A seam that bends trades a turn for a gap

Where a seam is allowed to go priced the constraint that governs a several-station landscape: a join between two bands, each drawn from its own station, stays invisible only where nothing of the depicted world crosses it. Moving the join further off shrinks the offset a crossing object shows and leaves the turn it takes at 23.2 degrees exactly, so emptiness is a painter’s only defence — and emptiness counted in the picture, rather than in metres, turned out to depend on where a landscape is crowded rather than how much.

Every one of those measurements took a join to be one depth, because that is what butting two bands along a horizontal line makes it. The essay ended by pointing out that nothing obliges a painter to do that. A join laid along a river bank, a ridge or a line of trees is a curve across the picture, and the two bands meet at whatever depth that curve reaches in each column. It would not have to avoid an object, only go round one — which sounded like a far weaker requirement, and would explain why a painter of this convention never appears to hunt for an empty depth.

Two parts of that proposal can be measured: what a curved join costs the way the two bands fit together, and how much room it actually buys. The second answer depends on the first, and both are smaller than the proposal expected.

One lift for a whole band #

The first fact is about what a join is. A level eye draws a ground point in a column set by where the point is across the view divided by its depth, and in a row set by the eye’s height divided by its depth. Two stations at different heights over the same ground therefore agree about columns and disagree about rows. At a straight join, one depth, the far band’s rows are all too high by the same amount — the focal length times the difference in eye height, over the join’s depth — and the painter lifts the whole far band by that amount, which is exactly the offset a seam breaks direction, not size found a painter can absorb by butting the bands.

A curved join sits at a different depth in different columns, so the amount each column needs lifting differs. But the far band is one picture and can be lifted by one amount only. The best single lift centres the mismatch, and what remains is a gap between the bands in some columns and an overlap in others. The worst column misses by

$$\text{gap} = \frac{f\,\Delta H}{2}\left(\frac{1}{D_{\min}} - \frac{1}{D_{\max}}\right),$$

where $\Delta H$ is the difference in station heights and $D_{\min}$ and $D_{\max}$ are the nearest and furthest depths the seam reaches. A straight seam has no spread of depths and no gap; a curved one has a gap set by how far it wanders, measured in one over depth. A landscape that changes its rule halfway up recovered each band’s station from its own marks; the gap is the same fact seen from the painter’s side, the two stations’ rows disagreeing by an amount that only one depth can make constant.

This corrects a small thing the earlier essay said. It described a curved join as one where “the riser’s height varies along the seam”. The riser is the difference in station heights and is the same everywhere; what varies is the riser’s image, which is the riser over the depth, and that is the whole of the gap.

How far a seam may wander #

The formula can be turned round to ask how far a seam may stray from a depth before its gap passes a size the painter is prepared to hide.

Not far. At the landscape’s first join, twenty-six metres out between stations 1.6 and 4 metres up, a gap of three pixels allows the seam to wander 1.15 m toward the eye and 1.27 m away. The second join, seventy metres out between stations 4 and 11 metres up, gets 2.88 and 3.13 m at the same three pixels. Twenty pixels of gap buys the first join 6.1 m toward the eye and 11.7 m away.

The asymmetry and the growth both come from working in one over depth. Equal steps in one over depth are larger steps in depth the further away they are, so a seam can stray further behind its centre than in front, and a join twice as far off can stray about four times as far for the same gap — against which the second join’s larger riser, 7 m rather than 2.4, takes some of it back.

So a river bank is available as a seam only if it keeps within a metre or so of one depth near the front of the landscape. That is a river bank seen from above, running across the view; a river that winds toward the viewer is not.

Going round an object is not a weak requirement #

The proposal’s central claim was that a curved join could go round an object — in front of it in the columns where it stands, behind it elsewhere — instead of having to avoid its depth altogether. The gap formula prices that directly: to pass in front of an object in some columns and behind it in others, the seam’s depth has to span the object’s full depth from front to back.

