A band lifted by columns bends its lines

A seam that bends trades a turn for a gap laid a several-station landscape’s join along a river bank instead of across one depth, and found it could go round what crossed it, at a price. The far band is one picture, lifted by one amount, and a seam that wanders in depth needs a different lift in each column: the best single lift centres the mismatch and leaves a gap. Three pixels bought the first join 1.2 metres of wander; going round a twelve-metre object cost sixteen pixels.

Everything in that essay rested on the far band being one picture. The essay ended by pointing out that a painter is not a camera, and nothing stops a far band being drawn as a family of columns each lifted by its own amount — drawn, that is, from a station whose height varies across the picture. The seam would then close exactly in every column. The cost would move: a band lifted differently in different columns bends every straight line that runs across it, so a road or a wall across the far band would come out curved. The question was how far into the band the bend reaches: whether it dies within a short distance of the seam, making a varying lift a cheap way to close any curved seam, or persists across the band.

It persists. There are two ways to lift a band by columns, one of them bends every line by the same amount all the way to the sky, and the other bends each line in proportion to its distance below the horizon — which dies away only as one over the depth, and then hands what is left to the next join.

Two ways to close a seam by columns #

The seam is the one the earlier essay found had to bend: at the first join, between a near band drawn from 1.6 metres and a middle band from 4 metres, wandering between 19.6 and 22.3 metres across the picture to get round the objects standing there. With one lift, its worst column gaps by 5.04 pixels.

The first way is to shift each column rigidly by the lift that column needs: the focal length times the difference in station heights, over the seam’s depth in that column. Each column of the band is still a picture from the band’s own station, moved up or down as a whole. The seam closes exactly. A straight road across the band at any depth is moved by the same column-dependent amounts, so it bends by the whole range of the lifts — 10.08 pixels, twice the gap one lift left, because one lift centred that range and left half of it either side. The horizon, being a row of every column, bends by the same 10.08.

The second way is to draw each column from its own eye. The band keeps one lift, and each column’s station height is chosen so that, with that lift, the seam closes in that column. Each column is then a pinhole column in its own right, with its own eye at its own height, and every column’s horizon is where every other’s is — at infinity every station sees the same row. A straight road across the band now bends by 8.13 pixels at 26 metres, 6.04 at 35, 4.23 at 50 and 3.02 at 70. The horizon stays straight.

These are not two arbitrary choices among many. A column of a picture drawn from a level eye puts a ground point at a row set by the eye’s height over the point’s depth. To close the seam each column needs a particular row at the seam’s depth. Changing the column’s offset while keeping its eye is the shift; changing its eye while keeping its offset is the station; and only the station keeps the horizon a straight line, since the horizon is where every eye’s rows meet whatever its height.

The bend dies as one over the depth #

The shift’s bend is the same at every depth, so its answer to the earlier essay’s question is immediate: it never dies. The station’s bend falls, and the figure below says how fast.

Drawn from a station per column, a road’s bend is the product of two things: how much the stations differ across the picture, and one over the road’s depth. The stations differ in proportion to how far the seam wanders; the one over the depth is how far below the horizon the road is drawn. So each column is stretched about the horizon by its own amount, and every line in the band bends in proportion to its distance below the horizon: 9.60 pixels at 22 metres, just beyond the seam, 5.28 at 40, 3.02 at the band’s far edge at 70.

That is the answer to the question. The bend does not die within a short distance of the seam. It dies as one over the depth, which on a band running from twenty to seventy metres means it is still more than a third of its size at the band’s far edge. A varying lift is not a cheap way to close a curved seam. It moves the gap from the seam into every line the band holds, in proportion to how low in the picture each line lies.

What the closing asks of the painter #

The station closing has a direct reading in terms a painter would recognise: it says where the eye was for each column of the band.

The heights run from 3.86 to 4.16 metres across the picture, tracking the seam’s depth column by column — higher where the seam is further away, lower where it comes nearer. A painter closing a curved seam this way is drawing the middle band from an eye that rises and falls across it by thirty centimetres: a station that is itself a curve, following the seam it was chosen to close.

That makes the convention concrete. A landscape that changes its rule halfway up recovered each band’s station from its own marks and found three stations, one per band. A band closed column by column would report a station that varies across the picture instead, and a reader recovering stations column by column — each column’s rows against the ground’s depths — would find the curve in the figure. The painter’s choice leaves its own signature, in the same marks, at the same precision.

The next join inherits the gap #

A band has two edges. The seam at its near edge is closed, but its far edge is the next join — a straight seam at one depth, where the next band begins — and the far edge is a line across the band like any road, bent by the columns’ different stations.

The band beyond the second join is one picture with one lift, as the middle band was before the painter began lifting columns. Meeting a bent edge with a single lift, it opens a gap of half the bend — and the bend at the second join’s depth is the station closing’s bend at seventy metres. For a seam wandering a stated distance about twenty-six metres, the gap the second join inherits is 37 per cent of the gap the first would have had: 0.92 pixels for a metre of wander, where the first join would have gapped by 2.49; 5.85 for six metres, where it would have gapped by 15.7.

The fraction is the ratio of the two joins’ depths, twenty-six over seventy, and it is the same at every wander because both gaps are proportional to it. So the painter who closes the first seam by columns has not removed the gap. The painter has carried it back into the landscape, to a join where it is smaller in the same proportion as the join is further away, and where — at under a pixel for a metre of wander — it may be small enough to hide. A painter willing to close the second join by columns as well carries it to the third, smaller again, and only at the horizon, where every station agrees, does it finally vanish.

