A layer is a meeting with the bank, not a corner turned

A reach set drawn from the river’s outline is a bound again ended with a drawing that needs no fan at all. Beside a river crossable only at bridges, it shades every point a walker reaches with at most one crossing — straight out from the centre, or straight to a bridge, over it, and straight on — and it is worked out from the river’s outline and the bridges alone. It claims no ground that cannot be reached, which the fan’s chords could not promise beside a bending river, and it misses 6.4 per cent of the ground a ten-kilometre walk truly reaches, where a sixteen-bearing fan misses 35.

That essay also said what the 6.4 per cent is. It is ground reached only by a route that turns somewhere other than at a bridge: round the tip of a bend, into a bay hidden from the centre by a spur, or along the far bank after crossing. And it named the way to recover it. Every place a route can turn is a place a ray can be cast from, so a drawing can be built in layers: rays from the centre, then rays from every point the first layer’s routes could turn at, and so on. Each layer is a lower bound, because each of its routes is a legal walk, and the layers end in the exact set.

Whether that is a practical construction depends on one number nobody had measured: how many layers it takes. If the answer is two, the layered drawing is a cheap and honest substitute for a shortest-path computation. If it is twenty, it is the shortest-path computation done slowly.

What a layer counts decides whether the depth means anything #

The exact reach set on this ground comes from shortest paths round the river, and a shortest path in a plane with polygonal obstacles turns only at the obstacles’ corners. So the obvious definition of a layer is a corner: zero layers is what the centre sees, one is what a route turning once reaches, and so on. It is also the definition that measures the wrong thing.

The river in the stated case is a sinusoid three kilometres north of the centre, swinging two kilometres either side of that line on a four-kilometre wavelength, eighty metres wide and crossed at bridges every two and a half kilometres. The computation draws each bank as a polygon, and a polygon has to be drawn at some resolution. Count layers by corners turned and the depth that exhausts the reachable ground is five with the bank drawn every 400 metres, eight at 200, twelve at 100 and eighteen at 50. It does not settle. A route that walks round a bend touches every vertex of it, and halving the segment length doubles the vertices on the bend.

That is a familiar failure. A line has a length only at a scale found a coastline’s measured length rising without limit as the ruler shortened, and the score is not stable at any scale found a compactness score still falling at two thousand points. A count of corners is a property of how the bank was drawn, and the river has no corners.

What the river does have is the structure a route takes past it. A route that must go round found that the shortest path past a circular exclusion is two straight legs and an arc of the rim between them: free, then in contact, then free. A route past a meander has the same anatomy repeated — free legs across open ground, runs along the bank, free legs again — and the number of runs is a property of the route and the river, not of the polygon. Counted that way, as meetings with the bank however many corners each one passes, the depth is three at every resolution from 400 metres to 50.

One detail of the counting matters and is stated. Where the bank is nearly straight, a fine polygon’s vertices are nearly collinear, and a route following the bank can cut a chord that skips one of them by a few centimetres. Two vertices of the same strip count as one run when every vertex between them lies within two metres of the chord joining them. Without that rule the 100-metre bank’s one-meeting share moves from 0.99 to 1.03 per cent, the two-meeting share by a hundredth of a point, and the depth not at all.

The one-meeting share itself still moves with the drawing: 1.4 per cent at 50 metres, 1.0 at 100, 1.6 at 200 and 4.2 at 400. Some of that is the river: a bank drawn every 400 metres cuts across the outside of every bend by up to a hundred metres, and its true reach set is 289 square kilometres rather than the finer banks’ 280. The coarsest drawing is a different river, and the depth that exhausts it is still three.

Most of what the bridges leave out is behind the near bank #

The first layer of the construction, counted by meetings, is not the one-crossing drawing. It contains it — a route over a bridge meets the bank at most once, at the bridge’s own end of the strip — but it also contains every route that meets the bank anywhere else and leaves it again. Its omission is 1.64 per cent against the bridges’ 6.40, and the difference is worth locating, because the earlier essay guessed wrong about where it was.

That essay pictured the one-crossing drawing’s remainder as mostly far-side ground: pockets behind the far bank’s bends that a walker off a bridge could only reach by going round a corner. Measured, nine tenths of what one meeting adds over the bridges is on the near side. It is ground in the bays of the near bank, hidden from the centre by a spur of the bank between them, and reached by walking to the tip of the spur and round it. The bridges were never going to find it, because nothing about it involves crossing. The one-crossing drawing was built to fix the fan’s real failure, which was the far side, and it left a smaller near-side failure of its own that the fan had shared.