At the first join, going round a three-metre object costs 3.7 px of gap, a six-metre one 7.6, a twelve-metre one 15.8 and a twenty-metre one 29.2. At the second join the same objects cost 1.5, 3.0, 6.0 and 10.2 px. The objects where a seam is allowed to go scattered through its landscape were three to twenty metres deep, so a seam that goes round them opens a gap between the bands that is several to tens of pixels wide, along the whole stretch where it detours.

That is not nothing and it is not everything. The offset a straight seam forces is 64.6 px at the first join, and a painter absorbs it by lifting the band; the gap from going round a twelve-metre object is a quarter of that, and it cannot be absorbed by any lift, because the lift has already been spent centring it. It has to be hidden — by mist, by a line of foliage, by whatever the painter lays along the seam. The requirement is not “avoid an object” and it is not “go round it freely”. It is “go round it, and hide a strip of this width wherever the seam changes depth”.

What bending actually adds #

That price decides how much room bending buys. To measure it the objects have to stand somewhere, since an object spanning the whole picture cannot be gone round: six objects of the earlier essay’s depth-extents, each as wide as it is deep, placed at seeded depths and at seeded places across what the picture sees at their depth, a hundred landscapes in all.

The picture is then a map. Every column is a line of ground running away from the eye, and an object closes the depths where that line passes through it. A straight seam is a row of the map with nothing closed in any column. A curved seam is a path across the map through open cells, confined to a window of depths as wide as the gap allows.

The first thing the map says is that straight seams have almost as much room among standing objects as among spanning ones: 80.5 per cent of the picture on average, against 80.6 when every object was taken to span the picture. That is not a coincidence. An object standing anywhere across a depth still closes that depth to a straight seam, because a straight seam crosses every column; whether the object spans the picture or stands in one corner of it makes no difference to a line that has to pass through all of them.

The second thing is where a bending seam gets through that a straight one does not. It needs a window of depths with no free row in it — so a straight seam could not have used it — through which a path still finds its way from edge to edge. In the landscape drawn, the window from 19.6 to 25.3 m is such a place: the seam passes in front of the object on the right and behind the one on the left, and its worst gap is 5.0 px. Eighty-one per cent of the same picture’s rows were free for a straight seam anyway.

Places like it are rare. Over the hundred landscapes, windows that only a bending seam can cross amount to 0.36 per cent of the picture at a 2 px tolerance, found in seventeen landscapes; 0.21 per cent at 5 px, in seven; 0.21 per cent at 10 px, in five; and 0.03 and 0.04 per cent at 20 and 40 px. The added room falls as the tolerance rises, which looks backwards until the reason is plain: a wider window almost always contains some row a straight seam could have used, and such a window is room the painter already had.

That also corrects a first reading of this measurement. Counting every window a bending seam can cross, rather than only those a straight seam cannot, reports the room rising from 80 to 93 per cent at 5 px. Nearly all of that rise is windows with a free row inside them — a straight seam slightly moved, not a curve. It is the kind of number that is true of its own definition and says nothing about the question.

The trade a curve offers #

So a curved seam does not buy a painter room. What it offers instead is a trade, and the trade is clearest on the one seam that had to bend.

Along it the gap runs from −5.04 to +5.04 px, a step where the seam changes depth to pass between the two objects. In every column the gap is exactly the offset a thing crossing the seam there would have shown — the curve has converted the offset into a gap. And it has no crossing, so there is no turn. A straight seam placed through either object would have let it cross, absorbed the 64.6 px offset with a lift, and shown the turn, 23.20 degrees for an eight-metre road between stations 1.6 and 4 metres up.

That is the real content of the proposal. A seam breaks direction, not size found that the offset is the symptom a painter can absorb and the turn is the one that convicts. A curved seam that goes round an object gives up the absorption — its offsets become a gap it must hide with mist or foliage — and gets rid of the turn entirely. For a painter who has an object standing exactly where the seam must go and a strip of mist to spend, that is a good trade. It is not a source of room.

What a painter would actually do #

Put together, the measurements say what a curved seam is good for and what the convention’s own paintings would show if it were used.

