Measuring a room off the page

The metrology field on this site is about reading a scene back out of a photograph of it, and every essay in it begins the same way: before anything on a plane can be measured, the plane has to be rectified — mapped back to its true proportions by a homography fitted from four marks.

That step is not free. It needs four points whose true configuration is known, it needs them in general position, and it propagates the error in every one of them into every subsequent measurement, unevenly, with the far corner of the plane worst affected.

An oblique picture of the same floor needs none of it.

An oblique picture of a plane is already a plan #

The map an oblique projection performs on the ground plane is

$$[u, v] = [x + d z \cos a,\; -(y + d z \sin a)]$$

which for a point on the floor, $y = 0$, is a linear map of $(x, z)$: a shear composed with a scale. It has no divide in it, so it is affine rather than projective, and an affine map of a plane is invertible by another affine map with no fitting at all — the shear angle and the depth ratio are properties of the drawing system, not of the picture.

The consequence is worth stating in one line. On an oblique drawing of a floor, a ruler laid on the paper measures the ground, up to one scale factor and one known shear. Two lengths in the same direction are in their true ratio anywhere in the picture; two areas are in their true ratio anywhere in the picture; and a length in one direction relates to a length in another by a constant that is the same everywhere.

A perspective picture offers none of those three without the rectification.

What the rectification costs, in numbers this site already has #

The metrology field measured this and the numbers transfer directly.

Rectifying a plane from four corners of a known rectangle recovers three lengths it was never given to $4 \times 10^{-16}$ relative — with exact marks. The interesting quantity is what happens with marks that are one pixel out, and the answer is the sensitivity per pixel: the error in a recovered length is the marking error amplified by a factor that depends on where on the plane the length sits and on how oblique the view is.

Two features of that amplification matter here.

It is not uniform. The far part of a receding plane is compressed in the picture, so a fixed marking error there corresponds to a larger world error — which is the same $1/Z$ that compresses the picture in the first place, appearing again in the error budget.

And it depends on the four marks. A rectification is a fit; a different choice of four corners gives a slightly different homography and hence slightly different answers everywhere. There is no such choice in an oblique drawing, because there is no fit.

The comparison, stated as a working procedure #

It is worth writing the two procedures side by side, because the difference is procedural rather than philosophical.

To measure a room off a photograph: find four points on the floor whose true configuration is known; check that they are in general position; solve the eight-equation system for the homography; apply it to every point of interest; and carry the marking error through the whole chain, remembering that it is larger at the far end.

To measure the same room off an oblique drawing: put a ruler on the paper. Multiply by the scale. If the length runs along the depth axis, divide by the depth ratio $d$, which is a property of the drawing system and is written on the drawing or inferable from any right angle in the plan.

The second procedure is what a scale drawing is, and it is the reason architectural drawing settled on parallel projection and stayed there for six hundred years after perspective was available. The convention in a Japanese narrative painting and the convention in a modern engineering drawing are the same convention, and the property both are exploiting is the one measured here.

Why the shear does not spoil it #

An obvious objection: an oblique projection shears the floor, so a right angle in the room is not a right angle on the page, and a reader measuring an angle gets the wrong answer. How can a sheared picture be a plan?

The answer is that a shear is an affine map and affine maps preserve exactly the quantities a plan is for.

Ratios of parallel lengths survive. Two lengths in the same direction are in their true ratio, wherever they are and whichever direction it is.

Ratios of areas survive. A shear multiplies every area by the same determinant, so two areas are in their true ratio even though neither is at true scale.

Parallelism and midpoints survive, which is the site’s own test for a measure-preserving system.

Angles do not, and neither do ratios of lengths in different directions without the depth ratio.

So the honest statement is: an oblique drawing of a floor is a plan in a sheared coordinate system, and undoing the shear needs one number. That number — the depth ratio and the shear angle, which the drawing system fixes — is known in advance rather than fitted, which is the entire difference from a homography.

