The stations are also a staircase

A landscape that changes its rule halfway up handed a reader nothing but the marks of a three-station landscape and got three camera heights back — 1.60 m, 4.00 m and 11.00 m, to 3.6 × 10⁻¹⁵ m. It then measured what that same reader cannot recover across a join: a common ground, which the next band’s marks read as sitting 2.40 m away from the true one at the first seam and 7.00 m at the second. The landscape in question is the one three distances in one landscape built, with a low station for the near ground, a level one for the middle and a high one for the far.

Those two numbers were reported as a disagreement between two honestly-fitted stations. They are that. They are also something else, and the something else is not an approximation to it.

Only one difference appears #

A level eye at height $H$ draws a ground point at depth $d$ and world height $Y$ on the row

$$v = p_y + \frac{f\,(H - Y)}{d}$$

and $H$ and $Y$ appear nowhere except in that difference. Raise the eye by a metre and lower the ground by a metre, and every row of the picture is the row it was — not nearly, but identically, at every depth, for every point.

So a picture drawn from several stations over flat ground is a picture drawn from one station over ground that steps down by the differences of those stations. The claim is not that the two are similar; it is that they are the same set of marks.

The arithmetic is worth running rather than trusting. Take the highest station as the single eye’s height — 11 m — and set each band’s ground at $11 - h_b$: 9.4 m for the near band, 7.0 m for the middle, 0 for the far. Project every one of the ninety-nine ground samples the three bands were built from, and the worst row anywhere disagrees by 5.7 × 10⁻¹⁴ px. That is the arithmetic floor. There is no fit here and no residual to report.

The disagreement was the terrace all along #

Set that beside the number A landscape that changes its rule halfway up measured and they are the same number.

The apparent ground drop at the first join was 2.40 m, and the terraces there are 9.4 m and 7.0 m — a step of 2.40 m. At the second join it was 7.00 m, and the terraces are 7.0 m and 0 — a step of 7.00 m. This is not a coincidence to be checked; it is forced, because both quantities are the difference of the two bands’ stations, and the algebra that produced the first reduced to exactly that.

What changes is what the number is a number about. Read as a station disagreement it is an artefact: the amount by which a reader who wrongly assumed one ground gets the ground wrong. Read as a terrace it is a measurement: the height of a real bank in a real scene that these marks depict exactly.

Both readings are available and the marks do not choose between them. That is the whole of the finding, and it sits precisely where a single view’s limits always sit — a quantity that the picture determines only once something outside the picture has been assumed.

It is worth setting beside the shape these essays keep arriving at from different directions. A divergent construction recovers a rectangle at every splay and leaves the tilt to an assumed focal length. A surface’s meeting points say whether one camera drew them only once the surfaces are assumed parallel. Here the eye’s height is fixed only once the ground is assumed flat. In every case the ambiguity is not in the measurement but in what the measurement is of, and in every case naming the assumption is the whole of the repair.

What the recovery was conditional on #

This puts a condition on that headline result, and the condition is worth stating in its own terms rather than as a caveat.

Recovering each band's own camera height recovers, from a band’s marks, the quantity $H - Y$ for that band, on the assumption that $Y = 0$. The assumption is not tested anywhere in the fit, and it cannot be: a band’s marks are a straight line in $1/d$, the slope of that line is $f(H - Y)$, and one slope cannot separate two numbers.

So the honest statement of the recovery is that a reader recovers each band’s eye height above its own ground, and gets 1.60, 4.00 and 11.00 m if the ground is one plane throughout. The same marks give 11.00 m three times over if the ground steps. They give 6.00, 8.40 and 15.40 m if the ground is one plane sitting 4.4 m below where the first reading put it, which is to say the whole family of readings is a one-parameter family and the marks fix only the differences.

That last form is the useful one. What a several-station landscape determines is the set of differences between eye and ground, band by band — and every one of the three recovered numbers is a difference, correctly recovered, described as a height.

