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

Range ring — where it appears

A circle drawn on a chart at a stated distance from a place. Its image is a circle only on an azimuthal projection centred there, and it is the set a reader is invited to read as the reach — which under any cost with a direction in it is a different set.

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

Everywhere within 3,000 km of London, drawn in Lambert cylindrical. The set of places exactly 3,000 kilometres from London by the shortest route, projected point by point. On the ground it is a circle — every point of it is the same distance from the centre, in every direction. On this page the longest radius from the drawn centre is 4.17 times the shortest, so a reader with a ruler measures two different distances for one ground distance depending on which way the ruler points. The two extreme radii are drawn.

A circle of a distance is not a circle

Eleven essays in this field have followed a route across a map. A range ring is not a route — it is the edge of a set — and drawing one exposes a failure the route essays cannot: the same ground distance comes out 4.17 times longer in one direction than another on a common projection, and 13.03 times at 70° north.

paths · Reach 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

Named alongside it

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

Great circleAnisotropyAreaAzimuthalBoundaryCostDijkstraEquidistanceEquidistantFlowGeodesicIsochrone

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