence long and is worth having before anything is measured, because everything that follows is a matter of finding out which part of the sentence is doing the work.

A pencil of lines through a point cuts two parallel lines in two ranges, and the map between those ranges is a homothety — a uniform scaling centred at the vertex. A homothety preserves ratios along a line. So the point one third of the way up the near end and the point one third of the way up the far end are joined by a line of the pencil, and dividing both ends into equal parts and joining like-numbered marks reproduces the pencil exactly.

Nothing in that argument mentions the vertex’s position. It could be a pixel outside the frame or a kilometre away and the construction is the same three operations — mark, mark, join — with the same result. That is the property being exploited, and it is not the same property as the vertex being far away is a small error. The vertex is not an input.

What the argument does mention, twice, is that the two ends are parallel. Both ends of this wall are verticals in the world, standing on a level floor, photographed by a level camera; verticals parallel to the picture plane stay parallel in the picture, and the two ranges are therefore two parallel lines. Remove that and the homothety becomes a general perspectivity, ratios stop being preserved, and equal divisions no longer correspond. The recipe as taught does not state the condition, and the section on tipping the camera below is what it costs.

Three pairs, and only two are given #

The construction is a map between two lines — the near end of the wall and the far end — and the right way to see what it needs is to ask what determines such a map.

A line is a projective space in its own right establishes the count: a projectivity of a line has three degrees of freedom and is fixed by three pairs of corresponding points. Two pairs leave a one-parameter family, and every member of that family agrees with both given pairs exactly.

The wall hands over two pairs and no more. The base of the near end corresponds to the base of the far end, because they are the two ends of the wall’s bottom course. The cap corresponds to the cap. That is two, and a projectivity wants three.

Thirty pixels on a 690-pixel sheet is not a rounding difference; it is the difference between two visibly different walls. And the six lines of that fan are not six degrees of carelessness. They are six exact constructions of six different masonry patterns, all of which have the same base and the same cap.

What “divide both ends into equal parts” is smuggling in #

So the recipe has to be supplying a third pair, and it does, in the sentence that looks like an instruction about arithmetic.

Dividing a segment into equal parts is an affine operation. It fixes the point at infinity of the line it is performed on — the ends of the segment go to the ends, and equal steps go to equal steps, which is a statement about the direction rather than about any drawn point. Doing it on both ends and pairing like-numbered marks therefore asserts that the two lines’ points at infinity correspond.

On this wall they do, because the two ends are parallel on the paper and two parallel lines share a point at infinity. That is the third pair, and it is a genuine pair of corresponding points rather than a convention: the last member of the fan above, the one at exactly zero, is the parallel choice, and the homogeneous layer the machinery works in holds it as an ordinary value rather than as a large number standing in for one.

Which makes the recipe honest and incomplete in a specific way. It is not that it omits a caveat. It is that its third input arrives disguised as a step, so a draughtsman applying it to two ends that are not parallel — a wall on a slope, a wall photographed from a tipped camera, a wall whose ends have been foreshortened differently — performs the same three operations and gets a different and wrong map, with nothing in the procedure to flag it.

How many courses the wall will take #

The construction has a range, and it is set by something other than the vertex — which is worth establishing because the vertex is what a reader is now watching.

The two ends of the wall are drawn at different lengths, because one is further away. Dividing both into $n$ parts puts the marks $n$ times closer together on the far end than the near one in the ratio of the two drawn lengths, and the joins that cross near the far end therefore cross at increasingly shallow angles. Push the count up and the limit arrives not as an error in the answer but as a pair of marks a divider cannot separate. On the wall drawn here the slider runs to twelve courses and the residual does not move at all, which says the arithmetic is untroubled; the drawing gives out first, and it gives out on the far end.

That is a different limit from the one the guessed vertex runs into, and the two are easy to conflate. A guessed vertex is a wrong map executed cleanly — every mark is crisp and every mark is in the wrong place. Crowded divisions are the right map executed on marks too close together to place, and its errors are unbiased and shrink with care. The first is not repaired by a sharper pencil and the second is nothing else.

Which is the distinction the ladder that reaches every denominator draws between a construction limited by how many times it is applied and one limited by what it is dividing. This one is in the second class: its cost is a fact about the wall’s foreshortening and not about the number asked for, so a shallow wall takes many courses and a steeply receding one takes few, whatever the vertex is doing.

A vertex guessed short #

The alternative a hurried draughtsman actually reaches for is to extend the base and the cap, see them closing, and put the vertex where the sheet ends. It is worth pricing, because it is the thing the construction exists to avoid and because the price is not obvious.

Three hundred millimetres is a course and a half of brick, on a wall a few metres long, from an error that consists of putting a point at the edge of the paper instead of a foot and a half beyond it. The dashed marks and the solid ones are close enough together that the drawing looks like a drawing. There is no configuration of lines in it that crosses badly, no crowding, nothing to squint at.

That is the shape of failure this collection keeps recording. Dividing depth by eye misplaces a post by metres and produces a perfectly convincing row; the two-point cube everyone is taught is a box rather than a cube and looks exactly like a cube. A construction that is wrong in a way the drawing shows gets corrected on the board. A construction that is wrong in a way the drawing conceals gets published.

The by-eye rule that is nearly right, and the reason it survives #

There is a second by-eye rule, and it was written into this essay as the control — the neighbouring case where the effect is absent, which is supposed to fail loudly and did not.

