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

Dijkstra — where it appears

The algorithm that finds the cheapest route to every place by expanding outward in order of cost. It is exact for non-negative costs and it measures the best path made of the steps it was given, so on a lattice its answer carries the lattice's own anisotropy.

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

Also named here as wavefront — the same set of essays touches all of them, so they are one junction rather than several.

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

Three rungs build reach sets out of a distance, which 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

Named alongside it

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

AnisotropyCostFlowIsochronePurposeReach setVerificationWavefrontGeodesicGreat circleMetricPartition

All concepts