Cartographic Projection
every map is wrong, and by exactly how much
Concept

Reach set — where it appears

Everywhere that can be got to from one place within a stated budget of distance or time. Under a distance it is a ball; under a cost that depends on direction it is not round, not the same forwards and backwards, and has to be computed rather than drawn with a compass.

Named by 11 essays across one field — each of them below, with the objects they name alongside it.

A reach set with a cost that depends on direction. Everywhere reachable in the time it takes to cover 3000 kilometres in still air, under a steady westerly, with the still-air set drawn inside it. The anisotropic set runs from 1653 kilometres against the flow to 4261 with it — a ratio of 2.58 — while the still-air set is round to 1.023, which is the lattice's own floor and not a shape. Drawn in an azimuthal equidistant centred on the source, so every radius on the page is a ground distance and none of the shape is the projection's.

A reach set with a cost that depends on direction

A reach set built out of a distance is symmetric and isotropic by construction. Nothing anybody travels is: in a flow at 45 per cent of a vehicle's own speed the same vehicle gets 4,261 kilometres one way and 1,653 the other — a ratio of 2.58 — while the ground it covers grows by ten per cent.

paths · Reach The set that can be reached is not the set that can reach. Two sets under a steady westerly: everywhere reachable from the marked place in a stated time, and everywhere from which the marked place can be reached in the same time. They have the same area to a fraction of a per cent — reversing a uniform flow is a reflection — and they overlap on only 32.3 per cent of their union. 67.7 per cent of the ground in one of them is not in the other.

The set that can be reached is not the set that can reach

The moment a cost stops being symmetric, two questions that read alike stop having the same answer. Under a flow at 45 per cent of a vehicle's own speed the set reachable from a place and the set from which the place is reachable have the same area to five significant figures and share 32 per cent of their union — so 68 per cent of the ground in one of them is not in the other.

paths · Reach The set a reach map shows, drawn from eight bearings. A geodesic disc of 4,000 km and the polygon a fan of eight bearings draws round it, on an equal-area azimuthal page centred on the disc so that the shaded ground is proportional to the ground it stands for. Every vertex of the polygon is on the true boundary and every edge between two of them is a chord, so the drawn set is inside the true one — always, at every count, for any convex reach set. The area it misses is 7.53% of 48,635,855 km², and it is not an error that care removes. It is what a finite fan is.

Every reach set ever drawn is too small

An isochrone is drawn by walking out along a finite number of bearings and joining the points, so its vertices are on the true boundary and its edges are chords — which puts the drawn set inside the true one, always, at every count, for any convex reach set. The deficit falls as the square of the count, and a spherical cap loses less than a circle by exactly cos t (1 + cos t)/2.

paths · Reach Everywhere reachable in 1 h 15 min on roads 5 km apart. The ground reachable in 1 h 15 min from a road crossing, with roads every 5 km in two directions at 80 km/h and the ground between them at 20 km/h, computed on a one-kilometre lattice. It reaches 100 km along a road and 67 km along the diagonal, and covers 18,873 km². The dashed diamond is the shape the set converges to as the roads get denser, whose area is 20,000 km². The dotted ring is what a range ring at road speed claims — 31,416 km², of which 39.9% cannot be reached — and the small dotted circle is what the ground alone would give.

The reach set takes the shape of the roads

On open ground a reach set is a disc. On a grid of roads it converges to the convex hull of a diamond and a disc, and once the roads are √2 times faster than the ground between them the ground's speed stops mattering at all. A range ring drawn at road speed then claims 57 per cent more ground than anybody can reach.

paths · Reach What 24 bearings draw beside a river crossable at eight bridges. Everywhere within a 10 km walk of the centre, with a river 3 km to the north crossable only at bridges every 2.5 km. Short of the river the reachable ground is the disc; beyond it, each bridge reaches a half-disc of whatever walk is left, and those half-discs are 30.3% of all the reachable ground. The solid outline is the polygon a fan of 24 bearings draws: every bearing that meets the river stops at the bank, so the polygon holds none of the ground beyond it and is short by 31.6% in total. The river is drawn wider than its 80 m.

