A yes in the table is a price

The comparison these essays have been built on is a table of nine drawing systems and five yes-or-no questions. A centre and a measure are exclusive found its one clean result: exactly one row has a centre of projection, and it is exactly the one row without a true measure. Each system answers its own question asked whether that result was manufactured by choosing the columns after the rows, and what perspective gave up priced the pinhole’s own losses on four quantities.

A yes-or-no column is a threshold, and a threshold throws away the thing it thresholds. Two systems that both answer yes, measure is kept might keep it on every edge of every object or on a third of them. Two that both answer no might lose a little or everything. The table cannot say, because it was built to say something else.

This essay replaces three of its columns with prices. The result keeps the exclusion — no row with a centre is free — and rearranges almost everything else.

The scene, and why boxes #

A price needs a scene, and the scene decides the prices. So it is worth stating exactly, and stating why it is this one.

Four hundred boxes stand on the ground, seeded so the same four hundred appear in every picture, each between 0.6 m and 2.4 m on every side, scattered across twelve metres of width and twelve of depth. Each system draws all of them. For each picture, the reader is then handed the single best ruler that picture admits — the scale that makes the average error of reading every edge as small as it can be — and the price is the average error that remains.

Boxes rather than segments pointing in random directions, because a drawing system is a convention for drawing rectilinear things aligned with it. A scene of random directions would price every system on a subject none of them was designed for, and the result would be a table of how badly nine tools do at a job nobody gives them. The aligned scene is the flattering one on purpose. What turning it does is the control, and it comes later.

Choosing the best ruler after the fact is also flattering, and equally on purpose. Every system gets the scale that suits it, so a price measures only what no ruler can repair: the fact that one picture draws some edges of equal length at different lengths. That is precisely what “keeps measure” was trying to say.

An edge, read with the best ruler #

The length prices sort the nine systems into four groups, and the groups are not the table’s two.

Free: isometric, cavalier and the plan oblique. Every edge of every box comes back at its true length under one ruler, to the arithmetic floor. In isometric the three axes are drawn at one scale; in cavalier the receding axis is drawn at full length; in the plan oblique the ground is drawn true and verticals at full height.

A sixth: dimetric and cabinet at 16.7%, trimetric at 15.9%. Each draws one axis at a different scale from the others — cabinet halves its receding edges — so a ruler that reads two thirds of the edges exactly misreads the third by half, and the average comes to a sixth. Trimetric’s three unequal scales land a little under that.

A third: the elevation, at 33.3% exactly. It draws every edge running away from the viewer with no length at all, and an edge drawn as a point is wrong by its whole length whatever ruler is used. A third of every box’s edges run that way.

Most: the pinhole at 39.5% and the handscroll at 39.2%, each drawing the same edge at a length that depends on how far away it is.

The table files the elevation with isometric. On price, the elevation is further from isometric than it is from the pinhole: 33.3 points separate it from the systems it shares a yes with, and 6.2 points separate it from the one system it does not.

Where the exclusion survives #

It would be easy to read that as the table being wrong, and it is not. The exclusion is exactly what survives the pricing.

Every row with a centre of projection — here, only the pinhole — pays a length price that no ruler can remove, and the price depends on the scene’s depth. Every row without one pays a price that is fixed by the system’s axis scales, independent of where in the scene an edge sits, and zero for the systems whose scales are equal. A yes in the measure column turns out to mean the price is a constant of the drawing rather than a function of the scene, and that is a real and exact distinction. It is simply not the same distinction as the price is zero.

So the column was answering a sharper question than its wording, and the wording invited the misreading. A reader told that the elevation “keeps measure” and asked to take dimensions off one would lose a third of them — every depth — and would have been told nothing false.

Why a sixth, and why a third, exactly #

The round numbers in the length prices are not coincidences of this scene, and seeing why they come out round makes the rest of the prices easier to trust.

The best ruler is chosen to make the average error smallest, and minimising an average of absolute errors picks a weighted median: the scale at which half of the edges’ combined drawn length is read long and half short. For a cabinet drawing that median is easy to find. Two thirds of every box’s edges — its widths and its heights — are drawn at full scale, and one third — its depths — at half. The full-scale edges carry more than half of the total drawn length, so the median sits at full scale, and the best ruler reads every width and height exactly and every depth at half its length. An error of one half on a third of the edges averages to one sixth, 16.7%. Dimetric has the same structure with a different pair of scales and lands on the same sixth; which axis scales are possible fixes those scales, and the price follows from them without the scene having any say.

