A seam breaks direction, not size

The stations are also a staircase ended by noticing that a join stays ambiguous only while nothing crosses it, and that a painter laying mist across a seam is doing what the ambiguity requires. That leaves a question it did not ask: what exactly happens to a thing that does cross.

The answer is sharper than “a discontinuity”. Some of a crossing object’s measurements come through the join completely unchanged, to the last digit, and others do not come through at all — and the split between them is clean enough to state as a rule.

The eye cancels out of a size and not out of a place #

A level eye at height $H$ draws a point at depth $d$ and world height $Y$ on the row $p_y + f(H-Y)/d$. Take two points at the same depth — the foot and top of a post, the two edges of a road — and subtract. The $H$ appears in both rows and cancels: a post of height $p$ is drawn $fp/d$ tall and a road of width $w$ is drawn $fw/d$ wide, and neither expression contains the eye at all.

So every size measured at one depth survives a change of station exactly, and every position does not.

At the first join a six-metre post is 161.5 px tall under the near band’s rule and 161.5 px under the far band’s. An eight-metre road is 215.4 px wide under either. Those are not agreements to a tolerance; the two expressions are the same expression.

And the post’s foot sits 64.6 px apart under the two rules — which is the riser the staircase reading has to draw, arriving from the other direction. A join does not distort what crosses it. It dislocates it: every part the right size, in the wrong place.

It is worth being exact about how large a class that is, because “every size at one depth” covers most of what a reader of a picture actually measures. The height of a figure, the width of a boat, the span of a bridge, the gap between two posts standing side by side, the diameter of a wheel seen edge-on: all of them are differences of two rows or two columns taken at one depth, and every one of them crosses a join untouched. A reader who checked a crossing object’s proportions for evidence of the seam would find none, however carefully they looked, because there is none to find.

The reason is the same one the staircase reading rests on: the eye’s height enters the row of a ground point only through $H - Y$, so any operation that differences two points at one depth removes it. That essay used the fact to show a reader cannot separate the eye from the ground. Here it shows a reader cannot detect the seam from proportions. One algebraic fact, two consequences, and both of them are about what the marks do not contain.

The turn is what nothing absorbs #

The dislocation splits again, and this second split is the one that matters to a painter.

Three distances in one landscape established the freedom a painter has at a join: the bands are not projected into one frame but laid out, each butted against the last, so a uniform vertical offset between two bands is free. That is why the measurable cost of a seam was reported as a change of rate rather than as an offset — the offset is absorbed by where the bands are placed on the sheet.

A post’s displaced foot is exactly such an offset, and butting the bands is exactly the operation that nulls it at the join depth. So the 64.6 px is not, on its own, something a reader sees.

An angle is different.

A road’s edge runs from the principal point outward: both its coordinates fall as $1/d$, $u$ as $f(w/2)/d$ and $v$ as $fH/d$, so the edge images on a straight line through the principal point whose slope is $2H/w$. The angle it is drawn at is a function of the eye’s height. The near band draws an eight-metre road’s edges at 21.8°, the far band at 45.0°, and the second join turns them again from 45.0° to 70.0°.

A translation moves a line without turning it. So the painter’s offset — the one freedom the layout provides — cannot touch the 23.2°, and a road crossing the first join arrives on the other side visibly bent. Every other continuous thing running in depth does the same: a river, a wall, a line of poplars, the sides of a valley.

The second join turns them further, and by a comparable amount — 45.0° to 70.0°, a turn of 25.0°. So a feature running through the whole picture, which is what a river in a towering-distance landscape is, is bent twice, by 23.2° and then by 25.0°, for a total of 48.2° between the direction it is drawn at nearest the viewer and the direction it is drawn at furthest away. A road entering the picture at 21.8° from the horizontal leaves it at 70.0°.

That is the placement-invariant signal a seam produces, and it is the reason the join has to be empty. Not because a discrepancy would be visible in principle, but because twenty-three degrees is not a discrepancy.

Why a size is not a length #

One qualification keeps the rule from being broader than it is, and it is the qualification that makes it a rule rather than a slogan.

A size survives only if it is taken at one depth. A length measured down the picture — the drawn extent of a stretch of road between two marked depths, the run of a river between two bends — is a difference of two rows at two different depths, and the eye’s height does not cancel out of that. It is exactly the quantity the convention exists to change: the far band gets more picture for its depth range precisely because the eye is higher, and that gain is 6.9 times what one camera would have allowed.

So the split is not between sizes and positions in any loose sense. It is between quantities that are differences at fixed depth — which lose the eye — and everything else, which keeps it. A stretch of road from 16 m to 26 m is drawn at one length under the near band’s rule and at quite another under the far band’s, and the ratio is the rate change the seam was reported as costing in the first place.

That is why a reader cannot use a crossing object’s proportions as evidence about the seam and can use its rate of foreshortening. The first is a fixed-depth difference and the second is not.

The worst thing to run across a seam #

The kink depends on the crossing object’s width, and not monotonically, which gives a painter something useful.

A very narrow thing is drawn nearly vertically under both bands, because $2H/w$ is large for both; a very broad one is drawn nearly horizontally under both, because $2H/w$ is small for both. The turn is largest in between — 24.8° at four metres, for eye heights of 1.6 m and 4 m — and falls away at both ends: 10.2° at one metre and 2.1° at a hundred and twenty-eight.

So the worst possible thing to run across a seam is something whose width sits near the two eye heights, which is to say a path, a stream, a track — exactly the features a landscape runs in depth. And the safest are a thread and a plain: a footpath narrow enough to read as a line, or a stretch of open water broad enough that its edges are nearly level anyway.

Whether painters knew that is not a question any measurement here answers. What can be said is that the geometry rewards the two extremes and punishes the middle, and that the middle is where most receding features of a landscape sit.