A drawn reach set stops at the river

A polygon drawn through a fan of bearings round a reach set is short by a chord deficit that falls as the square of the count — on ground with nothing in the way. Put a river three kilometres off, crossable at bridges, and the polygon misses 30 per cent of the reachable ground at every count from forty-eight bearings to a thousand, and a convergence check passes on it.

paths · Reach In a wind every circle moves upwind, and the route moves with them. New York to London with the sixty-minute allowance at a stated one-engine speed of 745 km/h. In still air every airfield's circle has a radius of 745 km, drawn faint and dashed. In a northerly of 150 km/h the places from which each airfield can be reached in sixty minutes form a circle of the same radius moved 150 km upwind, drawn solid. The dashed route is the shortest the still-air rule allows, 5,698 km; from one point of it the quickest diversion in this wind takes 68.7 minutes. The solid route stays inside the moved circles, 5,765 km, and its longest diversion takes 60.8 minutes — the sixty the moved circles stand for, to within what the Earth's curvature does to a wind that keeps one bearing.

A diversion allowance in a wind is the same circle, moved upwind

A diversion rule written in minutes is turned into a distance by assuming still air. In a uniform wind the places from which an airfield can be reached in sixty minutes are not a stretched region but the still-air circle moved upwind by the wind's run. A 150 km/h northerly makes the still-air route across the North Atlantic need a diversion of 68.7 minutes, a southerly makes it 54.0 — and the least radius at which the ocean can be crossed barely moves.

paths · Paths In a jet the sixty-minute sets stop moving together. The North Atlantic airfields with a southerly jet of 200 km/h whose core, the solid line, runs north past the southern tip of Greenland; dotted lines mark where it has slowed to 150, 100 and 50 km/h. Faint dashed circles are the still-air sets of 745 km. Solid outlines are the places within sixty minutes of each airfield in the jet, computed as quickest flights through it: each is a circle moved by the wind it meets, and airfields on opposite sides of a gap meet different winds. The least allowance at which New York is joined to London moves from 624.7 km of still-air radius to 587.7 (−36.9 km), pinned between Goose Bay–Narsarsuaq, whose sets first touch at the marked point. A uniform southerly of the same speed moves it +3.0 km.

A jet moves the floor that a uniform wind cannot

A uniform wind leaves the least radius at which the North Atlantic can be crossed almost where it was, because it blows the same way on both halves of every gap between airfields. A jet does not. A southerly of 200 km/h past the southern tip of Greenland lowers that floor by 36.9 kilometres, the same jet 400 kilometres further west raises it by 23.3, and one integral of the wind along each gap predicts every such move to within a few kilometres.

paths · Paths What 16 bearings draw beside a river that bends. Everywhere within a 10 km walk of the centre, with a river 3 km to the north meandering ±2.0 km on a 4 km wavelength, crossable only at bridges every 2.5 km. The reachable ground is shaded — pale where it is reached directly and darker where the walk had to go by a bridge. The outline is the polygon 16 bearings draw. It misses 34.1% of the true ground, as a fan beside a straight river does, and it also CLAIMS 0.14% that is not there — ground cut off by a bend, which a chord between two neighbouring bearings passes straight over.

A bend in the barrier puts the drawn polygon over water

A straight river leaves the near side of a reach set convex, and that is what makes a fan of bearings fail in one direction only: every chord cuts a corner, nothing is claimed that is not there. A meander breaks that: the near side is no longer convex, and a chord between two bearings can cross water. So the drawn polygon now over-claims as well as omitting — by a seventh of a per cent, at every bend measured, and more bearings do not reduce it. The question was whether the two errors partly cancel. They do not, and the reason is not their signs: the over-claim never reaches a hundredth of the omission, because one scales with the whole set and the other only with the river.

