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

Distortion criterion — where it appears

A rule for reducing a projection's distortion over a region to one number, of which several are in use and which do not agree about rankings. Four of them are computed here, and they rank the same projections over the same region differently, which is why a ranking has to name its criterion.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

The taught rule against the measurement, on 30 regions. Each cell is a region built to order — a box of the stated height and width-to-height ratio, centred at the stated latitude — with the family that actually scores best over it, and whether that is what the rule says. Every family is given its own parameters for the region: the conic its cone constant, the cylindrical its standard parallel and the choice of a normal or transverse axis, the azimuthal its centre. The rule is right on 19 of 30, and where it is wrong it is wrong in one direction: it keys on latitude, and what decides the answer is shape.

The rule of thumb, scored

Cylindrical near the equator, conic in the middle latitudes, azimuthal at the poles. It is the most repeated piece of practical advice in cartography and it has never been run against a population of regions. Run against thirty, it is right nineteen times, and every one of its failures has the same shape.

wrong · Audit The same map on four differently prepared pages. Mollweide at 30°E 20°N, then the same projection with a rotation and a magnification applied to the page, then with one axis stretched by 1.6, then with a shear of 0.5. The similarity changes nothing: flexion, skewness, ω and the anisotropy are identical to every printed figure, and only the last column — the same turning measured per unit of page arc rather than per unit of ground arc — moves, by exactly the magnification. The stretch and the shear change all of them, and flexion by 35 per cent.

The second derivative is not an invariant

The first-order ladder established which quantities survive a change of coordinates and which are artefacts of the parameterisation. Asked of the second order, the answer is that flexion survives a rotation and a magnification of the page exactly, and survives nothing else: a stretch of 1.6 in one axis moves it by eleven per cent, a shear of 0.5 by thirty-five, and a shear of 3 leaves a ranking of eight world maps with a rank correlation of −0.07 to the one it started with.

distortion · Flexion Where the projection's pole should go, and the answer the third rotation moves it to. Every dot is a pole position the search tried, sized by the best score it can reach there when the third rotation is also free — the score is the distortion of Robinson over Europe by Kavrayskiy's criterion, so smaller is better and the large green dots are the good regions. The circled mark is the two-parameter optimum and the square is the three-parameter one: they are 74 pixels apart on this map, which is a different aspect rather than a refinement of the same one. The search costs 8 times the evaluations of the two-parameter one. Drawn in Mollweide.

The third parameter, run

An aspect has three numbers and this site has been searching two of them, with a note admitting it. Searching all three is worth up to 2.1 times — and the obvious way to do it, starting from the two-parameter answer and letting the third move, finds a fraction of that or nothing at all.

choosing · Choosing 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 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 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

Named alongside it

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

Kavrayskiy's criterionOptimisationRankingAspectCompromise projectionInvariantProjection familyPurposeAiry's criterionAnisotropyConicRegion

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