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

Convex hull — where it appears

The smallest convex set containing a given set of points: every straight segment between two of its points stays inside it. It is what mixing a set of options can achieve, which is why a reach set built from several kinds of travel converges to the hull of what each kind reaches in unit time.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Four of the seven on the front can never be first. Every library projection over the whole sphere, with both errors normalised to the table's own range. The seven filled circles are the Pareto front — nothing beats them on both counts. The line through three of them is the lower convex hull, and a weighted sum of the two errors is a straight line in this space, so only a hull vertex can ever come first. Four projections — Web Mercator, Miller cylindrical, Equirectangular, Winkel tripel — sit in the dents, undominated and unchoosable.

Which projection a weighting can make best

The seven world projections nothing beats on both counts tell a reader with a preference that one of them is their answer. Four of the seven are not: they sit in dents of the front, undominated and unreachable, and no weighting of angle against area can ever put them first. Robinson can be first, on 2.8 per cent of the weight range.

wrong · Audit

Named alongside it

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

Angular deformationAnisotropyAreal factorCatchmentClosed formConvergence rateCostDiscretisationIsochroneOptimisationPareto frontPurpose

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