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

Equidistant — where it appears

A projection whose scale is true along particular lines — outward from one point, or along the meridians — and along no others. The azimuthal equidistant holds distances outward from one point; the plate carrée holds them along the meridians and nowhere else.

Named by 3 essays across 2 fields — 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 5 of 400 nearest-neighbour queries change answer in the plane. 40 sites and 400 queries over -20° to 40° east and 35° to 70° north, each query answered twice — once by geodesic distance and once by straight-line distance in the stored plane. They agree 98.8 per cent of the time, which is why the operation survives, and the 5 that differ are marked. The mechanism is not that the plane is wrong by a lot but that its scale factor varies: over this region it spans 139 per cent, and every disagreement is a contest closer than that — the worst margin measured is 6.4 per cent. The furthest a wrong answer is from the right one is 24 kilometres.

Nearest is a question about the metric

Asked in the plane the data is stored in, a nearest-neighbour query returns a different site for 5 of 400 queries over Europe on Web Mercator — and 103 of 400 on the plate carrée, with the wrong answer up to 309 kilometres further away. Every disagreement is a contest closer than the region's own scale spread, and over a city there are none.

applied · Dataset five planes a dataset might be stored in, scored on three operations. Each candidate measured over -10° to 30° east and 35° to 60° north: the worst areal error, the worst angular deformation, and the spread of the scale factor, which are what an area query, a shape and a distance respectively depend on. The best plane for areas is Gall–Peters, for shapes Lambert conformal conic, and for distances Lambert conformal conic — three different answers, and no fourth candidate would collapse them, because a projection exact in two of these columns has a = b = 1 everywhere and is the isometry Gauss's theorem forbids. area of a polygon costs 3.06× too large in the wrong plane; drawing a line between two points costs 194 km from the ground it claims.

The operation decides the coordinate system

Five candidate planes over one region, scored on the three things a spatial operation depends on. The conformal conic wins shape and distance and is 11.7 per cent out on area; the equal-area member is exact on area and 38.9° out on shape. No candidate is exact in two columns, and no candidate ever will be, because one that was would be an isometry.

applied · Dataset

Named alongside it

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

ConventionMetricPrincipal scale factorsPurposeRegional distortionVerificationAreaAzimuthalBoundaryConformalityCoordinate reference systemEqual-area

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