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

Weighting — where it appears

The relative importance a reader gives to two criteria that cannot both be optimised. It is a number nobody knows about themselves to three decimal places, and over a library of projections it decides the answer inside windows that narrow.

Named by 15 essays across 7 fields — each of them below, with the objects they name alongside it.

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 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

The seven world projections nothing beats on both counts tell 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 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

Every criterion in this subject optimises 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 The test the conjecture 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 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

Seven projections have no continuous symmetry, and they are almost exactly the set anybody would choose for a world map. That coincidence came with a conjecture and 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 Pins at one place move where the picture sits; pins at four reshape it. The best flat picture of four cities, fitted to every distance with a tolerance of 1 per cent, one bearing a pair to 0.01 radians, and pins to the sheet good to 50 km. Each ellipse is the error in a city's position relative to the picture's centre — its shape, which no shift changes — drawn 30 times its size. Solid: the part the distances and bearings put there. Dashed: the part the pins put there. Pinned at London alone, the pins put 0.00 km into the shape and the places 63.0 km. Pinned at all four, the pins put 30.5 km and the places 29.7 km, and the shape as a whole is 42.6 km against 63.0.

A second pin is a measurement of the places

Counted exactly, a picture of places is a direct sum: the distances and bearings spend its shape and two pins spend where it sits on the sheet. Given tolerances, the sum survives in one form and fails in another. Pins at one place put none of their error into the shape at any tolerance, and wherever the pins are, their average takes none of the distances' error. But every place pinned after the first is a statement about the geography, and held exact it makes the shape worse once its error passes about two thirds of the shape's own.

impossibility · Embedding Weighting the low sight turns the region the fix is in. Four bodies, three of them between 44° and 58° up on one side of the ship and one 5° up behind, with each sight's error made of a sextant part of 0.5′ and a tenth of its refraction correction: 0.94, 0.95, 0.93 km for the high bodies and 2.05 km for the low one. Solid: the 95 per cent ellipse of the fit that weights each sight by its variance, 2.30 by 1.55 km. Dashed: the ellipse a navigator draws who gives every sight the same variance, 2.63 by 2.02 km, its long axis turned 20.1° from the weighted one. Dotted: where that unweighted fix actually scatters, 2.98 by 1.57 km, turned 27.7° from the ellipse drawn for it. All three are centred on one point.

A low sight is worth keeping only if it is weighted

A body five degrees above the horizon is seen through so much air that its refraction correction is ten times a high body's, and a tenth of that correction uncertain doubles the sight's error. Given the same weight as the others, a sight below seven degrees makes the fix worse than throwing it away; weighted by its variance it never does. The ellipse drawn without weights is turned twenty degrees from the right one, and a navigator who must guess the weights loses far less by guessing high.

paths · Position fix Five readerships, four winners, one library. The best projection under each stated readership, with the share of the sphere carrying half that readership's weight. The criterion is Kavrayskiy's throughout, every map is in its normal aspect, and the library is the same twenty. Four different projections win. The flat weighting at the top is the one every earlier comparison used, and naming it as a readership rather than as the absence of one is the whole of what this changes: it is the assumption that everybody who will ever read the map is spread evenly over the globe.

The area weighting was a readership all along

Twelve measurements have taken a projection comparison apart and found the comparer's own freedom in every part of it. All twelve rested on one thing none of them examined: that a square kilometre of empty ocean counts exactly as much as a square kilometre of city. That is not the absence of an assumption about the reader — it is the assumption that everybody who will ever look at the map is spread evenly over the globe. Named as one readership among five, it produces a different winner from three of the other four, and against a North Atlantic readership its choice is fifth of twenty and 33 per cent worse.

wrong · Audit Weighted by their residuals, six sights hand the low one six times its share. Six bodies seen from one ship: five between 44° and 61° up between east-north-east and east-south-east, and one 5° up due south, alone in its direction. Along each bearing, two bars: the share of the fix's weight the sight's altitude earns it, and the median share it ends with when every sight is weighted by its own residual and the fit is repeated until the weights settle. The low sight earns 4.0% and ends with 25.0%; the five good ones earn about 19% each and end with 1.8% to 3.1%. Bar length goes as the square root of the share.

