A reach set drawn from the river's outline is a bound again

A bend in the barrier puts the drawn polygon over water found what a meander does to the simplest way of drawing a reach set. A fan of bearings is cast from the centre, each is followed until the walk runs out or the river stops it, and the ends are joined with straight chords. Beside a straight river that polygon is always too small: every chord cuts a corner of a set whose near side is convex, so nothing inside the outline is out of reach. Beside a river that bends, a chord between two bearings that both stopped on the near bank can span a trough of the bank — a stretch where the river swings towards the centre — and the outline then encloses water.

The amount is small: 0.15 per cent of the reachable area with sixteen bearings, beside an omission of 35, and more bearings do not reduce it. The essay argued it was nonetheless the more important of the two errors. An omission has a known sign and can be allowed for. An over-claim of any size means the drawing is no longer a lower bound, and a planner can no longer act on the rule that every point inside the outline is reachable. It also reaches the measurements built on a drawn set: the most compact shape depends on the paper scores a reach set by its area against its perimeter, and a chord that spans a trough shortens the perimeter as well as adding water.

It ended on a repair and a doubt. The repair: draw the boundary so that it fails in one direction on purpose, pulling each chord back to where the true set is. The doubt: finding where the true set is along a chord seems to need the shortest-path field the fan was meant to avoid computing, and a fan records very little about the river — only, along each of its own bearings, whether the bearing met the reachable ground once or more than once.

The case, and five drawings of it #

The ground is the one the earlier essay measured. A centre, a walk of ten kilometres, and a river three kilometres to the north meandering two kilometres either side of that line on a four-kilometre wavelength, eighty metres wide and crossable only at bridges every two and a half kilometres. The true reach set is computed exactly, as shortest paths round the river drawn as a polygon, and every drawing is audited against it along 240 bearings, each averaged over sixteen positions of the fan’s first bearing.

Five drawings share the fan’s sixteen radii and differ in what they draw between them, and in what else they are allowed to know.

• Chords, as drawn: the polygon of the earlier essay.
• The smaller radius: between each pair of neighbouring bearings, an arc at the smaller of their two radii. It needs nothing but the fan’s numbers.
• Pulled back where a bearing crossed twice: the smaller-radius arc beside any bearing whose own run met the reachable ground more than once — crossed the bank, met the river, and came back to reachable ground beyond it — and chords elsewhere. This is the fan’s interval structure, the one thing the earlier essay suggested might be enough.
• Cut at the first bank: chords, with each bearing of the drawing stopped where it first meets the river’s near bank. It needs the river’s outline as drawn on the map, and no shortest path.
• Kept where one crossing reaches: chords, kept at every point that a route with at most one crossing reaches within ten kilometres — straight out from the centre, or straight to a bridge, over it, and straight on. It needs the outline and the bridges, and again no shortest path.

What a bearing records is the bridges, not the bends #

The earlier essay’s hope was that a bearing which met the reachable ground twice had seen a bend. It had not. Seven of the sixteen bearings meet the set more than once beside the bending river, and seven beside a straight one, because what puts a second stretch of reachable ground on a bearing is a bridge: the bearing crosses the river, and the far-side ground beyond is within reach of the walk by way of a crossing somewhere near. That happens beside any river crossed at bridges. The count of intervals on a bearing records where the crossings are and says nothing about the shape of the bank.

A chord crosses water when it spans a trough of the bank, and a trough a chord spans lies between two bearings, by definition — a trough a bearing ran into would have stopped that bearing at the bank and put a vertex there. So even a bearing that did carry news of a bend would be carrying news of the wrong one.

Pulling chords back anyway, beside the bearings that met the set twice, removes seven eighths of the over-claim, and so does drawing every pair at its smaller radius. Neither removes it all: an arc at the smaller of two radii can still pass over a trough that reaches closer to the centre than either bearing. So the answer to the earlier essay’s question is no. A bearing’s interval structure cannot decide which chords to pull back, and no rule built only from the fan’s radii restores the bound.

The outline restores the bound #

The two drawings allowed to look at the river’s outline both restore the bound, and both do it by construction rather than by luck. A point on a bearing short of its first meeting with the near bank is seen from the centre across dry ground, so its walking distance is its straight distance, and it is reachable whenever that is under ten kilometres; the chord’s own radius never is more. Every point a one-crossing route reaches within ten kilometres is reachable because that route is a legal walk. Neither drawing can claim a point no walk reaches, and the audit agrees: both claim nothing, at every count and every position of the fan.