A perspective picture of the same floor is not a plan in any coordinate system, sheared or otherwise, because a projective map does not preserve ratios of parallel lengths at all. That is the gap: one is a plan needing a known correction, the other is not a plan.

Which oblique system is the best ruler #

An affine map is invertible everywhere with the same amount of work, and that does not make every oblique system equally good to measure on. The map has an anisotropy, it has a condition number, and the condition number picks out one system as the best ruler in the family.

The floor’s map is $[u, v] = [x + dz\cos a,\; -(y + dz\sin a)]$, which on the ground plane is the matrix $\begin{pmatrix}1 & d\cos a\\ 0 & -d\sin a\end{pmatrix}$. Its determinant is $d\sin a$ and its squared Frobenius norm is $1 + d^{2}$, so its two singular values satisfy

$$\kappa + \frac{1}{\kappa} = \frac{1 + d^{2}}{d\,\sin a},$$

with $\kappa$ the ratio of the larger to the smaller — how much a length in the worst direction is stretched relative to one in the best, and therefore how unevenly a marking error on the page propagates into the room.

Working the named systems through:

The last row is the finding. Setting $\sin a = 1$ and $d = 1$ makes the right-hand side exactly 2, so $\kappa = 1$ — the map is an isometry, and the floor is drawn at true scale in every direction with no distortion whatever.

That is the military or planometric projection, where the plan is turned in its own plane and the verticals go straight up the page. Its floor is the plan, at true scale, needing no depth ratio, no shear correction and no scale factor beyond the drawing’s own. Every other oblique system trades some of that away, and the table says how much: cabinet, the prettiest of them, is the worst ruler of the four, because halving the depth axis costs more conditioning than the shallower shear saves.

Three readings follow.

The conditioning is constant across the picture, which is the whole advantage over a rectified photograph and is worth saying as a comparison rather than as a property. A cabinet drawing’s worst direction is 3.2 times its best, everywhere; a photographed floor’s worst place is worse than its best place by a factor that grows without bound as the plane recedes. So the oblique drawing is not merely easier to measure — it is measurable to a stated precision, and the statement is one number for the whole sheet. A projective map’s amplification grows with depth without bound; an affine map’s is one number for the entire plane, and here it is a number a reader can look up from the drawing system’s name.

The optimum is not a compromise. $\kappa = 1$ is attained rather than approached, and only at $d = 1$, $a = 90°$ — so there is exactly one oblique system that is a perfect ruler for a floor, and it is the one draughtsmen reach for when a plan has to be measured off. That the geometry singles out the same system that practice singles out is the sort of agreement worth reporting.

And the price of the optimum is the elevation. Drawing the depth axis straight up the page means the picture’s vertical carries both height and depth, which is exactly the ambiguity a picture with no size–distance signal prices. So the best ruler for the floor is the worst arrangement for the things standing on it, and the systems that are used for narrative rather than for measurement are the ones that give some of the floor’s conditioning away to keep the two axes apart.

Which is the same trade the removed roof makes with its elevation, one rung over, arrived at through a condition number rather than through a coverage curve.

Where the oblique drawing gives up #

Two things it does not give, and the second is the one that would matter to somebody trying to use a fukinuki yatai painting as evidence.

It gives up depth as a free measurement. The depth ratio $d$ is a convention rather than a measurement — cavalier draws depth at full length, cabinet at half, and neither is a projection of anything, as this site’s parallel field measures. So a length along the depth axis is recoverable only if the ratio is known, and the ratio is a choice the draughtsman made. A drawing that does not state it is a drawing whose depths cannot be read.

And it gives up the guarantee that the picture is of anything. A cavalier drawing is not a projection of any solid from any direction; it is a construction. So a fukinuki yatai interior, measured, gives back a room that need not be the room — the measurements are self-consistent and correspond to a real building only if the draughtsman was consistent.

