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

Compromise projection — where it appears

A projection with no exact property, built to distribute the unavoidable failure rather than to eliminate any part of it. It is the honest response to the impossibility rather than an evasion of it, and it is what an atlas world map has almost always been.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

Three distances, two degrees of freedom. The Chamberlin trimetric construction with its three centres marked. A point is placed by intersecting circles of its true distance from each centre — but three circles in a plane do not meet in a point, and the three candidate positions obtained by taking the conditions two at a time are 17.0 km apart at the sample point. They are drawn here magnified 120 times about their own centroid, which is where Chamberlin's rule puts the point. The residual is not an error in the arithmetic; it is the amount by which the sphere refuses to be a plane, and it grows as the cube of the size of the region.

Three conditions are one too many

Two distances fix a point in a plane and a third has no freedom left to be satisfied with. Chamberlin's trimetric construction averages the three positions that satisfy two conditions each, and the spread between them — never zero anywhere, 22 km over North America, growing as the cube of the region — is the price of the extra clause.

choosing · Condition Eight world maps ranked by their second derivative. Each projection's root-mean-square flexion and skewness over the whole sphere, combined and ranked. The first-order ranking of the same eight by Kavrayskiy's criterion is given beside each bar; Spearman's rank correlation between the two orderings is 0.69. The two criteria are measuring different derivatives of the same maps and there is no reason for them to agree.

The second derivative has its own ranking

Rank eight world maps by how much they stretch and Mercator comes sixth of eight. Rank the same eight by how much they bend and it comes third. The two halves of the second-order score disagree with each other more sharply than either disagrees with the first-order one — Spearman 0.45 against 0.69.

distortion · Flexion Every library projection over Europe, on the two axes it can be wrong on. Each dot is one projection, scored over Europe on the two independent failures: how much it turns angles and how much it changes areas. The lower-left corner is the isometry that does not exist. The line joins the five projections nothing beats on both counts — the rest are inside it, and a reader who prefers either failure to the other should still not choose one of them, whatever weighting they hold.

The projections that are beaten on both counts

Two rungs of this ladder scored a rule of thumb over thirty regions and then forty-five. The same populations answer a harder question the ladder has never put: which library members are never the right answer at all. Two are beaten outright on both criteria everywhere, one is on no regional front in any population — and it is on the world's.

wrong · Audit Three implementations of one projection, and three different distortions. The spread in angular deformation between linear, Catmull–Rom and Aitken interpolations of Robinson's own table, at 60° of longitude. It is exactly zero at every fifth degree, because all three pass through the tabulated rows, and it is not zero anywhere else: 37 of 87 latitudes differ by more than half a degree and the worst is 8.42°. The graticules the three draw differ by 4.2 parts per thousand of the map's own span, which is under the width of a printed line.

A projection defined by a table has an interpolation in it

One member of this library has no formula: Robinson set nineteen pairs of numbers by eye and the table is the definition. Three published interpolations of it draw graticules that differ by four parts in a thousand of the map's span and report angular deformations that differ by 8.4 degrees.

families · Families Two members, their average, and the family's best answer to it. Two equal-area conics at cone constants 0.25 and 0.85, the average of the two, and the member of the family nearest that average — at 0.540, which is not the parameter midpoint. The average is not a conic at all: its parallels are still arcs but they are arcs of circles about different centres, so no single cone constant reproduces it and the residual is 0.1722.

A family is not closed under averaging

Nine rungs treat a family as a set of maps with a parameter running through it, and this collection's own compromise projections are averages of members. An average of two members is not a member: two conics far apart average to something 31 per cent of their own separation outside the family — and averaging within a family makes the map worse every time, while averaging across two makes it 16 per cent better.

families · Families The test the ladder asked for, and it refutes the conjecture. How much better the best asymmetric projection is than the best symmetric one, under weightings of four different symmetries, with every symmetric map allowed to re-aim its axis at twelve candidate poles. The conjecture rung eight recorded was that the seven earn their place by PLACING distortion where a symmetric map cannot, so their advantage should collapse under a criterion with no place preference. It does the opposite: the advantage is largest at 1.343 under the uniform weighting and smallest at 1.144 under a band, with the fully asymmetric concentration at 1.204 in between. The winner is named on each row and the map it beat is Equirectangular throughout. The seven are simply better maps.

The maps with no family are simply better

Rung eight found seven projections with no continuous symmetry and noticed they are almost exactly the set anybody would choose for a world map, then offered a conjecture with a test attached: their advantage should collapse under a criterion that does not care where anything is. Run, it does the opposite — 1.343 times under a uniform weighting and 1.204 under a concentration. The conjecture is refuted.

families · Families Two compromises along one path, and they do not agree. The angular deformation and the flexion of every blend between Mercator and Lambert cylindrical, each divided by the straight line between the two parents' own values. A value of one means the blend is exactly the average of the two errors and has bought nothing. The angular deformation dips to 0.649 at a weight of 0.13; the flexion dips only to 0.955, and it does so at 0.50. Anybody choosing a compromise is choosing a weight, and the two orders want different ones.

The best compromise for angle is not the best for bending

Every compromise projection in the library is an average of two others, and averaging is a first-order operation — so the second derivative was never part of the bargain. Swept along five ordinary blend paths, the weight that minimises angular deformation and the weight that minimises flexion are between a quarter and a half of the axis apart, and how much a compromise buys at one order predicts nothing about the other.

distortion · Flexion

Named alongside it

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

Winkel tripelDistortion criterionOptimisationTrade-offAngular deformationFlexionGeneral-purpose mapInterpolationObjectiveProjection familyPurposeRanking

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