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

Kavrayskiy's criterion — where it appears

A criterion for a projection's distortion over a region, formed from the mean square of the logarithms of the principal scale factors. It is the criterion the aspect searches here minimise, and choosing a different one moves the optimum.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

Mercator swung through every tilt, over Chile and the tropics. The regional distortion of one projection as its axis is tilted from the normal aspect at 0° to the transverse at 90°, each curve divided by its own value in the normal aspect so the two regions can share an axis. Chile is best at a tilt of 90°, a factor of 183.5 better than north-up; the tropics is best at a tilt of 0°, which is north-up. Rotating the sphere costs nothing and changes none of the projection's own properties, which makes this the cheapest improvement available and the one most often left unmade.

Fitting the aspect to the region

Choosing a projection is a choice among a few dozen named things. Choosing its aspect is a choice among a continuum, it costs nothing, it changes none of the projection's own properties, and for a long thin country it is worth a factor of 183.

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 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 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 Which projection wins, by the first derivative and by the second. Each column is a region, with the projections listed in the order Kavrayskiy's first-order criterion puts them in and the figure at the right of each row giving that projection's rank under the second-order criterion — flexion and skewness aggregated the same way. The rank correlations are Europe 0.83, the conterminous United States 0.81, the tropics 0.90, a cap of 30° radius 0.88, so the two orders agree broadly and disagree in detail. Where it matters is the winner: over a cap of 30° radius the choice moves from Albers equal-area conic to Lambert azimuthal equal-area, while Europe and the conterminous United States and the tropics keep theirs. A criterion that changed every answer would be suspect and one that changed none would be decoration.

The second derivative over a region

Flexion has been measured at points and over the whole sphere, and never over a region — which is the only unit anybody chooses a projection for. Doing it finds that the second-order criterion moves the winner in one region of four, and that one projection in the library has no second derivative at all.

distortion · Flexion Two rules, three populations. The taught rule keys on latitude; the replacement keys on shape. On the thirty regions the replacement was read from it scores 83 per cent against the taught rule's 63. On forty-five different regions built the same way it scores 91 — higher, not lower, so the generalisation the shortfall doubted is real. On the seven named regions this collection actually uses, both rules score 29 per cent, and following either costs a mean factor of 14.6. The third bar of each group puts the taught rule's polar clause back into the shape rule, which is what the out-of-sample failures ask for: it repairs four of them, breaks two that were right, and reaches 43 of 45.

The rule scored out of sample

A replacement rule was read off thirty regions and scored on the same thirty, and this collection recorded that as not being evidence about any other thirty. It is: on forty-five different regions the rule scores 91 per cent against the 83 it managed at home. What it cannot do is the seven regions the collection actually uses, where both it and the rule it replaced name the winner twice out of seven and cost a mean factor of 14.6.

wrong · Audit Every aspect of Mercator over Japan, and the line the old sweep searched. The regional distortion of Mercator over Japan for every position of the projection's pole — darker is better — with the meridian the site's one-dimensional sweep searches drawn on it. The sweep's best is a gain of 24.4× over the normal aspect; the two-parameter search finds 61.4×, which is 2.51 times better again, at a pole 45° of longitude away from anything the sweep could reach. The shortfall recorded when the sweep was written said this would happen for a region whose long axis runs diagonally, and this is the measurement of it.

The aspect has three numbers, not one

The site's aspect search has swung the projection's axis through one plane for a long time, and recorded that a region whose long axis runs diagonally has its optimum somewhere that plane never reaches. Searching the whole sphere of pole positions finds 2.6 times more improvement over Japan and 3.4 over the conterminous United States.

choosing · Choosing 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 One slice of the aspect objective, at the best γ. The Kavrayskiy score for Robinson over Japan, as the pole is moved over the whole sphere with the third rotation held at the value the search settled on. Dark is good. The marks are local minima of the full three-dimensional grid that happen to lie in this slice: there are 6 of them here and 58 in the cube, and a search that walks downhill from a random start reaches the best of them 7 per cent of the time.

The landscape the search walks on

The three-parameter aspect search was run and its answer recorded with a note admitting nothing proved it global. Mapping the objective finds 26 to 34 local minima for every projection and region tried, a downhill walk from a random start reaching the best of them 6 to 35 per cent of the time — and one seed from the coarse grid the search already uses reaching it in all four cases. The score is reproducible to two per cent across a sevenfold refinement; the pole it names moves 60 degrees.

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 Three projections that run in a circle. Mercator beats Sinusoidal beats Eckert IV beats Mercator, each on a majority of the same seven criteria over the whole sphere. Every margin is four to three, the narrowest a majority of seven can be, and the criteria that decide each edge are different ones. There is no way to place these three in an order that agrees with all three comparisons, and the obstruction is not a measurement error: every number is exact to the precision the sampler reaches.

The ranking is not an order

The previous rung showed that a weighting can make almost any projection best. Remove the weights entirely, let each of the seven criteria vote once, and the answer is worse: over the whole sphere Mercator beats the sinusoidal, the sinusoidal beats Eckert IV, and Eckert IV beats Mercator — four such circles, every margin four to three, with a Condorcet winner sitting above them all.

wrong · Audit 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 Two regions, three answers. Britain and New Zealand, 166° apart, with the pole of the best oblique conic under each of three objectives. Pooling the samples and taking an area-weighted score puts the pole in one place; refusing to let either region be worse than the other puts it somewhere else. The regions are drawn on Mollweide so that equal ground areas are equal page areas.

The pooled score abandons a region

Fifteen rungs optimise for one region. An atlas is several, and pooling their samples into one area-weighted score is what everybody does — which on Britain and New Zealand serves Britain 1.2 times worse than it could be served alone and New Zealand 125 times worse. The worst-case objective makes them equal at 33 and 59, and the cost of sharing rises with separation from 1.4 to 59.

choosing · Choosing Which of this collection's numbers are the sampler's. How much each of two published quantities moves when the sampler behind it goes from twenty samples a side to sixty, for eight projections over the whole sphere. The Kavrayskiy numbers — the summary means the rankings are built from — move by at most 1.15 per cent. The worst-point angular deformations move by up to 19.0 per cent, all in the same direction, because they are maxima over a sample and a maximum over a sample is a lower bound. The split is clean, and it says which numbers here need refining and which do not.

Which of these numbers are the sampler's

Four rungs have shown that a sampled maximum understates, a sampled mean can be a report on the sampler, no arrangement of points is neutral, and refining until the answer settles proves nothing. So the collection re-measured itself. The means move by at most 1.1 per cent, the worst points by up to 19, and the rankings — which is what the essays actually argue with — do not move at all.

distortion · Sampling

Named alongside it

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

OptimisationRankingRegional distortionAspectPurposeRegionDistortion criterionObjective functionProjection selectionToleranceVerificationAggregation

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