A walker who crosses at a bridge and then follows the far bank round its next bend is also a single meeting. The run starts on the near bank at the bridge, goes along the end of the strip, and carries on along the far bank without ever leaving it. The one-crossing drawing could not see that route, because it demanded a straight leg the moment the walker stepped off the bridge. It is the smaller part of what the first meeting adds, a tenth of the gain, because a route that keeps to a bend’s outside is longer than a straight leg and most of what it reaches is reached the straight way too.

After the first meeting the remainder falls fast. Two meetings leave 0.47 per cent, three and a half times less than one. Three leave nothing at all: no point in the true set needs a fourth. That is the practical answer to the earlier essay’s question for the ground it measured. A drawing that casts rays from the centre, then from every corner the centre can see, then once more from every corner those rays reach, is exact beside this river for a ten-kilometre walk, and stopping after two such layers costs under half a per cent.

Where the last half per cent lies #

The map shows why the depth stops where it does. Most of the ground — the whole southern half and the near-side lobes between the bends — is seen from the centre. One meeting covers the far side opposite every bridge the centre can see and the near-side bays. What needs two or three meetings sits in two small patches on the far side, each beside one of the outermost bridges the walk can afford, in the lee of a bend of the far bank.

The geometry is that of a walker at the edge of the budget. Crossing an outer bridge leaves a few kilometres of walk, and the ground behind the next bend of the far bank is hidden from the bridge’s far end. Reaching it means following the far bank round the bend — still one meeting with the bank — then striking off across a bight and meeting the bank again at the next bend’s tip, and sometimes a third time. Those routes exist only where the walk is long enough to go round two bends after crossing and short enough that the ground beyond them is still worth counting. At ten kilometres that is a strip near the rim of the walk on each side.

So the depth is a statement about the walk as much as about the river. A shorter walk cannot get round two bends after crossing; a longer one can get round many.

A longer walk needs a deeper drawing beside the same river #

Beside the stated river, a five-kilometre walk barely reaches the river at all, and one meeting covers everything but a sliver. A ten-kilometre walk needs three. Fifteen kilometres still needs three to bring the remainder under a tenth of a per cent, though its tail runs a little longer; twenty kilometres needs four. The centre’s own view gets worse as the walk grows, from missing a tenth of the ground to missing more than two fifths, because a longer walk puts more of its ground beyond the river where the centre cannot see.

The threshold matters to how those numbers are read. A tenth of a per cent of a twenty-kilometre walk’s reach is a square kilometre, and whether a planner cares about a square kilometre at the rim of a walk depends on what the walk is for. The curves are the honest form of the answer, and each falls by a factor of several per meeting; the depth at a stated tolerance is a summary of them, and it moves by one whenever the curve crosses the tolerance near a step.

Thirty-six rivers, and the depth each one needs #

The stated river is one case. Nine rivers — three wavelengths, three amplitudes — each walked to four lengths give thirty-six, with every bank drawn at 100 metres. The deepest drawing any of them needs at each walk is the clearest pattern in the grid: two meetings at five kilometres, four at ten, five at fifteen, six at twenty. No river in the sweep needs more than six, and no five-kilometre walk needs more than two.

Inside each column the order is weaker than might be expected. The tightest wavelength, two kilometres, is the deepest at the two longest walks, which fits the picture from the map: more bends within reach means more bends a walker might have to get round. But how far the river meanders orders nothing. On the eight-kilometre wavelength at twenty kilometres, the moderate meander needs five meetings while the gentle and the wild ones need two and three. The likely reason, which the sweep does not test, is that the middle case is the one whose far-bank bends are deep enough to hide ground from a bridge and shallow enough that the walk has budget left to go round several of them. A gentler river hides little; a wilder one puts so much bank between the walker and the ground that the ground is out of reach, and ground out of reach needs no layers.

Two cases sit on the threshold. The fifteen-kilometre walk beside the two-kilometre wavelength at ±2 km leaves 0.11 per cent after four meetings, and the twenty-kilometre walk beside the eight-kilometre wavelength at ±2 km leaves 0.10. Either could read one shallower at a finer quadrature. The second does not touch its column’s maximum, which is six. The first is the fifteen-kilometre column’s only five, so that column’s ceiling is four or five rather than certainly five.

