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

Plate carrée — where it appears

Longitude across and latitude up, plotted on ordinary axes — a projection with no exact property and the one a lon/lat file is implicitly in. Its page coordinates are the degrees themselves, so simplifying on its page and simplifying in degree space are one operation rather than two.

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

How Equirectangular distorts, by latitude. Angular deformation and areal scale plotted against latitude along the meridian at 0°. On Equirectangular the angular deformation reaches 114.2° and the areal factor reaches 11.5.

The plate carrée, the projection nobody chooses

Plotting latitude against longitude on ordinary axes is a projection. It preserves nothing, its angular deformation reaches 108° and its areal error eightfold, and it is probably the most widely produced map in the world because it is what happens when nobody decides anything.

wrong · Audit Two readings of the same triangle, and the band between them. The same three vertices, joined two ways: with straight lines in the plane the coordinates are stored in, and along the ground. Every dot is a point the two readings disagree about — inside on one and outside on the other. The band covers 29.5 per cent of the polygon, which is 4241 thousand square kilometres, and it is not an error in either reading: the file does not say which one it means. Densifying the stored boundary removes it, which is the only fix there is. Drawn in Equirectangular.

Inside is a claim about the edges

Whether a point is inside a polygon is not a property of the point and the polygon. It is a property of the plane the edges were understood to be straight in, and between two readings of the same file there is a band of disagreement — 29 per cent of one triangle's area, 4.2 million square kilometres, and the file does not say which reading it means.

applied · Dataset 10,000 observations of one place, averaged 120 times. Left: 400 single observations of the same place, with a 60-kilometre standard deviation east and north, drawn across ±400 km. They are scattered about the truth and their average is unbiased on the ground. Right: 120 independent averages of 10,000 such observations each, computed on the page and taken back to the ground, drawn across ±1.8 km. The cloud is tight, as averaging ten thousand things should make it, and it is not centred on the cross: it sits 489 m away, against 403 m predicted by the projection's second derivative alone.

The average of noisy positions moves

Average sixty thousand scattered observations of one place on a Mercator map and the answer is 404 metres too far north — at every sample size, because it is a bias and not noise. The same average on the Lambert cylindrical equal-area is 404 metres too far south, the two being ½ (σ²/R) tan φ and its exact negative, and on the plate carrée it is not displaced at all.

distortion · Precision

Named alongside it

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

Quadratic lawBiasBoundaryCentroidConservationCoordinate semanticsCovarianceData visualisationDefaultDensificationEqual-areaEquirectangular

All concepts