The turn is the riser, read sideways #

The two joins turn a road by 23.2° and 25.0°, and those numbers are not independent of anything already measured.

A post’s foot is displaced 64.6 px at the first join and 70.0 px at the second, and those are the riser heights the staircase reading has to draw. A road’s edge is a line through the principal point, and displacing every one of its points by the amount a post’s foot is displaced is exactly what turns it: the displacement is vertical and proportional to $1/d$, which is the same $1/d$ the edge’s own coordinates fall as, so the displaced line is another line through the principal point at a different angle.

So the three quantities reported about this join — the apparent ground drop in metres, the riser in pixels, and the turn in degrees — are one quantity in three currencies. A landscape that changes its rule halfway up reported it in metres of depicted ground; the staircase reading in pixels of a cliff face; here in degrees of a bent road. The reason to have all three is that only the last one is invariant under what a painter is free to do, and only the first is stated in the world rather than on the page.

How much room a painter actually has #

A join is one depth. An object blocks it only by spanning it, so an object of depth-extent $e$ rules out an interval of candidate join depths $e$ long, and the question of how constrained a painter is becomes arithmetic.

Scatter through the landscape’s own depth range — 8 m to 200 m — six features with depth-extents of 3, 3, 6, 6, 12 and 20 metres, which is a thinly populated scene rather than a crowded one, and 44.1 m of the 192 m range is blocked. Seventy-seven per cent of depths remain available, which is a good deal of room.

That is worth stating because the constraint is easy to overstate once its cost in degrees is known. A seam must fall where nothing crosses, and in an ordinary landscape most depths qualify. The constraint bites not on whether a join can be placed but on where — and it explains a practice rather than forbidding one.

There is a second reading of that figure worth taking. The blocked intervals are where a join cannot go, and they are set by the depth-extent of the features rather than by how many there are — six features of three metres block eighteen metres of range and one feature of eighteen blocks the same. So a landscape with a few large receding things in it is more constrained than one with many small ones, even though the second looks busier. A valley with one long river running its length is nearly unjoinable; a plain scattered with copses is almost entirely free.

That inverts the intuition a crowded picture suggests, and it is a consequence of the join being a depth rather than a region. Nothing is blocked by being near the join; only by straddling it.

It also explains the mist directly. A painter who wants a join at a particular depth, for reasons about the picture rather than about the scene, can always make that depth empty by putting something in front of it. Mist, cloud and a bank of trees do not hide a discontinuity; they remove the objects that would have crossed it, which is a different operation and a complete one.

The difference matters for how the convention should be described. Hiding a discontinuity is a patch over a defect, and a reader who found the patch would have found the defect. Removing the crossing objects leaves a picture with no defect in it anywhere: every band is an exact projection, every size crosses correctly, and there is nothing at the join to be inconsistent because there is nothing at the join. It is the same move a divergent picture’s surfaces make when they are drawn square to one another — not a flaw concealed but a case avoided.

What this does not settle #

It does not measure any painting. The landscape here is constructed from three stated cameras, and nothing about where any real painter put a seam is claimed or checked. The three stations are the ones this field has used throughout, chosen to make the arithmetic legible rather than to match a practice, and a convention worked at other heights would turn a road by other angles — larger ones, since the turn grows with the difference between the stations.

It does not say the turn is the only signal. It is the only one a vertical offset cannot absorb, which is a narrower claim. A painter has one freedom at each join and the turn survives it; whether some other freedom a painter might take — scaling a band, or shearing it — would absorb the turn as well is a question about a layout nobody here has drawn.

It does not price the horizontal. Everything above concerns objects running in depth. A thing lying across the picture rather than into it spans one depth, so it has no join to cross, and the whole question is about receding features.

It does not distinguish the two readings of the landscape. A crossing object’s turn is a property of the marks, so it is the same turn under the several-station reading and the staircase one. Under the staircase reading the road is not bent by a change of rule at all — it genuinely runs up a bank and its direction genuinely changes, which is what a road climbing a terrace does. The two accounts agree about the 23.2° and disagree about what it means, which is the shape the staircase reading is entirely about.

It does not settle what a viewer notices. Twenty-three degrees is large as a geometric quantity; whether a reader of a painting registers it as wrong, or reads it as a bend in the road, is about seeing rather than about projection, and nothing here bears on it. The claim is that the turn is present and that no placement removes it.

It does not model an object that is not straight. A road is taken as a straight run in depth, so its edges are straight lines through the principal point and a turn is a well-defined angle. A winding river has a drawn direction that changes anyway, and separating the turn a seam adds from the turn the river already has is a different measurement — and probably an easier one to hide a seam inside, which is worth someone’s attention.

And the free-depth share is a property of a scene rather than a result. Seventy-seven per cent is what six stated features leave; a crowded valley would leave less and an empty plain more. The number is here to say that the constraint has a size, not to say what the size is in general.

Still open: what a seam does to a shadow #

One kind of receding feature has not been asked, and it is the one whose geometry is already a projection of its own.

A shadow cast across the ground by a low sun runs in depth, so it crosses joins like a road — but a shadow’s direction on the page is set by the sun’s own vanishing point, not by the object that casts it, and that vanishing point is a property of the camera drawing it. Two bands drawn from two stations share a horizon, so they share the row a sun’s vanishing point sits on; whether they share the point is a different question, and it decides whether the shadows of two objects in different bands converge on one place or on two.

The measurement that settles it places a single sun over a several-station landscape, draws each band’s shadows by that band’s own camera, and asks whether a reader fitting a common light to the whole picture finds one — and if not, how far apart the two lights sit, in the same currency the station disagreement was reported in. A picture whose shadows point at two suns is a much louder failure than a road that bends, and it would put a second constraint on where a join can go.