Thematic mapping — where it appears
Named by 16 essays across 3 fields — each of them below, with the objects they name alongside it.
The projection that shows true size
There is no such thing, and the phrase hides a real question. Equal-area projections preserve area and destroy shape; nothing preserves size in the sense the phrase implies; and the tools that make the point best are not maps at all.
Equal-area on the wrong body
The site's headline is that Web Mercator puts geodetic latitudes into a spherical conformal formula and stops being conformal. The same sentence is true with "equal-area" in it and nobody says it: the areal factor is 1.00674 at the equator, 0.99332 at 88°, and averages to almost exactly one — so every check that adds up areas passes while every cell is wrong.
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.
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.
A slope is not a shape
Every map in this collection has carried geometry. An applied map far more often carries a field — elevation, pressure, a density — and the first thing anybody does with one is differentiate it. A gradient is a covector, it transforms by the inverse transpose of the Jacobian, and the ellipse that governs it is the indicatrix turned inside out.
The contour is right and the reading is wrong
There is exactly one thing about a field that no projection can get wrong: which points share a value. The contour lines on any two maps of the same field are the same set of points. Every quantity a reader takes off them — the spacing, the length, the area between two of them, the hypsometric curve — is not, and the two kinds of map get different ones wrong.
Water runs downhill on the ground, not on the page
A drainage network is the set of steepest-descent trajectories of a field, so it is built entirely out of directions. A conformal map preserves those directions exactly and therefore preserves the whole network; an equal-area map does not, and sends a route up to 892 kilometres away from where the water actually goes.
An edge has no order of convergence
On a smooth field the three resampling kernels converge at orders 1, 2 and 3 and the choice is obvious. Across a discontinuity they converge at 0.78, 0.58 and 0.60 — within a factor of 1.4 of each other, in an order that puts nearest-neighbour first, and a real raster is mostly edges.
The same data on two grids
Five essays have addressed, queried and ordered cells within one scheme and nobody has moved a number between two. Doing it exactly — area-weighted, both directions — preserves the total to 2 × 10⁻¹⁶ and loses 39 per cent of the field's own standard deviation in a single round trip; six round trips leave 23 per cent of its variance. The quantity that would reveal the damage is the one that never moves.
The attribute is a claim about the geometry
Fourteen essays price what a stored coordinate means and not one asks what the number stored beside it means. A rate is a quantity divided by an area, the area belongs to the geometry, and no format records which area — so a simplification that moves the outline by nothing visible moves the implied total by 1.55 per cent, an unweighted average of densities is 4.09 per cent out, and a choropleth gives a polar square kilometre fifteen times the ink of an equatorial one.
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.
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.
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.
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.
A cartogram keeps the shapes it inflates
Seven rungs build cartograms and none reads one back. Reading means recognising a region and dividing its drawn area by its base area, and the construction is against the reader twice: the correlation between how much a region grows and how much of its shape it keeps is −0.996, and the base area a reader has to divide by is the map the cartogram replaced.
A current drawn on a page has sources
Seven rungs project a scalar field and ask what the page does to its gradient. A wind or a current is the other half of what gets mapped, and the operator that matters for it is the divergence — which is preserved by an equal-area map exactly, by no other map at all, and by a conformal map least of anybody's expectation.
Named alongside it
The objects these essays reach for when they reach for this one.
Equal-areaAreal factorMercatorArea weightingBiasConformalityDensityGradientTest fieldVerificationAggregationAnisotropy