It lets a seam follow a feature near one depth. A shoreline, a ridge or a hedge that keeps within a metre or two of one depth near the front of the landscape can carry the join with a gap of a few pixels, and the gap is exactly where a reader expects a line anyway. The seam then looks like the feature rather than like a join.

It lets a seam dodge one object at the cost of a strip. The strip’s width is the object’s depth in one-over-depth, times the riser’s image: 3.7 px for a small building at the first join, 15.8 for a twelve-metre grove. A painter who does this has to lay mist or foliage along the stretch where the seam changes depth, not just where the object is.

It does not help with a shadow. A shadow decides which landscape it is found that a shadow crossing a join settles which of the two readings a picture is. A seam that bends round an object bends round the object, not round its shadow; the shadow lies on the ground to one side of it in depth, across whatever depths the sun throws it, and a seam that passes the object on that side passes through the shadow. How often that happens across suns and landscapes was not measured. So the curve that removes an object’s turn leaves its shadow to cross, and the reading the mist was hiding is decided anyway.

It does not free a painter from looking for empty depths. Straight seams already have four fifths of the picture in these landscapes, and bending adds a few tenths of a per cent. If a painter of the convention appears never to search for a gap, the reason is the four fifths — a sparse landscape leaves most depths empty — and not the curve. The stations are also a staircase is the reminder of why the empty depth matters: the seam is a step as tall as the band before it, and anything crossing it shows the step.

What the rectangles and the one join leave out #

The objects are rectangles in plan. Every object here has a straight front and back and square sides. A grove or a village has a ragged outline, and a ragged back edge offers a seam more places to slip behind; the pass costs above are for going round the whole depth, which a ragged object may not require.

The seam’s gap is taken as the thing to hide. Whether a gap of three pixels between two painted bands is visible depends on what the painter lays there. The tolerances here are a sweep, not a claim about any painting’s mist.

One landscape’s stations and joins. The riser, the join depths and the focal length are the standing landscape’s: stations at 1.6, 4 and 11 m and joins at 26 and 70 m, chosen in three distances in one landscape so that each band gets back the picture depth takes from it. A convention with closer stations has smaller risers and cheaper curves, in proportion.

And the second join is only reported. The room measurements are for the first join, where the riser’s image is largest. At the second join, 70 m out, every pass costs about a third as much in pixels, which should let bending open more there; the objects there are also a smaller share of the picture, so there is less for a curve to go round. The balance was not measured.

A curve, costed #

A join that is a curve rather than a depth cannot be matched by one lift of the far band, so it opens a gap equal to the focal length times the riser over two, times the spread of one over depth along the seam. At the first join, three pixels of gap allows 1.15 m of wander toward the eye and 1.27 m away; going all the way round a twelve-metre object costs 15.8 px, and a three-metre one 3.7.

Among standing objects a straight seam still has 80.5 per cent of the picture, the same as among spanning ones, because a straight seam crosses every column. Bending adds 0.36 per cent at 2 px of gap and less at wider tolerances, since a wider window almost always holds a row a straight seam could use. What a curve genuinely offers is a trade on the rare seam that must bend: the offset becomes a gap to be hidden, and the turn that convicts a crossing disappears.

Still open: whether a band can be lifted by more than one amount #

Everything above rests on the far band being one picture, lifted by one amount. That is what makes a curved seam open a gap: the lift that fits one column does not fit the next.

A painter is not a camera, though — a map along, and a picture across describes a handscroll whose eye moves continuously and has no join at all — and nothing stops a far band being drawn as a family of columns each lifted by its own amount — which is to say, drawn from a station whose height varies across the picture. The seam would then close exactly in every column. The cost moves elsewhere: a band lifted differently in different columns bends every straight line that runs across it, and a road or a wall running across the far band would come out curved.

The measurement that settles it takes the seam that had to bend, lifts the far band column by column to close it, and measures how much a straight horizontal feature in the far band is bent as a result, against how far into the band it lies. If the bend dies away within a short distance of the seam, a varying lift is a cheap way to close any curved seam and the gap above is not a real constraint; if it persists across the band, the painter has only moved the gap from the seam into every line the far band holds.