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

Isochrone — where it appears

A curve joining the places reachable in the same time. It is a level set of a travel-time field, so it is a circle only when the cost of a step is the same in every direction, which nothing anybody travels obeys.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

The shortest route and the quickest one, in a zonal jet. A craft making 20 km/h through the medium, from 40° north, 34° west to 52° north, 6° east. The great circle is 3313 km and takes 111.6 hours in this flow; the quickest track is 169 km longer — 5.1 per cent further — and takes 99.1 hours, saving 11.2 per cent of the time. The strokes are the flow at its own scale, and the whole difference is that the track bends into the helping part of it. Drawn in Lambert conformal conic.

The quickest route is not the shortest

In a medium that does nothing, length and time are the same question divided by a constant. Once the water moves, they are different questions with different answers: the quickest track sails five per cent further and arrives eleven per cent sooner, and the journey back takes four times as long as the journey out.

paths · Paths 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

Named alongside it

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

Reach setVerificationClosed formAnisotropyCatchmentCostDiscretisationReachToleranceAreaConvergence rateDijkstra

All concepts