A bend in the barrier puts the drawn polygon over water

A drawn reach set stops at the river put a straight river three kilometres north of a ten-kilometre reach set, crossable only at bridges, and measured what a fan of bearings draws over it. The polygon misses about a third of the reachable ground and claims almost nothing that is not there — the error is essentially all in one direction.

That one-sidedness is not a property of rivers. It follows from a straight barrier leaving the near side convex, and convexity is what every reach set ever drawn is too small derives the whole one-sided result from: a chord between two points of a convex set stays inside it, so a polygon inscribed in one can only cut corners.

Real barriers bend. A river meanders, a coastline has inlets, a motorway curves — and a bent barrier breaks the hypothesis. The near side of the set is no longer convex, and a chord between two neighbouring bearings that both stopped at the bank can pass over water.

The model has to be a shortest path, not a formula #

The straight case has a closed form because the shortest legal route to any point is one of two things: a straight line, or a straight line to a bridge and another from it. Nothing else can help, because the banks are straight and the only openings are the bridges.

With a bend neither is true. The shortest route round an inlet hugs the bank and turns at its corners, so the reach of a point is a geodesic in a plane with a polygonal obstacle — and that is exactly a visibility graph. The river is drawn as a closed strip between its two banks with a gap at each bridge; two points see each other when the segment between them crosses no bank and its midpoint is not inside the strip; a shortest legal path turns only at bank corners; and Dijkstra over those corners, run once from the centre, gives the reach of every point as the smallest $d_0(v) + |P - v|$ over the corners $v$ that $P$ can see.

The strip needs width, and the reason is a defect the first version had. A zero-width polyline is not a barrier at all: a path may turn at any of its vertices, and a vertex lies on both banks at once — so every ray crossed the river for free by passing exactly through one. The whole disc came back reachable and the audit reported a true area of $\pi R^2$ to four digits, which is what that mistake looks like from outside. With the river drawn as a strip, a path that turns at a bank corner is still legal, which is right — it is walking along the bank — and one that steps across is not, because the segment’s midpoint lies inside the polygon.

The reachable ground along this bearing is in two pieces, and the fan records only the end of the first — so the second is ground the polygon has no way to know about. That much is true beside a straight river too: a bearing that crosses it and finds far-side ground within reach of a bridge meets the set twice, and 40 per cent of bearings do. What a bend adds is the other half of the picture — a chord between two bearings that stopped on the near bank, spanning a stretch where the bank swings towards the centre, passing over the river between them.

Two errors, and they are not the same size #

The omission is what a straight river already produced, and it grows only a little with the bend: 33.6 per cent for a straight river, between 33.8 and 35.7 at every amplitude up to two kilometres either side. That plateau is the earlier result and its cause has not changed — every bearing that meets the river stops at the bank, so the polygon holds none of the ground beyond it, and adding bearings cannot help with ground no bearing reaches.

The over-claim is new, and it behaves quite differently. For a straight river it is exactly zero: a straight bank leaves the near side convex, and a chord between two points on it cannot leave it. With the first bend it appears — 0.128 per cent at a meander of four hundred metres either side — and it then stays where it is, between 0.13 and 0.15 per cent at every amplitude up to two kilometres, where it is 0.154.

At every bend measured the over-claim is under half a per cent of the omission — 0.37 to 0.44 per cent. So the two do not cancel, and the reason is structural rather than accidental.

The omission is the far side of the river and the corners the chords cut, over a set ten kilometres across — it scales with the whole reach set. The over-claim is where a chord between two bearings that stopped on the near bank spans a trough of the bank, the stretch where the river swings towards the centre, and the ground the chord then claims is almost all the river itself. At sixteen bearings 0.34 of the 0.38 square kilometres claimed in error is water, 0.04 is far-side ground, and less than a hundredth is the near side. It lives in a strip as wide as the river along the bank the set touches — it scales with the river — and on any reach set much larger than the river is wide, the second cannot catch the first.

That is a statement about geometry rather than about this river, and it says when the conclusion would fail: a reach set whose size is comparable with its barrier’s own bends. A one-kilometre walk beside a river meandering half a kilometre is that case, and nothing here measures it. A reach set with a cost that depends on direction is the nearest thing to it in this collection, and it makes the set lopsided without making it disconnected.

A bend too tight or too loose is not one to fall into #

The over-claim is largest at a wavelength of about six kilometres and smallest at fourteen, and only the second of those has a reason that can be stated with confidence. A meander much longer than the gap between neighbouring bearings is locally straight over the length a chord spans, so the near bank is nearly convex at the scale the polygon works at even though it is not convex globally, and a chord seldom spans a trough.