The gap, before the painter lifted anything #

The earlier essay’s own figures are the baseline this one is measured against.

With one lift, the gap along the bending seam runs to about five pixels either side of nought, and in each column it is exactly the offset something crossing the seam there would show. The shift closing turns that gap into a bend of twice its range in every line; the station closing turns it into a bend of the same size at the seam, dying away into the band, and a smaller gap at the next join. None of the three makes the mismatch disappear, because none of them changes the fact the earlier essay rested on: two stations at different heights draw the same ground at rows that agree at only one depth.

The wander budget the earlier essay computed carries over to the station closing with one change of meaning. There, three pixels of gap allowed the first seam about a metre and a quarter of wander either way. Under the station closing the same wander leaves no gap at the seam at all, a bend of about six pixels in the roads just beyond it, and a gap of about a pixel at the second join. A painter who would rather bend lines than open seams can take a seam two or three times further from one depth before the next join’s gap reaches three pixels. What they cannot avoid is that every line in the band bends by the amount the seam would have gapped, near the seam, and less further back.

The scroll that has no join #

The earlier essay’s suggestion came with a reference: a map along, and a picture across describes a handscroll whose eye moves continuously along the scroll and has no join at all. It is worth being clear about why that is not the station closing, because the difference says what the column-wise station can and cannot do.

A handscroll’s eye moves sideways — along the scroll’s length — as the reader unrolls it, and each column is drawn from where the eye was when that column was painted. The stations differ in position across the view, not in height, and a ground line running along the scroll is drawn at one row because every station is at one height. A scroll is not a panorama found that such a picture has no single centre and keeps every row of the ground level. The station closing is the other case: stations that differ in height across the view, at one position. Its rows are not level, because each column’s eye sees the ground from its own height, and so its lines bend.

So the handscroll escapes the join by moving its eye in the one direction that leaves the rows alone. A several-station landscape stacks its stations vertically, one per band, precisely because that is what gives each band its own tilt of the ground — the stations are also a staircase found that a level eye’s rows depend on the eye’s height and the ground’s only through their difference, which is why the bands read as a rising ground. A painter who varies the height column by column to close a curved seam has turned the staircase’s riser into a slope that runs across the picture, and a slope across the picture is a bend in every line.

What the station closing leaves alone is worth adding. A level eye draws every set of parallel ground lines to one vanishing point on the horizon, whatever its height, so the band’s receding lines — a road running into the picture, a furrow, a shadow cast by a low sun — still converge where they did. The bend is in the lines that run across the band, the ones whose rows the eye’s height sets. A shadow decides which landscape it is read a several-station landscape from the directions its shadows take across a join; a band closed by stations keeps those directions, and changes only how long each shadow is drawn.

Where the mismatch is allowed to be #

This sequence has now placed the mismatch between two stations in every position a painter can put it. A seam breaks direction, not size found that at a straight seam it is an offset, absorbed by butting the bands, which leaves a turn in anything that crosses. Where a seam is allowed to go found that the offset is smallest at a join put further off, and the turn the same wherever it is put. The bending seam converted the offset into a gap, column by column. And lifting by columns converts the gap into a bend of the band’s lines, which dies as one over the depth, and passes a fraction of itself to the next join.

The quantity is conserved throughout. It is the difference between two stations’ views of the same ground, and it depends only on the stations’ heights and the depths involved; a painter can decide what form it takes — offset, turn, gap, bend — and where it sits, but not whether it exists. A single-station picture has none of it, which is what makes a several-station picture recognisable as one: the mismatch has to be somewhere, and a reader who knows the forms can find it.

What was assumed #

The stations are level and differ only in height. Every station here looks level and stands at the same place across the view; the convention is about elevation. A station that also moves sideways between columns — a handscroll’s moving eye — adds a mismatch in columns as well as rows, and a column-wise closing then has two things to close.

The ground is flat. Every row here is a row of a level ground plane. A far band containing hills has lines that are not at one depth, and their bend under a station closing varies along them with their depth; the one-over-depth rule applies point by point.

The lift is one number for the band. The station closing keeps a single lift and varies the eye. A painter could vary both, and a two-parameter closing has room to spread the bend differently — but any closing that keeps the horizon straight and each column a pinhole column is the station closing, because the horizon’s row and the seam’s row together fix both numbers in every column.

Still open: whether a reader can recover a station that is a curve #

The section on what the closing asks of the painter pointed at a test that has not been run. A band closed by a station per column carries its station curve in its marks: each column’s ground rows, against their depths, give that column’s eye height, and the heights across the picture trace the seam’s depth. A band closed by one lift has one station, and a band closed by a column-wise shift has one station with a column-wise offset.

The measurement that settles whether a reader can tell them apart takes the marks a band’s columns carry — the rows of ground features at known or recoverable depths — reads them with a pixel of error, and fits each column’s station height and offset, asking two things. First, how precisely a column’s station is recovered, and so how large a station curve a reader could detect: whether a thirty-centimetre rise and fall across the picture stands out of the noise. Second, whether the three closings are distinguishable at all from one band’s marks — whether a shift’s constant-station, varying-offset signature and a station curve’s varying-station, constant-offset signature separate — so that a reader of a several-station landscape could say not only that its painter closed a curved seam, but how.