The control has terraces too, and they are zero #

A control was drawn alongside that recovery: three bands butted the same way, from one camera at one elevation throughout, whose recovered elevations agree and whose ground-seam statistic comes out at 8.9 × 10⁻¹⁶ m.

Under the stepped reading that control says something slightly different and rather clearer. Its terraces are the differences of its stations, and its stations are all the same, so its terraces are all zero — the stepped reading and the several-station reading coincide, because the staircase has no steps in it.

That is the right behaviour for an ambiguity: it should vanish exactly where there is nothing to be ambiguous about. A picture drawn from one station over flat ground has one reading, and the second reading is the same reading. The ambiguity appears precisely with the convention, and in proportion to how hard the convention is working.

What a reader would have to be shown #

Two readings, one set of marks, and no way to choose. The natural next question is what evidence would choose, and the answer turns out to be large rather than subtle.

Under the stepped reading, the ground drops at each join. A drop has a face, and a face at depth $d$ is drawn $f \cdot \text{drop}/d$ tall.

At the first join a 2.40 m drop draws 64.6 px, which is 67% of the near band’s entire height on the page. At the second a 7.00 m drop draws 70.0 px, which is 103% of the middle band’s — a cliff face taller than the whole stretch of landscape above it.

So the two readings are not separated by a subtlety that a careful eye might catch. They are separated by the largest single object in the picture, and the picture does not contain it.

That makes the convention’s own habit at a join into something more than decoration. A painter who lays mist, cloud or a bank of trees across the seam is doing exactly what is required for both readings to remain available — not by hiding a small discrepancy, but by omitting the one object whose presence or absence decides the question. The ambiguity is not a weakness of the reading; it is what the convention is for.

Why a band cannot tell on its own #

There is a tempting objection, and running it down is what makes the ambiguity firm rather than merely announced.

A band drawn from a low station over flat ground and one drawn from a high station over ground raised to match are the same rows — but surely the rest of the band differs? A vertical post, a building, the horizon: something in a band ought to know how high its eye really was.

Nothing does, and each case fails for its own reason.

A post of known height gives nothing. A post of height $p$ standing at depth $d$ has its foot on the row $p_y + f(H-Y)/d$ and its top on $p_y + f(H-Y-p)/d$. The drawn height is $f p/d$ — the eye’s height cancels. A row of posts fixes the depth scale and says nothing about the station.

The horizon gives nothing. The ground plane’s vanishing row is the limit of $p_y + f(H-Y)/d$ as $d$ grows, which is $p_y$ whatever $H$ and $Y$ are. The horizon sits at eye level in both readings, on the same row, because raising the eye and raising the ground with it moves the horizon not at all.

And the band’s own extent gives nothing, because that is the slope again. How much page a band spends on its depth range is $f(H-Y)(1/d_{\text{near}} - 1/d_{\text{far}})$, which is the one quantity the fit already reads.

So the free parameter is genuinely free within a band, and it is only across bands that anything is determined at all — the differences. A single band of a landscape determines the height of its eye above its own ground and nothing else whatever, and three bands determine three such heights and two differences between the grounds. Every number in the picture is one of those.

How bold a scheme can be before the reading fails #

There is one thing that does bear on the choice, and it is not geometric.

The terraces are the differences of the stations, so a convention that buys its far band more picture by raising the eye further is asking a reader to accept more relief.

At the convention’s own numbers the largest step is 7.0 m, which is a bank that a great many landscapes hold without remark. Raise the far station to 40 m and the step becomes 36 m; to 160 m and it becomes 156 m, which is not a bank but a mountain wall, and one standing exactly at the depth where the picture happens to change bands.

So the stepped reading has a range of validity that the geometry does not set. Within it, two readings; beyond it, one reading and an absurdity. Where the boundary falls is a judgement about what landscapes look like, and it belongs to whoever is looking at a particular picture rather than to any measurement made here.