At the short end the pattern is not a clean peak. The tightest meander measured, two kilometres, over-claims nearly as much as the six-kilometre one, and three kilometres less than either. Troughs of the bank and bearings of the fan are two periodic things of different periods, and at sixteen bearings the ground a chord spans at three kilometres from the centre is about a kilometre wide, the same order as the meander’s own half-wavelength; the over-claim at any one wavelength is where the two happen to fall against each other. That is a statement about the instrument and the barrier together rather than about either alone — the same shape of finding as the score is not stable at any scale, where a shape measure turned out to depend on the resolution it was computed at rather than on the shape.

More bearings help, unequally #

Against a convex set a polygon of $n$ bearings has an area deficit falling as $n^{-2}$, which every reach set ever drawn is too small derives and measures. Nothing here does.

The omission falls with a fitted exponent of $-0.24$, and the straight river’s with $-0.23$: from 42.3 per cent at eight bearings to 35.0 at sixteen and 30.6 at sixty-four. Both are far shallower than a square law because most of what is missing is not a cut corner at all — it is the far side, which no bearing reaches at any count.

The over-claim does not fall. Its fitted exponent is $+0.03$, with no trend to fit: from twelve bearings to sixty-four it stays between 0.14 and 0.18 per cent. More bearings do make each chord that spans a trough of the bank shorter, and the water under them falls, from 0.34 square kilometres at sixteen bearings to 0.20 at sixty-four. But more bearings also means more of them passing straight through a bridge, running on across the far side and dragging the chords either side of them over far-side ground no walk reaches — 0.04 square kilometres at sixteen, 0.38 at sixty-four. The two roughly cancel.

So doubling the count buys a slowly shrinking omission and nothing at all against the error the bend created. A fan of any count beside a bending river crosses water.

That last figure is the one to be careful with, and it is the practical sting.

A user of a drawn reach set sees an area, and the area is the difference of the two errors. The over-claim is too small to move it: at sixteen bearings the bending river’s drawing is short by 34.9 per cent against the straight river’s 33.6, and at sixty-four by 30.5 against 30.6. The total moves by a point or so with the bend, in whichever direction the fan happens to fall, and says nothing about whether the polygon crosses water.

It does. The drawn area is a fair estimate of the true one in both cases, and in one of them it is a lower bound and in the other it is not.

What is lost is a bound, not accuracy #

The sizes are the visible part of the result and they are not the important part.

Every reach set ever drawn is too small is not a statement about how wrong a fan polygon is. It is a statement that the polygon is a lower bound: whatever the count, whatever the cost, whatever the shape, the drawn set is contained in the true set and every point inside the outline really is reachable. That is worth far more than a small error, because it can be used. A planner who needs to be sure can act on the drawing; a planner who needs the true extent knows which way to allow.

A bend destroys that, and it destroys it at an amplitude far below where the error becomes numerically interesting. Four hundred metres of meander against a ten-kilometre reach — an eighth of a per cent of over-claim, invisible in any total — is already enough for the outline to enclose water. The guarantee is gone at the first bend and the quantity is still negligible, so there is no threshold at which the bound fails; there is only the question of whether the barrier is exactly straight, and no real one is.

That reverses which of the two numbers a reader should care about. The omission is a third of the ground and correctable in principle, because its sign is known. The over-claim is a seventh of one per cent and not correctable from the drawing, because a drawn boundary gives no way to tell which part of it is over land.

The same distinction runs through the neighbouring measurements once it is stated. A circle of a distance is not a circle is about a drawn shape that is wrong in a way the page produces and the ground does not, which is knowable; a corridor has a width the page cannot keep is the same for a band. Both are one-sided and both can be allowed for. A bite cut by a bend is not, and nothing in a fan’s own output says it is there.

Where the non-convexity reaches the rest #

Two other measurements of reach sets assume, without saying so, that one is the kind of thing a fan can describe.

The most compact shape depends on the paper scores a reach set by area against perimeter. A polygon that crosses water has too little perimeter as well as slightly too much area: the bites it spans are exactly the wiggles a compactness score is measuring. So a reach set beside a meander will read as more compact than it is, and by more than the area error suggests, because perimeter is the sensitive half of that quotient.

