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

Inverse — where it appears

The map from page coordinates back to the body, which most projections here have only numerically. Newton's method supplies it, and on a projection with more than one branch the starting guess decides which branch is found rather than whether one is.

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

What the drawn line costs, projection by projection. The ground length of the page-straight route from London to Tokyo, as a percentage above the shortest route. Two of these have names. On the gnomonic, centred on the route, the drawn line IS the shortest route, at -0.0000 per cent. On Mercator it is the rhumb, matching the rhumb's own length to 1 parts per million — which is why that projection exists. On the other eight it is a curve with no name and a cost between 0.0 and 19.6 per cent.

The line drawn straight on the page is a route

Eleven essays draw the route on the map. Nobody has drawn the map's own proposal: the ground curve somebody follows by laying a ruler on the page. It has a name on exactly two projections and is 18.2 per cent long on the one where it is famous.

paths · Paths Going out and coming back, at three orders. A point three degrees from the central meridian, taken forward into the projection and back again with the series truncated at the same order both ways, and the ground distance between where it started and where it returned. The second-order pair is out by a tenth of a metre at some latitudes; the third by half a millimetre; the fourth — the order every national grid formula in ordinary use is written to — by 0.39 mm. The dips are where one of the two series passes through a node, not where the map is better.

The inverse of the series is not the series of the inverse

Transverse Mercator on an ellipsoid has no closed form, so every national grid computes it as a truncated series. There are two of them — one out and one back — and they are separate approximations. Composing them does not give the identity: at the order every grid formula in ordinary use is written to, a point three degrees from the central meridian comes back 0.39 mm north of where it started.

families · Ellipsoid Every projection against a picture that is not a map. The worst relative distance error over London, New York, Tokyo, Sydney, for each projection in the library and for the best flat arrangement of the same distances. Each azimuthal member is centred on the set's own centroid, which is the fair comparison. The free picture reaches 0.63% and the best projection, Azimuthal equidistant, reaches 8.28% — a ratio of 13.17. The free picture cannot lose, because every projection's own layout was handed to the search as a starting point; what the figure measures is how much the freedom is worth, and it is worth different amounts at different sizes.

The best flat picture is not a map

A set of dots whose separations are as nearly right as separations can be made beats every projection in this collection — by a factor of thirteen on four world cities, and by eight per cent on sixteen. The collapse between those two numbers is not about cartography. It is that a picture of n places has 2n − 3 free numbers and n(n − 1)/2 distances to spend them on.

impossibility · Embedding The density every real cartogram is handed. Two islands in an ocean that is not lightly populated but empty: the density is exactly zero over 83.7 per cent of the sphere. Four rungs of this anchor assume a positive density, because the construction divides by it — and the conditional cumulative of a column with no mass in it is zero over zero. Drawn on the equal-area base, so a cell's ink is a density and its area is ground.

A density that asks for no room at all

Five rungs assume the density is positive everywhere, because the construction divides by it. Every cartogram anybody draws has an ocean, and an ocean is not sparsely populated but empty — 83.7 per cent of the sphere, exactly zero, and the construction returns nothing at all for a third of the probes.

distortion · Cartogram

Named alongside it

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

AzimuthalPurposeAngular deformationAreal factorAuxiliary latitudeBoundary conditionCartogramChartConformalityConvergenceDegeneracyDegrees of freedom

All concepts