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

Catchment — where it appears

The set of places that belong to a site under a stated cost. Under a symmetric cost there is one of them; under a directed cost there are two, and which one a planner wants is decided by whether the traffic runs towards the site or away from it.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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 The ground two divisions disagree about. seven places across the world dividing the sphere twice under a steady westerly. Outbound, a place belongs to the site that can reach it soonest; inbound, to the site it can reach soonest. The pale tints are the outbound division where the two agree; the dark ground is where they do not, and it is 50.7% of the sphere — 258,530,666 km². Every site's own share of the world barely moves between the two divisions, by at most 3.8%. The two divisions assign nearly the same AMOUNT of ground to each site and assign completely different ground.

A partition under a directed cost has two versions

Dividing a surface among several sites is one question when the cost is a distance and two questions when it is not. Under a steady flow at 0.45 of a vehicle's own speed, the division by who can reach a place soonest and the division by which place can be reached soonest disagree about 50.7 per cent of the sphere — while no site's own share of the world moves by more than 3.87 points.

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.

ReachVerificationAreaClosed formDiscretisationIsochroneReach setToleranceAnisotropyConvergence rateCostEstimator

All concepts