What they differ in is what they give up. Cutting every bearing at the first bank loses half a point of ground against the chords — it misses 35.5 per cent where the chords missed 35.0 — to remove 0.15 points of over-claim, so it pays three and a half times what it removes. Keeping the chords wherever one crossing reaches loses 0.21 points, one and a half times. The two fan-only drawings, which do not even restore the bound, give up three and four points.

The reason cutting at the bank costs anything at all is that the chords which cross the river do not stop at its far bank. A chord spanning a trough passes over the water and then over far-side ground; so does every chord reaching out to a bearing that happened to pass straight through a twelve-metre bridge and run on across the far side. Much of that far-side ground is reachable by a bridge. Cutting at the first bank throws it away with the water. The one-crossing drawing keeps it, because it knows where the bridges are.

More bearings do not change the order #

At every count from eight bearings to sixty-four the order holds. The one-crossing drawing stays within half a point of the chords — 42.8 against 42.3 at eight, 30.8 against 30.6 at sixty-four — and cutting at the bank within a point and three quarters. The two fan-only drawings lose between about one and six and a half points more than the chords, and they keep some of the over-claim, so they are dominated at every count.

The price of the bound falls as bearings are added, because the chords get shorter and span less of anything; by sixty-four bearings both drawings that cross no water are within two tenths of a point of the chords. What does not fall is the omission all of them share. The chord polygon goes from 35.0 per cent missed at sixteen bearings to 31.8 at thirty-two and 30.6 at sixty-four. It never approaches the dotted line, which is what the next figure is about.

The bend sets the price, and a straight river charges nothing #

With a straight river the chord polygon is already a lower bound, and a drawing that restores a bound it never lost should cost nothing. The two drawings built from the outline cost exactly nothing there: beside a straight bank no chord leaves dry ground, and no bearing that stops at the bank has anything beyond it to keep or cut. The smaller-radius arc costs two and a half points with no bend at all, because it pulls in every chord, bend or none. That is the whole case against building the repair from the fan: it cannot tell a chord that needs pulling back from one that does not, so it pulls back all of them.

As the meander grows, the outline-built drawings’ price grows with the over-claim they remove. At two kilometres either side the one-crossing drawing gives up 0.21 points to remove 0.15, and the remainder is ground the chords covered correctly and a one-crossing route cannot reach — far-side pockets behind a bend of the far bank, where a walk from a bridge has to go round a corner and a straight leg would cross the river again.

The fan was the wrong instrument #

The one-crossing set does not need the fan. It is worked out from the river’s outline and the positions of the bridges — straight lines from the centre, straight lines from each bridge the centre can see — and it is a drawing in its own right. Drawn alone, it misses 6.4 per cent of the reachable ground and claims none that is not there. The fan’s chord polygon, at sixteen bearings, misses 35.0 per cent and claims 0.15; at sixty-four bearings it still misses 30.6.

That turns the earlier essay’s question round. It asked whether a drawn boundary could be made to fail in one direction without computing the shortest paths the fan was meant to avoid, and the measurement says yes — but the useful part of the answer is that the one-direction drawing does not come from the fan at all. A fan is an instrument for a set whose boundary is what a ray from the centre meets first. Beside a barrier crossed at bridges, most of the reachable ground on the far side is not met first by any ray from the centre; it is met first by rays from the bridges. The information that finds it is where the crossings are, and a fan cast from the centre throws that away.

A drawn reach set stops at the river found the fan missing a third of the ground beside a straight river and concluded that extra bearings cannot help with ground no bearing reaches. The remedy that implies — cast rays from the crossings as well as the centre — is exactly the one-crossing drawing, and beside a bending river the same remedy is also what restores the bound.

What the one-crossing drawing still misses #

The 6.4 per cent is not noise, and it is the one-crossing drawing’s own omission rather than a fan’s. It is ground reachable only by a route that turns somewhere other than at a bridge: the far bank’s own troughs, entered from a bridge by walking round the tip of a bend; and near-side ground in a bay of the bank, hidden from the centre by a spur of the bank and reached by walking round its end. A route that turns once at a bridge and once at a bank corner is legal and short, and a drawing restricted to one crossing and no other turn cannot see it.