The rule is to keep the angle between successive courses even. It is the only quantity about a pencil a draughtsman can judge without reaching its vertex, and it is wrong in principle: the angles of a pencil are not linear in where its lines cut a transversal. The cross-ratio is what is linear, and an eye has no access to a cross-ratio — that is the whole of why the invariant is worth having.

Wrong in principle, and on this wall it misses by less than a quarter of a pixel.

The reason is geometric and is the better half of this essay’s finding. A pencil whose vertex is far away is nearly a parallel bundle, and a parallel bundle’s angles really are even. So the by-eye rule converges on correctness in the same limit that makes the correct construction necessary. Where the vertex is comfortably on the paper the rule is at its worst and nobody needs it; where the vertex is off the board it is nearly exact and the draughtsman who used it all his life had no way to find out that he had been relying on a coincidence.

That is why a rule can be wrong and survive four centuries of workshop practice. It is not that nobody checked. It is that the cases in which it is checkable are the cases in which it fails, and the cases in which it is used are the cases in which it very nearly works. The control had to be replaced with the guessed vertex above, which does fail, and the near-miss turned out to be the more interesting number.

The condition nobody states, and what a tipped camera costs #

The homothety argument needs the wall’s two ends to be parallel on the paper. A level camera photographing verticals on a level floor supplies that. A camera tipped down — which is every photograph of a pavement, every view over a balustrade, and every drawing with a third vanishing point in it — does not.

Twelve degrees is not an extreme camera. It is the tilt of a hand-held photograph of a building’s lower storeys, or of a drawing laid out with a vertical vanishing point a few sheet-widths below the paper — the arrangement a three-point layout is built around. Past it the construction is contributing more error than the pencil is, and the recipe has said nothing.

The control here reads exactly zero. At pitch nought the residual is the arithmetic floor and at thirty degrees it is 6.4 px, a ratio of some sixty trillion, which is what makes the measurement discriminating rather than a statement about a tolerance. A check whose two branches read the same number is not a check, and this site has shipped three of those.

Exact and drawable, against exact and undrawable #

It is worth putting this construction beside the one it most resembles, because they arrive at opposite conclusions from the same starting point and the difference is the useful part.

Carrying a height across the room transfers a known height from one place on a floor to another with two lines. It is exact at every camera and every pair of positions. And one of its two lines has to be drawn to the vanishing point of the join of the two feet, which for feet standing abreast runs off to infinity — thousands of canvas widths, half a kilometre across the desk on a printed page.

Both constructions are exact everywhere. The difference is that the height transfer’s vanishing point is an input — a line must actually be drawn to it — and the wall’s is not. So the height transfer has a range of arrangements in which it cannot be executed at all, and the courses construction has none: it is exact over the whole sweep of wall angles and drawable over the whole sweep, and the vertex may run three canvas widths out without costing anything.

Which sharpens the rule this collection has been circling. A construction is undrawable when a point it must reach leaves the paper, and not when a point it is about does. Those are different relations to the same absent point, they look identical in a diagram, and only the first has a repair that involves the size of the sheet.

What the measurement does not settle #

Three limits, and the first is the one most likely to be misread.

The residual quoted throughout is a statement about geometry executed in double precision, not about a person with a pencil. Every join here is drawn between two marks made with dividers, and dividers on paper are good to perhaps a fifth of a millimetre. The construction’s own contribution to the error is fifteen orders of magnitude below that. What the number says is that the method is not what limits the drawing — which is exactly the claim worth having, because the alternatives measured above contribute errors a hundred times a pencil’s width.

The second is that dividing each end into equal parts uses a ruler. This is not a straightedge-only construction and does not pretend to be. The ruler is legitimate because it is used on a line lying in the picture plane, where the scale is uniform, which is the same licence the measuring point takes on the ground line and the same one Alberti’s section takes on the panel’s edge. A ruler laid along a receding line would be measuring the wrong thing entirely.

The third is that the courses come out in the right place and in no particular relation to the world. The construction reproduces the pencil the picture already implies; it does not know how many courses of brick the wall has. Six equal divisions of the drawn end are six equal divisions of the world’s vertical only because a vertical at a fixed depth images with its heights in proportion. Move to a course pattern that varies in depth and the licence is gone, which is the question the bays that are not equal takes up on the receding direction rather than the vertical one.

The absent point, as a class #

The vertex of this pencil is the first of four points this collection measures that are exactly where a construction needs them and nowhere a draughtsman can reach.

The measuring point sits at the eye’s own distance from the receding direction’s vanishing point, which for any reasonable focal length is off the sheet — and the classical response, to move the vanishing points closer together until it fits, quietly widens the lens and is what putting both vanishing points on the paper prices in centimetres of the reader’s room.

The distance point of a pavement construction is the same quantity under another name, drawn on the horizon of every such construction since the fifteenth century and named as a distance in none of them.

The vanishing point of a join is the height transfer’s, above, with no bound at all.

And the vertex of a pencil of courses is this one, which is the only one of the four that can simply be ignored. That is not a difference of degree. The other three are reached, or approximated, or worked around with an intermediate mark; this one is never an input, and the construction that avoids it is not a workaround but the natural way to execute a homothety with two rulers and a straightedge.

The general form is the one the bay repeated by a straightedge states from the other direction: a construction made of joins and meets is exact at any length, and what varies from case to case is not whether it is right but whether the paper is big enough to contain the marks it needs. Asking which points a construction reaches, rather than which points it is about, sorts the taught constructions into two piles quickly, and the pile that needs no repair is smaller than the manuals suggest.