paths · Reach The floor against where the streak is, and the stretch of it a crossing sweeps. The diversion floor for the New York–London chain with a streak 200 km/h faster than a 100 km/h jet core, against where along the jet the streak's centre lies, computed exactly on a grid of positions 125 km apart and read between them. With the streak far off the floor is the plain jet's, 624.7 km. With it over the Goose Bay half it rises by up to 41.1; over the Narsarsuaq half it falls by up to 44.8. The bar is the stretch a streak starting at -1400 km sweeps in a four-hour crossing at 250 km/h; the floor to be met is the worst on it, 666.0 km. The shaded band either side is a forecast position error of ±200 km at one standard deviation.

A forecast's error in where the streak is becomes a margin

A jet streak drifting along a gap moves the diversion floor by up to forty kilometres during a crossing, and an operator knows where the streak is only from a forecast. Planned to the worst over the forecast's own crossing window, the floor needs a margin of about eight kilometres for every hundred of error in the streak's position, and stops paying for itself against simply ignoring the streak near an error of two hundred and fifty. Planned to the departure analysis alone, it is worse than ignoring the streak at any error at all.

paths · Paths Beside a bending river, the fan's chords cross water; a route drawn from the river's own outline never does. Everywhere within a 10 km walk of the centre, shaded pale, with a river 3 km north meandering ±2.0 km on a 4 km wavelength and bridges every 2.5 km. Left: the polygon 16 bearings draw, with the ground it claims and should not in solid ink — chords spanning the bends. Averaged over where the first bearing falls, it misses 35.0% of the true ground and claims 0.15% that is not there. Right: every point reachable by a route with at most one crossing — straight out, or straight to a bridge and straight on — worked out from the river's outline and the bridges alone, with no fan and no shortest path. It misses 6.4% and claims nothing.

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

Beside a bending river, a fan of bearings drawn with straight chords claims a strip of water, and the drawing stops being a lower bound. The fan's own numbers cannot repair it: what a bearing records about the river is where the bridges are, not where the bends are, and the best a rule built from them does is cut the over-claim by seven eighths while giving up twenty times as much ground. The river's outline repairs it exactly. Cutting each bearing at the first bank costs three and a half times the over-claim it removes; keeping the chords wherever a route over one bridge reaches costs one and a half. Drawn from the outline and the bridges with no fan at all, that route set misses 6.4 per cent of the reachable ground where the fan misses 35.

paths · Reach Beside a bending river, every reachable point is reached by a route that meets the bank at most three times. Everywhere within a 10 km walk of the centre, beside the stated river meandering ±2 km on a 4 km wavelength with bridges every 2.5 km, shaded by the fewest times a shortest route to it has to meet the bank — a contact being one run along the bank, however many corners it passes. 70.3% of the ground is seen from the centre across dry land. One contact reaches 28.0% more, 85% of it beyond the river and the rest in bays of the near bank hidden from the centre by a spur; two reach 1.18% and three the last 0.47%, 70% of those two beyond the river, in the lee of the far bank's bends beside the outer bridges. Nothing needs a fourth.

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

A reach set drawn in layers — rays from the centre, then from every place a route can leave the bank — ends in the exact answer, and how deep it must go depends on what a layer is. Counted by corners turned, the depth belongs to the polygon the bank is drawn as: five turns on a bank drawn every 400 metres, eighteen on one drawn every 50. Counted by meetings with the bank, it is three at every resolution. One meeting anywhere leaves 1.6 per cent of a ten-kilometre walk unreached, where rays from the bridges alone leave 6.4; two leave 0.47; three leave nothing. A longer walk needs more — up to six meetings at twenty kilometres across thirty-six rivers — and the bend's steepness, which prices the first layer, ranks the depth at 0.24.

paths · Reach

Named alongside it

The objects these essays reach for when they reach for this one.

VerificationToleranceEstimatorIsochroneClosed formDiscretisationGreat circleAnisotropyBarrierCatchmentConventionCost

All concepts