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

Area weighting — where it appears

Weighting each region in a statistic by how much ground it covers, rather than counting each one once. It is the right correction for regions of unequal size, and it inherits a projection whenever the area used is the one the stored geometry has.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

One quantity, one region, and the exponent left free. The scale departure of six projections over the world, aggregated as a p-norm, against p on a logarithmic axis. At p = 1 the best is Eckert IV; at p = 64 it is Winkel tripel. Nothing about the maps changed between the two ends of the axis — only how much of the region a bad point is allowed to spoil. Drawn in no projection: the axes are an exponent and a score.

The average was a choice of norm

Ten projections, one region, one measured quantity, and the only free decision left is how to turn a field into a number. Over the world's scale departure the ordering at the mean and the ordering at the worst case have a rank correlation of −0.04, all ten maps change position, and the exponent that produced each answer is stated nowhere.

wrong · Audit 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 The same totals, drawn as symbols, on two pages. A band field, largest in the mid latitudes carried by each region and drawn as a circle whose area is proportional to the total. Every symbol is right: a total is a total wherever it is drawn, and the two maps carry identical numbers. What differs is the ground under each symbol. On Mercator the symbol at 60° sits on a region drawn 4.0 times larger than it would be on an equal-area sheet, so the density a reader forms — symbol against region — is out by that factor.

A symbol has a size on the page and an area on the ground

A proportional symbol is right as a total wherever it is drawn — a count is a count. Read against the region beneath it, which is how a reader forms a density, it is wrong by exactly the reciprocal of the areal factor: 0.083 at 73° north on Mercator, a factor of twelve, with the correction available as one multiplication that no atlas makes.

distortion · Thematic The same count, scattered two ways. 22 dots in every cell of a 30-cell covering, drawn on Mercator. The upper panel places them uniformly on the GROUND — uniform in longitude and in the sine of latitude, which is what uniform on a sphere means — and the lower places them uniformly on the PAGE, which is what a drawing routine handed a polygon does. Both panels carry exactly the same number of dots in exactly the same regions, so both are honest as totals. They are different pictures, and a reader reads a dot map by density.

A dot map's density is partly the projection's

A dot map carries the right number of dots in every region whichever way it is drawn, so it is honest as a total under both placements. It cannot be honest as a density under both: ground on a uniform field reads 0.099 of its equatorial density at 72° north on Mercator, and scattering inside the polygon on the page moves 64.3 per cent of a cell's dots into its northern half without one of them leaving the cell.

distortion · Thematic One field, one classifier, two sets of breaks. A field that varies with latitude put into 5 classes by area-weighted quantiles, on Mercator. The upper panel weights each region by its ground area and the lower by the area it occupies on this page, which is what a classifier handed projected geometry does. 60 of 150 regions land in a different class, marked in the lower panel. The data has not changed and neither has the number of classes.

The class breaks were computed on the page

The three rungs below price what a reader does with a finished map. A classifier is software, it runs on the geometry it has, and the geometry it has is projected: a five-class quantile classification of one stated field puts half of the three hundred and eighty-four regions in a different colour on Mercator, and 87.5 per cent of them at nine classes.

distortion · Thematic One map, two samples, two answers. The same Miller cylindrical projection carrying two sets of sample points of the same size. On the left the points are uniform over the SPHERE — equal ground area between them, which is what every mean in this collection integrates against. On the right they are uniform over the PAGE, which is what a raster, a pixel loop or any figure that walks its own canvas produces. The right-hand set crowds where the map stretches, and the mean angular deformation it returns is 18.0° against the left-hand set's 7.2°.

The sample was drawn on the page

Every mean in this collection integrates over the sphere, because that is where the ground is. A raster, a pixel loop and any figure that walks its own canvas integrate over the page instead, and the difference is exactly the covariance between the quantity being measured and the map's own area distortion — 7.2° of mean angular deformation on Miller becoming 18.0°.

distortion · Sampling

Named alongside it

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

Areal factorEqual-areaBiasMercatorThematic mappingDensityQuadratureAggregationChoroplethCovarianceEstimatorObjective function

All concepts