Weighted by what it disagrees with, the fix trusts the sight it leans on

With six sights and two unknowns, the residuals seem to offer a way to find the bad sight without knowing the weather: weight each line by how little it disagrees, and solve again. One step of that gives a low body almost exactly the weight its altitude would, 4.5 per cent against 4.0. Every further step walks it away. Iterated until the weights settle, the fit is worse than weighting nothing in all four arrangements tried. Where the low body stands alone in its direction it ends with six times the weight it earns, because the fix leans on its line and its residual shows only 46 per cent of its error.

paths · Position fix A week of twilights learns the refraction share, and one twilight learns almost nothing. The share of the refraction correction that is uncertain, estimated from the residuals of equal-weight fixes pooled over one to fifty-six twilights, each residual corrected exactly for its own fix's geometry; the truth is a tenth. Shaded: the middle eighty per cent of 1,500 seeded weeks for a navigator who always takes one body between 3° and 8°, with its median solid. Dashed: the median for one who takes six bodies anywhere between 4° and 70°. After one twilight the first navigator's middle eighty per cent runs from 0.000 to 0.181; after fourteen, from 0.055 to 0.131; after fifty-six, 0.079 to 0.116. The medians climb to the truth from below: 0.049 and 0.020 after one twilight, 0.092 and 0.088 after fourteen.

A week of twilights learns what a sight is worth

One fix cannot tell how good its low sight was, but every residual of every fix is a point on one straight line whose slope is the square of the refraction share, once each is corrected exactly for its own fix's geometry. After one twilight the estimate is zero four times in ten; after a week it is within half again of the truth nearly four times in five. And the lesson is worth little: a guess within a factor of two already costs under 4 per cent of the fix, a week's estimate 1.3, and a share that changes from night to night by half again costs more than the learning buys.

paths · Position fix Where on the land the homolosine beats the sinusoidal, read from Goode's own meridians. The land of the world outside Antarctica on a plain latitude–longitude grid, in two-degree cells, each shaded by how much less angular deformation Goode's homolosine gives it than the plain sinusoidal does, both cut into Goode's lobes and each point read from its own lobe's central meridian (dashed); solid lines are the cuts, and the dotted parallels are Goode's join at 40.7°. The key names the two inks: one where the homolosine wins, the other where the sinusoidal does, deeper for a larger difference. Weighted by land, the homolosine averages 16.51° and the sinusoidal 17.34°. Antarctica, drawn pale, is not in this score.

On the land it was drawn for, the homolosine wins by its meridians

Scored over the whole sphere and cut into Goode's lobes, the homolosine loses to the plain sinusoidal; for it to win on latitude alone, a reader would have to care about the band from 32° to 61° forty per cent more than its area. The land does not do that — the band holds 1.03 times its share of the continents. Weighted by land anyway, and each point read from the meridian of the lobe it lies in, the homolosine wins: 16.51° against Craster's 17.00 and the sinusoidal's 17.34. The win is in where Goode put his meridians, which lift his map from third to first; count Antarctica as land and it falls to fourth.

families · Families Told the control has an error, the test must see a larger move before it names a station. The local cluster at the centre and its 5 control stations ten kilometres away, drawn on their true bearings with the distance shortened. The shaded ellipse at each station is its published error, 53 by 31 mm on average. Outward along each tie, at the same scale, the smallest move of that station its test finds four times in five at a size of 0.001: the thick bar for a test that carries the published error, 74, 79, 78, 78, 78 mm; the thin dashed bar for a test that holds the control exact, 48, 48, 49, 48, 48 mm.

Honest about its control, the test needs eight centimetres where it needed five

A test that holds the control exact reads the control's published error as movement, and at five stations flags a good survey sixteen times in a hundred. Carry the published covariance into the adjustment and the false alarms fall to half a per cent — and the smallest move the test can find grows from 48 millimetres to 77. A five-centimetre move is then named 29 times in a hundred instead of 63, the naming rate of perfect control takes twenty-four stations, and above ten centimetres the two tests agree. What the honest test buys is a flag that means what it says.

practice · Reduction The land wants the join at fifty-two degrees; Goode put it at forty-one. The homolosine rejoined at every latitude from 20° to the pole, the Mollweide cap rescaled so each parallel stays continuous, each composite cut into Goode's lobes at its own best proportions. Solid: mean angular deformation over the land outside Antarctica. It falls from 17.33° at 20° to its least, 16.29°, at a join of 51.7°, and rises again to the sinusoidal's 17.34° at the pole; Goode's join at 40.7° scores 16.51°, and every join sampled from 50° to 60° is within a tenth of a degree of the best. Dashed: with Antarctica counted as land, and dotted: over the whole sphere by area — both fall all the way to the pole, where there is no cap at all.

