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

Gall–Peters — where it appears

The cylindrical equal-area projection with its standard parallels at 45 degrees, and an affine image of every other member of that family. Its standard parallels are a scale factor, which the fit that identifies a projection removes, so it cannot be told apart from Behrmann's or Lambert's.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

How much each projection inflates a cell, by latitude. Five patches of the sphere, each 20° by 10°, and the factor by which each projection enlarges them relative to the equatorial one. An equal-area projection sits flat on 1. Mercator reaches 15.4× at 70°, which is the mechanism behind every complaint about the size of Greenland.

Mercator against Peters

The most-argued question in cartography, conducted almost entirely without anyone measuring anything. Both projections are exactly what they claim, each destroys what the other keeps, and the numbers are computable in either direction.

wrong · Audit The half-extent at which each rival stops fitting a Gall–Peters graticule. For each candidate, the size of region at which its best fit to a Gall–Peters map first leaves a residual of a fifth of a per cent of the map's width. Below that size the two are the same picture. The numbers are half-extents in degrees of latitude, at 20° north, with the plane affine transformation removed; a bar at 90° is a rival that never separates at all within the range searched.

What a careless copy hides

A photocopier that stretches one axis is a nuisance to remove before a projection can be identified. Removing it costs more than it looks: an affine fit absorbs the entire difference between the cylindrical equal-area projections, so Gall-Peters and Behrmann become the same picture at any size, to sixteen decimal places.

wrong · Identify The same field, the same statistic, two pages. A single northern concentration — v = 10 + 90 exp(−d²/(25°)²), d the distance from 20°E 55°N — drawn twice, with the shading rule and the data identical. The ground's own area-weighted mean is 14.009. A reader weighting by the area actually on the page takes 17.490 off Mercator, which is +24.85%, and 14.009 off Gall–Peters, which is +0.00%. Neither map has misdrawn a single cell.

A choropleth is read by area

Every cartography course states the rule — use an equal-area projection for a thematic map — and states it as advice. It is a theorem, and it has a residual: the error a page puts into a reading is exactly the covariance of the value with the areal factor, which is 24.85 per cent for a northern concentration read off Mercator and 0.00 per cent for the same field read off a map that spreads area by 7.7 to one.

distortion · Thematic

Named alongside it

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

Equal-areaMercatorStandard parallelAffine transformationAngular deformationAnisotropyArea weightingAreal distortionAreal factorBiasCartographic controversyChoropleth

All concepts