The elevation’s third is simpler still. Its depth edges are drawn with no length, so whatever the ruler, each is misread by its whole length — an error of exactly one — and a third of the edges run in depth. Every other edge is read exactly. The average is a third, with nothing left for the scene to decide.

Trimetric, with three different axis scales, has no such arithmetic. Its median lands on one of the three, and its other two axes are misread by amounts that average to 15.9% on this scene, a little under a sixth. That is the one parallel price in the table that depends on the mix of box proportions, and so it is the one another scene could move.

The pinhole’s 39.5% and the handscroll’s 39.2% are of a different kind entirely. Their ratios are not a handful of constants but a continuous spread set by each edge’s depth, and that is exactly what makes them functions of the scene rather than of the drawing.

A face, and a right angle #

Length is one currency. The same four hundred boxes can be priced in two more.

Area splits the free-for-length group. Isometric stays free: its three families of faces are drawn at one area scale, because its three axes are symmetric. Cavalier charges 13.8% and the plan oblique 19.2%, because each draws one family of faces true and shears the others, and a shear that keeps every edge’s length does not keep every face’s area. The elevation charges two thirds — two of the three face families are edge-on and have no area at all — and the handscroll overtakes the pinhole as the dearest system, at 71.4%.

Angle turns the table over. The price here is the share of right-angled corners a protractor does not read within five degrees of 90°, and the three axonometric systems — isometric, dimetric, trimetric — miss every one. No face of an axonometric box is drawn with a right angle, because no face lies in the picture plane. The four oblique systems keep one face family square and miss the other two thirds. And the two systems with the worst length and area prices are the best at angles: the pinhole misses 60.9% of right angles, and the handscroll only 36.0%, because a box whose front face happens to be square to the eye is drawn with right angles by both.

So the cheapest system for a reader who wants lengths is the most expensive for a reader who wants right angles, and the handscroll, which is last or nearly last on two currencies, is first on the third. That is the substance each system answers its own question was reaching for, now with the size of each answer attached.

Why a scroll reads right angles best #

The angle result looks like a quirk, and it has a precise cause that says something about the handscroll that had not been put this way.

A scroll draws every point’s column as its position along the track, multiplied by a constant, whatever its depth. So an edge of a box that runs straight away from the eye — a receding edge, at one position along the track — is drawn as a vertical line on the paper: every point of it has the same column. An edge running along the track is drawn horizontal, because both its ends are at one depth and one height and so on one row. A vertical edge is drawn vertical.

That settles two of a box’s three families of faces. A top face has an edge along the track and a receding edge, drawn horizontal and vertical: every corner a right angle, at every position along the roll. A front face has an edge along the track and a vertical edge: every corner a right angle again. The third family, the side faces, have a receding edge and a vertical edge, and both are drawn vertical — collinear on the paper, a face with no width, whose corners no protractor can read. One family of corners in three is lost, and the measured 36.0% is that third plus the handful of corners drawn too short to read.

A pinhole keeps the front faces square wherever they sit, since a face parallel to the picture plane is drawn as a similar rectangle, and draws a top face square only where one of its receding edges happens to be drawn nearly vertical, close to the middle of the frame. That is why it misses more. A scroll is a camera that moves, and the same motion that costs it straight lines — a straight line in a scroll is a hyperbola whenever it recedes obliquely — gives every receding edge a single column and every top face its right angles.

It also explains the axonometric systems’ clean 100%. An isometric drawing turns every family of faces away from the picture plane by the same amount, so none of its corners is drawn square, and the symmetry that makes that true is the same one that lets the drawing not say which corner is nearer.

A price that refused to be one #

The angle price is a share, and it is worth saying why it is not the obvious alternative: the average departure of a drawn corner from 90°.

That average is the same for every system above, and it is the same by a theorem rather than a coincidence. The three drawn directions of a box’s axes divide a half-turn into three gaps that sum to 180°. A face spanned by two axes is drawn as a parallelogram whose corners all depart from 90° by the same amount, |90° − gap|. So long as no gap exceeds a right angle, the average departure over the three face families is (270° − 180°)/3 — exactly 30°, for isometric’s three equal gaps of 60° and for cavalier’s 45°, 45° and 90° alike.

