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 both hypotheses at once. A bearing can leave the reachable ground and re-enter it further out, and a chord between two neighbouring bearings 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.

That picture is the broken hypothesis made visible. 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, and the chord to the next bearing passes straight over its outer part.

Two errors, and they are not the same size #

The omission is what a straight river already produced, and it does not systematically grow with the bend: it sits between 28 and 33 per cent throughout, with large scatter depending on where the fan’s bearings happen to land. 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. It starts at 0.019 per cent for a straight river — which is the quadrature’s own floor rather than a measurement, since a straight barrier cannot produce one at all — and rises steadily with the bend to 0.376 per cent at a meander of two kilometres either side.

At the widest bend measured the over-claim is 1.34 per cent of the omission. So the two do not cancel, and the reason is structural rather than accidental.

The omission comes from chords cutting corners at every one of sixteen bearings, over a set ten kilometres across — it scales with the whole reach set. The over-claim comes only from the bites a bend cuts out of the near side, which live in a band as wide as the meander’s amplitude and as long as the stretch of bank the set touches — it scales with the barrier. One is an area of order $R^2$ over $n^2$ and the other of order $A \times L$, and on any reach set much larger than the bends in its barrier 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 smaller at two and at fourteen, and neither end is an accident.

A meander much shorter than the spacing between bridges is averaged away by the walk: every bight has a bridge near it, so the route into a bight is short and the ground inside it is reachable after all. A meander much longer is locally straight over the distance a chord spans, so the near side is convex at the scale the polygon works at even though it is not convex globally.

What is left in between is bites whose size is comparable with the gap between two neighbouring bearings where they cross the bank. 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. None of these three does.

The omission falls with a fitted exponent of about $-0.3$, and the straight river’s with about $-0.2$. 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 falls faster, at about $-0.75$, because it is a chord effect: a bite is missed only when a chord spans it, and halving the chord length halves the chance.

So doubling the count buys least where a reader would most want it and most where the error was already smallest. A fan fine enough to reduce the over-claim to nothing still misses a third of the ground.

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. So the over-claim makes the drawn area a slightly better estimate of the true one while making it a considerably worse description of which ground is in it. At sixteen bearings the bent river’s net shortfall is smaller than the straight river’s; a figure reporting only the total would call the bend an improvement.

It is not an improvement. The polygon is over water.

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. Half a kilometre of meander against a ten-kilometre reach — a fifth 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 thirty per cent and correctable in principle, because its sign is known. The over-claim is a third of one per cent and not correctable at all, 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 0.08 per cent — 310.56 km² against 310.31.

They do not agree on how a fan’s error splits: the closed form gives an omission of 33.98 per cent at sixteen bearings and the visibility graph 31.33. That difference is real and is recorded rather than tuned away. The closed form allows a route of two straight legs through a bridge; the visibility graph also allows a route that turns at a bank corner on the way. The second is the correct shortest path, so the earlier number is a small over-estimate of the omission — about two and a half points on this configuration.

A straight barrier must leave the near side convex, so the over-claim is required below half a per cent of the omission. It gives 0.06 per cent, which is the quadrature’s floor.

A bend must produce a real over-claim, at least five times the straight case’s. It gives twenty times.

Some bearing must meet the true set more than once. A third of them do. That is the hypothesis a straight barrier satisfies and a bent one does not, so a construction in which no bearing met the set twice would not be modelling a bend at all.

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 1.3 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 large and 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 ten 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 says what that trade would be worth: at sixteen bearings it would remove 0.4 per cent of over-claim and add rather less than that in omission, because 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.