Nearest of many is a partition and a partition under a directed cost has two versions divide ground between several centres by which is nearest. With a bent barrier the cells are not connected: a pocket on the far side of a bend can belong to a centre it cannot reach directly, and its nearest centre by legal walk may be one whose cell is elsewhere entirely. A drawn partition would give that pocket to the wrong centre and the boundary between cells would be in the wrong place — an error of the same kind as the one measured here, and one no amount of extra bearings removes.

Neither is measured here and both follow from the same broken hypothesis, which is worth saying because it changes what the shortfall is. The bend does not add a small correction to three earlier results. It removes the property all three were relying on.

What each number was checked against #

With no bend the visibility graph must reproduce the closed form. The straight case is the one configuration where an independent answer exists, and the two constructions must describe the same set. They agree on its area to 0.06 per cent — 310.49 km² against 310.31 — and on how a fan’s error splits: an omission of 34.01 per cent at sixteen bearings against the closed form’s 33.98.

That second agreement was not there when this essay was first written, and the disagreement was published as a finding. Each bearing’s reachable intervals were found by sampling the walking distance at 110 points over ten kilometres, 91 metres apart, and the river is 80 metres wide. A bearing crossing it squarely stepped clean over it about one time in eight, read the far bank as reachable, and put its vertex on the far side, so the audit of a straight river read an omission of 31.33 per cent against the closed form’s 33.98, and the gap was attributed to routes that turn at a bank corner. It was the sampling. Every bearing is now sampled at three points to the river’s width, and every number in this essay has been measured again that way: the over-claim at a bend of two kilometres, first reported as 0.376 per cent, is 0.154; the claim that more bearings reduce it, and that the bend makes the drawn area look better, did not survive.

A straight barrier must leave the near side convex, so its over-claim must be below half a per cent of the omission. It is exactly zero.

A bend must produce a real over-claim, at least five times the straight case’s. The straight case’s is nothing, and the bend’s is 0.136 per cent at the fan position this control uses.

Some bearing must meet the true set more than once. Nearly half do — 45 per cent of bearings beside the bending river — but so do 40 per cent beside the straight one, because a bearing that crosses the river and finds far-side ground reachable by a bridge meets the set twice whatever the bank’s shape. The control is kept because a construction in which no bearing met the set twice would not be modelling a river at all; it does not separate a bend from a straight bank, and nothing about the count of intervals does.

And the over-claim must stay an order of magnitude below the omission across the whole sweep, which is the finding stated as a control. The worst ratio is 0.44 per cent.

Where the model stops #

Kilometres on a plane. As in the straight case: the ground is flat, the cost is distance, and nothing here is on a sphere. At ten kilometres that is exact to well below everything measured.

One barrier, of one shape. A sinusoid is the simplest curve with an amplitude and a wavelength, and a real river’s bends are neither regular nor symmetrical. What the sinusoid buys is that the two parameters can be varied independently, which is what makes the resonance in the wavelength readable; what it costs is that a real meander’s bights vary in size, so its over-claim would be a mixture of the wavelengths measured here rather than one of them.

The amplitude stops at two kilometres against a river three away, and that is a limit of the case rather than of the method. Beyond it the nearest bend comes within a few hundred metres of the centre, and the configuration stops being a reach set beside a barrier — the barrier is then through the middle of it, which is a different question and probably a more interesting one.

The phase scatter is shown rather than smoothed. Sixteen bearings against a four-kilometre meander is two periodic things beating against each other, and the omission moves by up to three and a half points depending on where the fan’s first bearing falls. Every number quoted is a mean over sixteen positions of the fan, and the bars are the spread.

And the walk is the only cost. The reach set takes the shape of the roads is where a cost field with structure enters this collection, and a real barrier usually comes with one: a river has a road along it, a bridge has an approach. Both would change where the bites are, and neither changes that there are bites.

Still open: whether a drawn boundary can be made to fail in one direction on purpose #

The result suggests a repair rather than only a warning, and it is worth setting out because it is cheap.

The over-claim exists because the polygon is drawn through the fan’s vertices with straight chords. Nothing forces that. A boundary drawn as the inner envelope — each chord pulled back to the nearest point of the true set along it — would be one-sided again by construction, at the cost of a little more omission. The measurement above suggests the trade would be small: at sixteen bearings there is 0.15 per cent of over-claim to remove, and the chord already lies mostly inside.

What is not obvious is how to find the pull-back without the shortest-path field the drawing was supposed to avoid computing. A bearing’s own interval structure is available — the fan already knows whether its bearing met the set once or twice — and whether that alone is enough to decide which chords to pull back is a question a fan that records only its vertices cannot ask.