A landscape that changes its rule halfway up

Three distances in one landscape built a piecewise picture from three stated cameras — 1.6 m, 4 m and 11 m up — and measured what the arrangement buys and costs entirely from the construction’s own numbers: a stated gain of 6.9 times the picture for the far band, a stated jump in the rate of depth of 2.50 and 2.75 at the two joins. Every one of those figures was known in advance, because the essay built the landscape and simply read its own bill back off it.

This essay asks the harder version of the same question: handed nothing but the marks — no captions, no construction notes, only the drawn ground — what can a reader actually recover, and where does the recovery run out? A total-least-squares fit through each band’s own ground samples returns all three camera heights exactly, to 3.6e-15 m, with nothing about the true elevations supplied in advance. The same honestly-recovered bands then disagree about the ground at every join: a reader who fitted one band’s rule and assumed it continued reads the next band’s marks as a ground point 2.40 m away from the true one at the first seam and 7.00 m away at the second. Recoverable within a band and unrecoverable across one, in the same picture, from the same kind of fit.

How the elevation is read off nothing but the marks #

recoverBandElevation treats a band’s own ground samples the way the horizon is at eye level treats a set of uprights: not as a picture to be looked at, but as a scatter of numbers with a known shape. A ground point at depth $Z$ under a level eye of height $h$ images at a height proportional to $h/Z$ above the horizon, so plotting a band’s drawn ground height against $1/Z$ ought to trace a straight line through the origin with slope $h$ — and fitting that line by total least squares, the identical routine two stations in one picture uses to fit a horizon through several points, hands back the slope, and the slope is the elevation.

Nothing about the true camera height is given to the fit. It is handed a set of drawn positions and a set of depths, and it returns 1.60 m, 4.00 m and 11.00 m — the actual heights each band was built at — off by 3.6e-15 m at worst, which is the same arithmetic floor every exact recovery on this site reaches when nothing is standing between the fit and a genuine, noiseless projection. A reader with a ruler and the picture’s own scale, and nothing else, is in exactly the position this fit is in: three numbers, recovered rather than read off a caption.

Total least squares rather than an ordinary fit of drawn height against $1/Z$ is the same choice fitting a line through a horizon’s own scattered points makes, for the identical reason: an ordinary regression treats the depths as exact and the drawn heights as the only noisy quantity, and here neither axis is more trustworthy than the other — a ground sample’s depth is a modelling choice as much as its drawn height is a measurement. Total least squares minimises the perpendicular distance from the fitted line to every point rather than the vertical one, which is what lets the same routine serve a horizon fit in one essay and a band’s own elevation fit in this one without being told which axis to trust more. And within any one band, the samples that feed it are themselves found the way the same person, twice on one panel finds a horizon from repeated figures — a ground plane sampled at enough distinct depths that a line, rather than a single point, is what comes out the other end.

The ground that is not there #

Recovering a station within a band is one thing; asking what happens where two honestly-recovered bands meet is another, and it needs its own statistic, because a raw pixel offset at the join conflates two different things — the layout choice a painter makes when butting the bands, and a genuine fact about the ground.

groundSeamMetres removes the layout freedom by working entirely in one band’s own units: it takes the near band’s fitted rule, asks what ground height that rule implies at the depth where the far band begins, and compares it with the ground height the far band’s own rule implies there. Since both bands are drawn of the identical, motionless ground, any difference between the two readings is not a fact about the terrain — it is a fact about the camera having moved between them, expressed in the currency a viewer would actually misread it in in the field: metres of apparent ground level. 2.40 m at the first join and 7.00 m at the second are what “the elevation jumped” costs when translated out of a rate-of-depth ratio and into a length.

Working within each band’s own units is not a stylistic choice; it is the only option available. Three distances in one landscape already found that putting all three bands into one shared frame overlaps and inverts them — the middle band’s near edge would land below the near band’s far edge, which a painter avoids by placing the bands rather than by drawing them consistently. A statistic that needed one shared picture frame to be defined would be undefined on the very object this essay is measuring; reading the seam as a difference of ground heights, each computed inside its own band’s own honest geometry, sidesteps the need for a frame that does not exist.

