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

Distortion distribution — where it appears

How a projection's error is spread over a region rather than what its worst value is. Two maps with the same mean can be uniformly mediocre or excellent everywhere except one corner, and a summary that reports only a mean cannot tell them apart.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Angular deformation around three standard parallels. Three equal-area cylindrical projections differing only in where they are exact — standard parallels at equator, 30°, 45°. Each has zero angular deformation at its own standard parallel and grows away from it in both directions. Choosing a standard parallel is choosing which latitudes to treat well, and there is no choice that treats them all well.

What a standard parallel buys

A standard parallel is a line where the projection is exact. Choosing one does not reduce the distortion — it decides where the distortion is zero and lets everything grow away from it.

families · Families London to Tokyo, seen four ways. The same two routes on four projections. The gnomonic projection renders every great circle as an exactly straight line, which is what it is for; Mercator renders every rhumb line straight instead. Neither path changed — only the map did.

The aspect is a free choice

A projection's distortion pattern is fixed relative to its own axis, and where that axis points is entirely up to the cartographer. Rotating it is the cheapest available improvement and it is the one most often left unmade.

families · Families 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

Named alongside it

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

AspectMercatorNational GridZoneAggregationAiry's criterionArea weightingAzimuthalCylinderDistortion criterionEstimatorExponent

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