That second point is the honest counterweight to the whole essay. A photograph is evidence in a way a scale drawing is not: it was made by light, and the geometry of the picture constrains the scene whether anybody was careful or not. A drawing constrains the scene only as far as its maker did. The measurement advantage is real and it is an advantage in convenience and error propagation, not in evidential force.

Area, which is where the difference is largest #

The single sharpest number in the comparison is not a length but an area.

Under the parallel system, a square metre of floor images to the same area everywhere: the ratio of the largest imaged square metre to the smallest across the whole building is 1.000000000000. Under the eye that frames the same building it is 1.535.

Areas matter because they are what a reader estimates without meaning to. A room drawn half again as large as another reads as half again as large, and in a perspective picture of a long building that impression is an artefact of position. In an oblique one it is a measurement.

This is also the quantity that makes the convention work for its actual purpose. A narrative painting showing a scene in each of several rooms is asking a reader to compare them — who is where, how many people, how much space — and a system in which the same space is drawn the same size wherever it is makes that comparison free.

What a reader has to be told, and what they can find #

A useful way to compare two measuring instruments is to list what each one needs supplied from outside.

A perspective picture needs: four points on the plane whose true configuration is known, a decision about which four, and — for anything metric rather than proportional — a reference length. It needs nothing about the camera, which is the remarkable part and is what makes single-view metrology work at all: the homography absorbs the intrinsics.

An oblique drawing needs: the depth ratio, and a reference length. It does not need any points on the plane, because the map is known in advance rather than fitted.

Those are close to equal in count and very different in kind. The photograph’s four points have to be found in the picture and marked, and every marking error propagates; the drawing’s depth ratio is a single number that either is written on the drawing or is not. One instrument’s uncertainty is spread over four measurements a user makes; the other’s is a single fact a user either has or does not.

There is also a difference in what can be checked. A photograph’s four points over-determine nothing — four is exactly the number a homography needs — so there is no residual to inspect and no way to tell a bad marking from a good one. Mark a fifth point whose configuration is known and there is: the homography becomes a fit, and the residual reports. That is worth doing and is rarely done, and it is the same redundancy argument the multi-view field makes at a larger scale.

An oblique drawing offers the same check for free in a different form: every right angle in the plan is a check on the shear, and every parallel pair is a check on the projection having been kept.

One more thing the affine map gives #

A last property, easy to miss and genuinely useful: parallel edges stay parallel.

Under a homography they do not — that is the point of a vanishing point — so a rectangular room in a photograph is a quadrilateral, and deciding whether the room is really rectangular means fitting something. Under an oblique projection a rectangle is a parallelogram, always, and a parallelogram whose sides are not in the drawing system’s own two directions is a room that is not rectangular.

So an oblique drawing lets a reader check the shape of a room by eye, immediately, in a way a photograph does not. That is the same property the site’s wrong field exploits in the other direction: a perspective drawing made by hand can be asked what solid it depicts and will usually answer not the one intended, precisely because the projective map has enough freedom to hide the error. The affine map has less freedom and therefore less room to be wrong in.

Two numbers that make the case concrete #

To finish with something a reader can carry, here are the two comparisons in the form of a task.

Task one: how long is the far room? On the oblique drawing, measure it and divide by the scale. Done, exactly, with an error equal to the measuring error.

On the photograph: find four points on the floor whose true configuration is known — which usually means assuming a rectangle somewhere — mark them, solve the homography, transform the two ends of the room, and take the difference. The error is the marking error amplified by a factor that at the far end of a receding plane runs to several times the near-end value.

Task two: is the far room the same size as the near one? On the oblique drawing, compare them directly: the area spread across the whole building is 1.000000000000, so the drawn areas are in the true ratio.

On the photograph, the same comparison is wrong by up to 1.535× before any measurement is taken, and correcting it requires the rectification from task one.

That second task is the one a reader performs without noticing, on every picture, all the time. In an oblique drawing the impression is a measurement. In a photograph it is an artefact of where the rooms are.