The land puts the homolosine's join eleven degrees higher

Goode joined the sinusoidal to the Mollweide at 40° 44′ because that is where their parallels have the same length — a fact of geometry, not of any objective. On the land his map was drawn for, read from his own lobes, the join has an objective at last, and its best value is 51.7°: a composite joined there gains a fifth of a degree over Goode's, and every join from about 46° to 60° does nearly as well. A smooth curve fitted to the land with no join in it at all stays within three per cent of flat to 50° and then closes, gaining twice as much again. Goode found the right shape and put its knee too low; with Antarctica counted, the land wants no knee at all.

families · Families Tonight's residuals cost the fix a steady three per cent; the typical share costs more every time the nights differ more. Six bodies a twilight, 3 of them between 3° and 8°, and tonight's refraction share the typical tenth times a factor drawn afresh each night; the horizontal axis is that factor at one standard deviation. The fix error, root-mean-square over 40,000 seeded twilights, as a percentage above that of a navigator who knows tonight's share. Dashed: weighting with the typical share, 0.8 per cent at a factor of one and a quarter, 10.3 per cent at two and 36.2 per cent at three, off the top of the chart. Solid: weighting with tonight's share estimated from tonight's own residuals, 3.6 per cent, 3.2 per cent and 1.9 per cent. Dotted: an even blend of the two, 1.2 per cent, 2.3 per cent and 2.0 per cent. Tonight's estimate beats the typical share from a factor of about 1.56, and the blend from about 1.33.

Tonight's residuals measure tonight's air badly and weight tonight's fix well

A week of twilights learns the typical refraction share and nothing about tonight's, and a share that changes from night to night costs more than the week buys. With the sextant's error taken from the week, one twilight's residuals estimate tonight's share alone — badly: within half again of the truth 43 times in a hundred with three bodies near the horizon, and zero 22 times. Used as a weight, that poor estimate costs the fix between two and four per cent whatever the night, so it beats the typical share once nights differ by more than half again, and a blend of the two, leaning on tonight as the nights grow more variable, beats both. A weather predictor has to explain most of the nightly change to do better.

paths · Position fix From one survey an optimistic report and a moved mark look alike. The scale on the regional network's published covariance, estimated from the residuals of one survey of five control stations, over 3,000 seeded surveys each: honest report, nothing moved, mean 0.92 and median 0.53; report twice too optimistic, mean 1.91 and median 1.43; honest report, one mark moved 5 cm, mean 4.84 and median 4.48. One survey's estimate has a standard deviation of 3.80, and a five-centimetre move along a tie adds between 3.7 and 4.2 to it: about one standard deviation, and four times what a report twice too optimistic adds.

One survey cannot tell an optimistic report from a moved mark

A local survey that carries its control's published covariance can estimate, from its own residuals, how far that covariance understates the truth. With five control stations the estimate has a standard deviation of 3.8 times the published scale, and a single monument moved five centimetres adds about as much again — four times what a report twice too optimistic would add. Run at the scale it estimated for itself, the honest test stops flooding good surveys with alarms, and names a moved mark seven times in a hundred instead of twenty-five: the move inflates the estimate and the estimate absorbs the move. The report can be checked only across surveys, and telling one twice too optimistic from an honest one takes about two hundred of them with five stations each, or fifty with twelve.

practice · Reduction Eight hundred surveys, each a poor estimate of the scale, find the patch where the published covariance is four times too optimistic. The middle one of nine agencies, by how much of the patch its map flags: a region covered by 800 local surveys at random, each tied to five control stations of a regional network whose published covariance is honest everywhere except a disc covering 15% of the region (dashed), where the control's true errors are 4 times its published variance; one survey in ten has a monument moved five centimetres. Filled dots: the 357 surveys whose own residuals put the scale above 2. Shaded: the 18 of 441 grid cells where the count of such surveys within 0.2 of the region's width exceeds what an honest region exceeds there one time in twenty, over the whole map; 18 of them lie in the disc.

An agency can map where its published covariance is optimistic

The local surveys tied to a regional network each estimate, badly, how far the network's published covariance understates the truth, and the agency that receives them can pool what they say. When the understatement is confined to one part of the region, pooling it into one number dilutes it: a patch covering 15 per cent of the region with errors four times their published size reads as 1.30 for the whole. Counted place by place, the same surveys find the patch — two agencies in three find it with 400 surveys and nearly every one with 800, flagging almost nothing outside it. Fed back as one number the scale removes little of the false alarms inside the patch and costs nearly a third of the test's power outside; fed back as a map it cuts the false alarms inside from 20 per cent to 3 and costs little outside.

practice · Reduction

Named alongside it

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

Least-squaresVerificationCovarianceEstimatorPurposeControl pointsNavigationNoiseOptimisationRankingStandard errorTrade-off

All concepts