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

Winkel tripel — where it appears

The arithmetic mean of the equirectangular and Aitoff projections, and the compromise most atlas world maps of the last century used. It is a compromise projection with no exact property, and it is what most atlas world maps of the last century were drawn in.

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

6 projections of the same sphere. The same graticule under Robinson, Winkel tripel, Mollweide, Mercator, Gall–Peters, Eckert IV. Every one of them is a faithful drawing of the same object and no two agree, because each has chosen a different thing to preserve.

Compromise projections

A projection that preserves nothing exactly can distort everything less than one that preserves something exactly. For a general-purpose world map that is the right trade, and it is why the two most widely used ones today have no exact property at all.

choosing · Choosing 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 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 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 Four of the seven on the front can never be first. Every library projection over the whole sphere, with both errors normalised to the table's own range. The seven filled circles are the Pareto front — nothing beats them on both counts. The line through three of them is the lower convex hull, and a weighted sum of the two errors is a straight line in this space, so only a hull vertex can ever come first. Four projections — Web Mercator, Miller cylindrical, Equirectangular, Winkel tripel — sit in the dents, undominated and unchoosable.

Which projection a weighting can make best

Rung eight finds the seven world projections nothing beats on both counts and tells a reader with a preference that one of them is their answer. Four of the seven are not: they sit in dents of the front, undominated and unreachable, and no weighting of angle against area can ever put them first. Robinson can be first, on 2.8 per cent of the weight range.

wrong · Audit 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.

Compromise projectionGeneral-purpose mapOptimisationAngular deformationRobinsonTrade-offCompromiseDistortion criterionFlexionJacobianObjectivePareto front

All concepts