A quantity that takes the same value for every system cannot rank any of them. It was computed, found to be constant, and refused as a price, which is the reason the measurement above counts corners read true instead. The episode is small and it is the whole discipline in miniature: an average that looked like the natural measure turned out to be an identity, and an identity measures the arithmetic rather than the drawing.

Turn the boxes #

The flattering scene has done its job. The control is the scene the conventions were not built for: the same four hundred boxes, each turned about the vertical by up to 45° either way.

Of the three systems that were free for length, only the plan oblique stays free. Isometric charges 10.9% and cavalier 15.8%, because a box turned in plan has edges that no longer run along the directions those systems draw at true length. The plan oblique draws the whole ground plane without distortion and every vertical at full height, and a turned box’s edges are all either horizontal or vertical — so its yes holds for any box standing on the ground, in any orientation. It is the only unconditional yes in the column.

Every other system’s price moves, and not always upward. The pinhole falls from 39.5% to 35.1% and the handscroll from 39.2% to 31.9%, because turning a box moves some of its receding edges into directions those projections draw less unequally. The elevation falls from 33.3% to 24.2%, because a turned box no longer has a whole third of its edges pointing straight at the viewer.

What the turn establishes is that three of the four free yeses were bought with an alignment. A drawing office that draws rectilinear buildings square to the sheet pays nothing in isometric; one that draws a street laid out at an angle to the sheet pays a tenth of every dimension. That is not a property of isometric in the abstract. It is a property of isometric together with a choice about how the subject sits, and the yes in the table silently assumed the choice.

Where the pinhole is headed #

One more sweep, because it answers the question the exclusion invites: what does a pinhole’s price do as the scene recedes, and what does it become?

What perspective gave up wrote every one of the pinhole’s costs as a function of a single dial — the scene’s depth range divided by its distance — and found that at a dial of zero a pinhole is a parallel projection. That predicts the price should fall as the scene recedes. It does not say to what.

It falls to the elevation. Moving the field of boxes from 2 m to 100 km takes the pinhole’s length price from 39.5% down to 33.36%, monotonically, and the elevation’s price is 33.33%. A distant pinhole is a parallel projection along its own line of sight, and for boxes standing square to that line, the parallel projection along it is an elevation: every edge pointing at the viewer is drawn with no length.

So the exclusion has a limit, and the limit is not a system that keeps measure. The pinhole’s price approaches the price of a system that the table files as keeping measure, and that price is a third of every edge. The row the table puts on its own is, at the far end of its dial, identical to one of the rows it puts with isometric — which is the precise, priced form of parallel projection is not primitive.

What the prices do not settle #

Three limits, each stated because a table of prices looks more final than a table of yeses.

These are prices of this scene. Boxes, standing on the ground, aligned or turned by up to 45°, seen from one elevation. A scene of cylinders, of sloping roofs, or of boxes seen from above would give different numbers, and there is no scene on which the numbers are the prices of the systems themselves. The table’s yes-or-no answers are scene-free because they are thresholds; a price is not.

The best ruler flatters every system equally, and that is a choice. A reader handed a fixed ruler — the scale printed on a drawing — would pay more on every system whose scale differs from the printed one, and a cabinet drawing read with its front-face scale would charge a third for depths rather than a sixth. The prices above are the least a reader can pay, not what one typically does.

And the currencies are not commensurable. A 10% length error and a 36% share of misread right angles cannot be added. Which currency a reader pays in is set by what the drawing is for — a builder takes dimensions, a surveyor takes angles, a buyer looks at areas — and the viewer’s place enters none of them. The prices sort the systems for each purpose separately and deliberately decline to sort them overall.

Still open: what fits the world on a page #

The pinhole’s price has one more face that none of these currencies measures, because it is not about any object in the scene. It is about the page. A parallel system draws a ground plane stretching to a distance D over a page area that grows as the square of D, so an unbounded ground needs an unbounded sheet; a pinhole draws the whole of an infinite ground inside a bounded band below its horizon. That looks like the reason a centre was worth its prices, and measuring it finds that the centre is not what does it. A page is bounded by a divide, not a centre sums the drawn area of the ground for each system out to a thousand kilometres, and sets beside them a camera with no centre at all whose page is bounded anyway.