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

Airy's criterion — where it appears

A measure of a projection's distortion over a region, formed from the squared logarithms of the two principal scale factors and averaged over the area. It is one of several criteria in use, and it ranks the same projections over the same region differently from the others.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

Angular deformation against latitude, four projections. The same quantity for mercator, gallPeters, winkelTripel, robinson, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both.

Every projection minimises something

A projection is the solution to an optimisation problem, and naming the objective explains more than naming the family. Some objectives are exact constraints, some are least-squares fits, and one is a table of numbers a man adjusted until it looked right.

choosing · Choosing The same six projections, ranked over Europe and Chile. Each column orders the projections by Kavrayskiy's regional criterion — the root-mean-square departure of the two principal scales from unity, integrated over the region with the area element. The lines cross, which is the point: Lambert conformal conic leads over Europe and comes fifth over Chile. A table of projections ordered by distortion is a table about somebody's region.

Distortion over a region

An indicatrix describes a point and every practical question is about a country. Going from one to the other means integrating, integrating means choosing a weighting, and the weighting is the step that turns a measurement into somebody's opinion.

distortion · Tissot Every mixture of Equirectangular at 50.46° and Aitoff. Airy's criterion over the whole sphere, for every weighted average of the Winkel tripel's own pair. Each mixture is normalised to its own best constant scale first, so the comparison is about shape rather than size. The curve has its minimum at w = 0.45, where the criterion is 0.3037 against 0.4114 and 0.4290 at the two ends. That is a projection nobody constructed beating both projections somebody did, by 26 per cent.

The average of two projections

The Winkel tripel is literally the arithmetic mean of two other projections, and this site's implementation of it agrees with that mean to the last bit. Averaging beats both ingredients by 26 per cent — and it preserves conformality exactly, destroys equal-area completely, and can turn eighteen per cent of the world inside out without either distortion measure saying so.

choosing · Choosing Four averages of one region's indicatrices — Robinson over the whole sphere. The faint ellipses are the indicatrix at ninety sampled places; the four drawn over them are four candidate averages of exactly that set. The arithmetic mean of the tensors gives semi-axes 1.1621 and 1.0182; the log-Euclidean mean gives 1.0053 and 0.9608; the Karcher mean gives 1.0042 and 0.9619; and taking the mean of a and the mean of b separately — which is what a published average distortion usually is — gives 1.2335 and 0.8200. Their maximum angular deformations are 7.57°, 2.60°, 2.47°, 23.23°, against a mean of the pointwise deformations of 20.67°. Five numbers, one set of ellipses.

An average of ellipses is not an ellipse

Rung three integrates distortion over a region and finds the weighting is somebody's opinion. It never asked what was being averaged. An indicatrix is a positive-definite matrix, matrices form a cone rather than a vector space, and the arithmetic average of an equal-area map's indicatrices comes back inflating area by up to 11 per cent.

distortion · Tissot 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.

CompromiseConformalityKavrayskiy's criterionObjective functionDistortion criterionEqual-areaMercatorOptimisationRankingRegional distortionVerificationAggregation

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