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

Centroid — where it appears

The centre of area of a shape, computed by a formula that belongs to a plane — so the answer depends on which plane the coordinates were in. The gap between the answers computed in two different projections is a real distance on the ground, and it grows with the region rather than with the coordinates.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

A feature across nine tiles at zoom 4, and the nine labels it gets. A rectangle from -10° to 30° east and 36° to 62° north, drawn in the projected plane with the tile boundaries over it. A renderer that draws each tile from the geometry inside that tile puts the label at the centroid of the piece, marked hollow; the whole feature's centroid is the filled mark. The furthest piece's label is 1863 kilometres from it. The pieces' areas sum to the whole to 2.2e-16 of a relative part, so the clipping is exact and the displacement is the operation rather than an error in it.

A tile is drawn without its neighbours

Independence is what makes the scheme scale: one request touches one square of the world. It also means a feature crossing nine tiles is nine features, each labelled at its own centroid — up to 1,863 kilometres from the whole one — and the displacement is bounded by the size of the feature rather than the size of a tile, so it grows as the reader zooms out.

applied · Screen The image of a circle on Mercator, against its own indicatrix. A circle of three radii on the ground at 30°, 40°, projected exactly — the solid curve — against the ellipse the indicatrix predicts for it, dashed. The filled dot is the image of the circle's centre and the hollow one is the centre of area of what was actually drawn, which is not the same point. The departure runs from 3.14 per cent at 4° to 15.80 per cent at 16°, so it grows in proportion to the radius rather than to its square: halving the circle halves the relative error and does not quarter it.

The indicatrix is a limit

Tissot's ellipse describes an infinitesimal circle, and every published one is drawn finite. The error of the description falls as the radius rather than as its square — halving the circle halves the lie — and on the Robinson projection at a table entry it does not fall at all: sixty metres and six kilometres are both 1.17 per cent wrong.

distortion · Tissot Where the middle of a 20° × 20° region is, in five planes. The region is drawn in longitude and latitude — which is itself a projection, and one of the ones being compared. Each filled mark is the shoelace centroid computed in one projected plane and inverted back to the ground; the hollow mark is the centre of area on the sphere, by integration. They spread over 273 kilometres. The equal-area member is 66 kilometres out, because a centroid is a first moment and preserving area says nothing about where the area sits.

A centroid belongs to a plane

Every renderer labels a region at its centroid, and every centroid is a shoelace over coordinates as stored — which is a statement about the plane they are in. Six planes put the middle of one 20° × 20° region up to 273 kilometres apart, the equal-area member is 66 kilometres out, and the disagreement falls as the square of the region's size.

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 lawToleranceVerificationEqual-areaLocalityNumerical differentiationThematic mappingAreal scaleBiasBoundaryClippingConvention

All concepts