The outline ranks the depth, and the steepness does not #

The earlier essay’s last question was whether the depth a layered drawing needs could be read off the barrier’s outline before anything is drawn. Four numbers that need nothing but the outline and the walk were tried, each ranked against the depth across the thirty-six cases.

The bend’s steepness, amplitude over wavelength, is the natural candidate, and it is the one that fails. Its rank correlation with the depth is 0.24. It is a good predictor of something else: it ranks the one-crossing drawing’s deficit at 0.86, because a steeper bend hides more ground from straight legs off the bridges. The number that prices the first layer says almost nothing about how many layers are needed, and the difference is the one the depth grid already showed. A steep bend makes the first layer expensive; it does not make the later ones necessary, since the ground a steep bend hides is often ground the walk cannot reach at all.

The walk does better. Its length alone ranks the depth at 0.71, the number of bends of the river within the walk at 0.74, and the walk beyond the river divided by the wavelength at 0.77 — how many wavelengths of far-side ground a walker might have to get round. That is a usable statement but not a rule. A correlation of 0.77 orders most of the cases and still leaves a twenty-kilometre walk needing two meetings beside one river and five beside its neighbour. What can be read from the outline is a ceiling, set by the walk, and the measured ceiling across these rivers is two, four, five and six — the five resting on the one case at the threshold.

What would have shown the count wrong #

Every layer together must give back the exact set. The ground reached by any number of meetings, summed, must be the ground the unconstrained shortest paths reach. It is 280.11 square kilometres both ways, to the quadrature’s floor.

The first meeting must contain the bridges. Every one-crossing route meets the bank at most once, so one meeting must reach everything the one-crossing drawing reaches and possibly more. Its remainder is 1.64 per cent against 6.40, and the measurement samples both at the same points, so a single point the bridges reached and one meeting did not would have shown.

Counted by meetings the depth must not move with the polygon, and counted by turns it must. At banks drawn every 400, 200, 100 and 50 metres the meeting count that exhausts the stated case is 3, 3, 3 and 3, and the turn count is 5, 8, 12 and 18. Had the meetings moved too, the distinction this essay draws would be a preference rather than a measurement.

The quadrature must not decide the layers. With twice the bearings and twice the radial samples, the 100-metre bank’s one-meeting share moves from 0.99 to 0.93 per cent and its two-meeting share from 0.45 to 0.46. The one-meeting figure is good to about a tenth of a point; the two-meeting figure and the depth are not affected.

What this river leaves out #

One shape of river. A sinusoid of stated amplitude and wavelength. A real river has bends of many sizes, and the depth grid suggests that a mix would be deeper than any of its parts at the same walk, since the depth followed the number of bends within reach more closely than their size.

Distance only. The walk costs its length. A reach set with a cost that depends on direction is where a current or a slope makes the cost lopsided, and there a free leg is no longer straight, so a ray from a corner is no longer a straight line either.

What a layer costs to draw. A meeting is a property of a route; a ray is cast from a corner. Counting layers by meetings makes the depth a property of the river, but each layer still casts rays from every corner its routes can reach, and the number of corners still depends on how the bank is drawn. The drawing’s depth is three; its cost in rays is not measured here. Every reach set ever drawn is too small found the fan’s error falling as the square of its count on open ground, which is the cost side of the same question for a single layer.

Bridges as points, and one centre. A ford or a ferry is a crossing with a price, and the reach set takes the shape of the roads found what a network of fast routes does to the set’s shape. With several centres, nearest of many is a partition, and each cell boundary beside a river is a layered drawing of its own.

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

Still open: whether a ceiling can be stated before drawing #

The sweep gives a ceiling by measurement — two meetings at five kilometres, six at twenty — and a correlation of 0.77 with how many wavelengths of far-side ground the walk covers. What it does not give is a rule. A walker’s route meets the bank once per bend it has to go round, and the number of bends it can go round is bounded by the budget left after crossing divided by what a bend costs to round. That suggests a ceiling with a closed form in the walk, the river’s distance and its wavelength, and an amplitude only through what a bend costs. Whether such a bound holds on every river of the sweep, whether it is tight on the cases that need the most, and whether it survives a river whose bends are not all one size, are questions a grid of thirty-six cases can test but not answer.