The fix has the same shape as before. Every bank corner the centre or a bridge can see is a place a route can turn, and adding their rays to the drawing, one level at a time, recovers the pockets in order of how many turns they need. That is the visibility graph the exact computation uses, built outward a layer at a time and stopped at a stated depth; each layer is still a lower bound, and the omission is what the layers not yet drawn would add. Every reach set ever drawn is too small found the fan’s error falling as the square of its count on open ground. Nothing here measures how fast the layered drawing’s error falls with its depth, only that the first layer beyond the centre takes it from 35 per cent to 6.4.

How the drawings were checked #

A drawing built from the outline must never over-claim. Cut at the first bank and kept where one crossing reaches, at eight, sixteen and thirty-two bearings and at every position of the fan, the largest over-claim must stay below a hundredth of a per cent. It is zero.

A drawing built from the fan alone must still over-claim. If the smaller radius or the flagged pull-back restored the bound at any count, the question of whether the fan could find its own troughs would have been answered by accident rather than measured. They claim 0.018 and 0.019 per cent at sixteen bearings, and at thirty-two the smaller radius still claims 0.004 — small, and not zero, which is the whole difference between a drawing that happens to be close and one that cannot be wrong.

With no bend, the one-crossing drawing must be the chord polygon. A straight bank leaves every chord on dry ground, so pulling back to one-crossing routes should change nothing; the two miss 33.56 per cent each.

The river the drawings see must be the river the walk sees, at a resolution that sees it. Two defects were found this way, and both were the same kind. The cut at the bank first used the smooth curve of the meander while the shortest paths go round the river as a polygon of straight segments, and on the outside of each bend the two differ by up to twenty-five metres. And every bearing — of the fan, of the audit, and of the one-crossing set — was first sampled at points 42 to 91 metres apart along a river 80 metres wide, so a bearing crossing it squarely could step clean over it. The second defect was also in the measurement the earlier essay published, and it had put the chord polygon’s over-claim at more than twice its size; that essay has been measured again. Every bearing is now sampled at three points to the river’s width.

Where the walk stops #

One river, one shape. A sinusoid of stated amplitude and wavelength, as in the earlier essay. A real river has bends of many sizes, and the one-crossing drawing’s price depends on how many far-side pockets sit behind a bend from every bridge.

Distance only. The walk costs its length. A reach set with a cost that depends on direction is where a flow or a slope makes the cost lopsided, and there a straight leg from a bridge is no longer the shortest route to anything; a one-crossing drawing would need its legs to be the flow’s own least-time paths.

Bridges are points. Each is twelve metres wide and crossed in its eighty-metre length. A ford or a ferry with a timetable is a crossing with a cost, and a drawing from crossings would weight each one — the arrangement the reach set takes the shape of the roads measured for a grid of fast roads, where the set stops being a disc and takes the shape the roads give it.

One centre. Nearest of many is a partition divides ground between several centres by which is nearest, and beside a river the same drawing question arises for every cell boundary: a boundary between two centres’ cells drawn from fans can cross water just as a single fan’s outline can, and a boundary drawn from the crossings would inherit the one-crossing drawing’s guarantee only cell by cell.

The plane. Ten kilometres on flat ground, exact to well below every number here. At the size where a circle of a distance is not a circle begins to matter, every straight leg in this essay would have to become a geodesic.

Still open: how deep the drawing has to go #

The one-crossing drawing is the first layer of a construction that ends in the exact answer: rays from the centre, then from every point a ray can reach and a route can turn at, and so on. Each layer is a lower bound and each is cheaper than the whole shortest-path field, because it stops.

Whether a planner needs more than one layer depends on what fraction of reachable ground sits behind a turn other than a crossing, and that is a property of the barrier’s shape — the size of its bends against the walk — rather than of the drawing. Beside this meander one layer leaves 6.4 per cent. How that remainder falls with each further layer, whether two layers are enough for any river whose bends are small against the walk, and whether the depth at which a layered drawing is good enough can be read off the barrier’s outline before anything is drawn, are questions one layer cannot ask.