It is worth noticing which way that cuts. The convention’s own numbers sit well inside the range where the stepped reading is perfectly plausible — a 2.4 m bank and a 7 m one, at 26 m and 70 m out, are ordinary ground. The ambiguity is not a pathological case constructed to embarrass the recovery; it is the situation the convention actually produces.

What this does not say #

It does not say the landscape was drawn that way. Nothing here claims that any painter drew terraced ground from one station, and the arrangement is not being proposed as the true account of the convention. What is claimed is that the marks permit it. A convention is a practice rather than a proposition, and the practice this one describes — move up for the far distance — is recorded in what painters did rather than inferred from pictures.

It does not extend to a change of pitch. The equivalence is about elevation, and it is about elevation only. A scheme whose bands differ in how far the eye is tilted is not a one-station picture of stepped ground, because a pitch moves the row the horizon lands on and no movement of the ground does — put the same stepped ground under an eye pitched a quarter of a metre’s worth over the same distance and the marks are pixels out rather than at the arithmetic floor.

It does not touch the rate measurements. Three distances in one landscape reported a gain of 6.9 times the picture for the far band and jumps of 2.50 and 2.75 in the rate at which depth runs. Those are properties of the marks, and the marks are the same marks under either reading. What the stepped reading changes is the story about where the marks came from, not the marks.

It does not touch the smooth alternative either. A continuously climbing eye was already priced against the three abrupt stations and found the smooth one gives the far band’s mountain 20.2 px against 71.5. A continuously climbing eye is equivalent, by the same algebra, to a level eye over ground that falls away continuously — a slope rather than a staircase — so the comparison between the two conventions survives the reinterpretation unchanged, with both sides restated.

And it does not make the recovery useless. A reader who has independent reason to believe the ground is one plane — a river running through every band, a shoreline, a road — has fixed the free parameter and the three heights are then exactly right. That is the same service a reference length performs in every single-view recovery: one fact from outside the picture collapses a family of consistent answers to one. The recovery is sound and its input includes one thing the picture does not contain.

The transferable form #

Stripped of the landscape, the result is a statement about what a fit can separate, and it recurs wherever a projection has two quantities entering through a single combination.

A band’s marks determine the number $H - Y$. A fit that returns $H$ has been told $Y$; a fit that returns $Y$ has been told $H$; and a fit that reports either without saying which it was told has reported an assumption as a measurement. Nothing about the fit’s quality bears on this — the fits here are exact, to fifteen decimal places, and being exact is precisely what makes the error easy to miss. An exact recovery of a combination looks identical to an exact recovery of a quantity.

The useful habit that follows is to ask, of any recovered number, which combination the marks actually fix and what was supplied to split it. Recovering the camera from a photograph answers that question explicitly and is the model; a recovery that does not answer it has an assumption in it whether or not anybody has looked for one.

And the second half is about where such a thing gets caught. It was not caught here by a residual, because there is none, nor by a control, because the control has no steps and behaves identically under both readings. It was caught by writing the projection down and noticing that two symbols appear only as a difference — which is an inspection of the formula rather than of any output, and is the only kind of inspection that finds this class of thing.

Still open: whether a seam can be placed where nothing crosses it #

The two readings stay available because the picture omits the riser, and the picture omits it because the convention lays something across each join.

That suggests a constraint on where a join can go that nobody has stated. A join must fall where no object of the depicted world spans it — no river running out of the near band into the middle one, no road, no line of trees — because any such object would be drawn by two rules at once and would show the discontinuity the mist hides. A painter choosing where to put a seam is therefore choosing among the depths at which the landscape happens to be empty.

The measurement that would settle it takes the convention’s own bands and asks how much of a landscape’s depth range is actually available for a join: place objects of stated depth-extent through a scene, and compute the share of depths at which no object spans. If that share is small, the joins in a real painting are not placed by aesthetic judgement but by where the scene permits them — and the depths at which real landscapes of this kind put their seams would be a testable prediction rather than a matter of taste.