The two independent checks matter for the same reason the arithmetic-floor controls do throughout this site: a number computed one way could be a mistake in that one way. Reading the seam off the actual drawn pixel through the near band’s own formula, and reading it as the plain difference between the two bands’ independently recovered elevations, are different calculations that happen to be forced to agree by the underlying geometry — the ground truly does not move, so any two honest routes to “how far does it appear to move” must land on the same number, and they do, to the resolution both checks carry.

Naming the disagreement #

The shape of this finding is worth setting beside a sibling result from elsewhere in this collection, because the two are the same kind of disagreement produced by two different mechanisms.

Two stations in one picture measures a station-recovery routine handed a single sheet drawn under two different depth rules, glued together with no real second eye anywhere in it, and finds a recovered-horizon gap that grows continuously as the rules pull apart. This essay’s disagreement has a real second eye in it — the far band genuinely was drawn from a higher station than the near one — so it is not the same mechanism, but it produces the identical shape of result: two individually exact recoveries, each correct on its own patch, that disagree the moment their answers are compared across a boundary neither recovery was told exists. Calling the two joins’ 2.40 m and 7.00 m a station-disagreement is accurate in that sense — not a fault in either band’s own fit, which is exact, but a fact about what happens when two honestly-fitted stations are asked to describe one continuous ground.

The two essays differ in what they hold fixed while producing that shape, and the difference is worth being exact about. There, one sheet, one nominal rule, and a dial that measures how far the second half has drifted from it — the disagreement is manufactured by degree, and it is a statement about how sensitive a recovery routine is to its own assumption failing, growing continuously from nothing as the dial turns. Here, two real stations and two rules that are each individually a perfectly ordinary pinhole projection produce a disagreement fixed the moment the scene itself is fixed — 2.40 m and 7.00 m, with no dial between them and the arithmetic-floor case that essay’s own control demonstrates. A station-disagreement, in other words, is not one phenomenon with one cause; it is a shape two different mechanisms can both produce, one by degrees and one all at once, and telling them apart requires asking whether a real second station is actually present, which this essay’s landscape has and the earlier essay’s blended rule does not.

The smooth alternative, and why it is not used #

The seam is a cost, and it is fair to ask whether it is a necessary one — whether the far band’s gain could be had without any join at all, by raising the eye gradually rather than in three abrupt steps.

A continuously climbing eye is not a hypothetical — it is exactly the pushbroom this site already models for a handscroll’s own direction of travel, with a height that varies along the track rather than staying level. Run over the far band’s own depth range, reaching 11 m only at the very end rather than holding it throughout, it gives the mountain 20.2 px and 1.65° of subtended angle — a real, seamless, entirely available picture with no join anywhere in it. The three-station convention, stepping abruptly to 11 m for the whole band, gives the same mountain 71.5 px and 5.83°, 3.5 times the smooth alternative’s picture, for the identical total change in elevation.

That is the finding the seam is the price of. A smooth join exists, costs nothing in continuity, and draws a mountain roughly a third the size — too small, in the terms three distances in one landscape already uses, to hold the shape a mountain is being drawn for. The abrupt convention is not a compromise forced by geometry; it is a choice to spend the gain concentrated at the moment it is needed rather than spread thin across the whole climb, and the seam measured above is exactly what that concentration costs a reader trying to read the ground as continuous.

The comparison is worth making precise about what stays fixed and what does not, because a naive version of it would not be a fair trade at all. Both cameras cover the identical depth range and reach the identical final elevation of 11 m; what differs is the path the eye takes to get there — instantaneously, at the band’s near edge, in the convention actually used, or gradually across the whole band, in the alternative. Fixing the endpoints and the depth range is what makes 71.5 px against 20.2 px a statement about how the gain is distributed rather than a statement about how much gain is available in total, which is a different, larger number three distances in one landscape already reports on its own terms.

A straight line in a scroll is a hyperbola is the cost the smooth alternative would actually be paying elsewhere on the sheet, for a track that runs sideways rather than in depth: a pushbroom’s own straight world lines bow into curves under a translating eye, which is a price this essay’s landscape does not have to pay because its bands are stacked in depth rather than laid out along a moving track. That the smooth alternative is available and unused here, while a structurally similar smoothness elsewhere in this collection is used and does cost something, is a reminder that “seamless” is not a single virtue with one uniform price — each convention trades a different discontinuity for a different curve, and which trade is worth making depends on which one a painter is trying to avoid.

The control: one eye, no seam #

Every seam measured so far depends on the bands genuinely coming from different elevations. The control removes that at the source.

8.9e-16 m is what groundSeamMetres returns when there is genuinely nothing for it to find: one elevation throughout means every band’s own fit recovers the identical height, so the near band’s rule and the far band’s rule are the same rule, and asking whether they agree at a join is asking whether a number equals itself. The far band’s own mountain comes out at 15.6 px under that one, unraised elevation — not compared against anything, because a control that has not paid the convention’s price has nothing to be a gain relative to; it simply states what an unremarkable single-camera view of the same ground looks like, for a reader to hold against the raised readings above.

What a single camera would have cost anyway #

The seam this essay measures is a cost specific to changing elevation partway through a picture. It is worth setting beside a cost that a single, unchanging elevation pays throughout — the ordinary price of depth under one perspective camera, which the three-station convention exists to avoid paying at the far end.

A single perspective camera does not merely compress a distant band’s mountain; it moves every interior point of anything drawn across a range of depth, and the two prices are the same fact read two ways. A map along, and a picture across measures the midpoint drift directly; this essay’s own far-band compression is the identical one-over-depth-squared law applied to an interval rather than a point. Raising the eye for the far band, as the actual convention does, is one way to fight that law locally; keeping one eye throughout and accepting the drift, as the control above does, is the alternative the seam is traded against.

The honest limit #

Nothing here recovers a station from a real photograph of a real landscape — every band’s ground samples are built precisely, at the resolution a fitted line reads to the arithmetic floor, and a real drawn scroll offers nothing so clean. recoverBandElevation’s fit needs several samples along one band’s own ground, visibly distinguishable and correctly assigned to that band rather than a neighbour, which is a description of a diagram rather than of an inked or painted line with a brush’s own width to it.

Nor does the seam measured here say anything about the offset a viewer’s eye actually sees at a real join, which three distances in one landscape already separates from the rate this essay’s own statistic is a metric cousin of: a painter absorbs the offset by where the bands are placed on the sheet, and neither that essay’s rate nor this one’s ground-metres statistic is the raw pixel gap a viewer’s eye would register. Both are quantities invariant under the placement freedom a painter has and sensitive only to what that freedom cannot remove, which is why they are the right things to measure and not the only things a real viewer experiences.

And the recovery throughout assumes a level eye and a flat ground within each band — the same assumption the horizon is at eye level states its own condition for. A pitched camera, which real towering-distance views often use to look up at a near mountain, would need its tilt recovered alongside its height, and nothing here measures how the fit degrades once that assumption is dropped.

Finally, the smooth-versus-abrupt comparison holds the total elevation change fixed and asks only how it is spent, which is the right question for pricing the seam but not the only question a real painting answers. A continuously climbing eye is harder to draw by hand than three stated stations, because it has no single vanishing point a straightedge can be laid to and no single elevation a painter can hold in mind while inking a band — the abrupt convention may be a compromise with the medium as much as a deliberate spending of the elevation gain where it is most needed, and nothing measured here separates those two motives.

What this joins #

Three distances in one landscape measured the bill this essay’s own picture was built to owe; recovering the same numbers from the marks alone, with nothing supplied, is the round-trip check that says the bill was real rather than a property of how the essay chose to describe its own construction. Recovering the camera is the general statement of that check, run here on three cameras sharing one sheet rather than one camera alone, and no surface keeps everything is the wider family this piecewise strategy belongs to — a single map paying one price everywhere, against several maps paying the price only at their joins.

The transferable form is the same one two stations in one picture reaches from a rule glued to a rule rather than a station glued to a station: within any one honestly-drawn patch, a fit recovers exactly what drew it; across the boundary between two such patches, exactness within each side is no guarantee of agreement between them, and the size of the disagreement is itself a real, computable, transferable quantity rather than a failure of the method.