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

The thread: The trade-off is forced — page 5

Conformal means the principal scales are equal; equal-area means their product is one. Both at once forces them both to one, which is an isometry, which cannot exist. The trade-off is two lines of algebra. Essays 97 to 120 of 146.
Forty-two starts, forty-two answers. Every local optimum coordinate ascent reaches from 42 starts on the same region and the same criterion — a 25° spherical square at a tolerance of ±0.5% — plotted at the share of area it keeps. They run from 36.98 per cent to 67.50, a spread of 30.5 points, and no two of them are the same map: clustered on their scale fields rather than their coefficients, 42 of the 42 are distinct. The two marked starts are the ones anybody would actually use — Chebyshev's map reaches 36.98 per cent and the mean-square map 64.05, and neither is the best this search found. What each projection optimises

A criterion worth using is one whose answer is not unique

Forty-two starts on one region and one tolerance give forty-two different maps, keeping from 36.98 per cent of the area to 67.50. The two starts anybody would actually use — Chebyshev's map and the mean-square map — end 27.1 points apart, which is more than the criterion buys over either of them. And the spread is widest at exactly the tolerance where the criterion is worth using at all, because both facts are the same fact: an objective with a flat top and a sharp edge has somewhere to hide answers.

 What a folded net is accurate to, beside what the map it carries is wrong by. For a cube globe of 100 mm radius: each tolerance of cutting and folding a paper net, as a displacement on the assembled sphere, and the seam corner of each of the three face maps beside it. The fold's own error is the largest at 0.955° — 106 km at Earth scale — because a degree of dihedral error moves the far edge of a face by a degree of the face's own radius, which for a cube is 54.7°. The conformal face map's corner is 0.769°, below the whole budget of 1.000°; the gnomonic's is 20.2°, twenty times above it. The families

The map is finer than the paper it is folded from

Thirteen measurements price a polyhedral map in angles, areas and lengths, and none of them is about paper. A hand-folded net is accurate to about a degree of dihedral angle, which displaces the far edge of a cube's face by 0.955° of arc — 106 km at Earth scale — and that floor does not shrink when the globe is made larger, because a degree is a degree. Against it the conformal face map's seam corner is 0.769° on a cube and 0.0040° on an icosahedron, where it is 165 times finer than the sheet can be folded to.

 A fast stretch over one half of a gap, with nothing changed across the jet. A jet of 100 km/h blowing along the Goose Bay–Narsarsuaq gap, carrying a streak that reaches 300 km/h over a stretch about 1,000 km long, centred 500 km from the gap's midpoint on the Goose Bay side. The dotted bands are where the wind has slowed to 150, 100 and 50 km/h, and they are drawn from the jet's own cross-section — nothing about it changes across the flow. The solid outlines are the places within sixty minutes of each airfield. The least allowance at which New York is joined to London is 665.6 km of still-air radius against the plain jet's 624.7 — a move of +40.9 km, pinned between Goose Bay–Iqaluit. Paths and directions

A streak along the jet moves the floor while the crossing is flown

The diversion floor answers only to the difference between the wind on a gap's two halves, and every case so far produced that difference by moving a jet sideways. A jet carrying a streak — a thousand kilometres of core markedly faster than the rest — produces the same difference with nothing changed across the flow at all, and moves the floor by up to 43 km where the same jet at one speed along its whole length moves it by ten metres. Plotted against the half-integral difference the two geometries fall on one line: the floor cannot tell them apart. And because a streak drifts with the flow, the allowance an operator must satisfy is 30 km higher than the one the departure analysis reported.

 The curve that does the whole job has no kink and does not turn over. The parallel spacing along the central meridian, as a share of the equator's, for the best single smooth curve found and for three maps it is being compared with. Goode's composite is flat to its join at 40.7° and then follows the Mollweide's, and the kink where the two meet is the join every essay before this one has been about. The fitted curve has no kink, and it is not the shape the question expected either: it is flat only through the first fifteen degrees (97.2 per cent at 15°), then closes monotonically and faster than the Mollweide — 54.3 against 77.3 at 60° — and it never turns over. The families

One curve beats the three maps it would replace

Goode's composite is a sinusoidal, a Mollweide and now a splice strip between them, and the question left open was whether one smooth spacing curve could do the whole job or whether the joins are doing work no single curve can. One term does it: a curve with a single coefficient carries 6.8 degrees less angular deformation than the composite, two beat every named equal-area pseudocylindrical in the library, and the composite turns out to be worse than the Mollweide it is half made of.

 Where the correction overtakes the sphere it is correcting for. The worst angle error of four substituted spheres against the size of the triangle, with Legendre's spherical excess on the same axes. The two that are wrong at first order in the flattening are flat in the side, because the error belongs to the latitude and not to the distance. The two that are right at first order have no such floor, so what is left is a term rising in proportion to the side. The excess rises as its square, so each pair meets exactly once, and the meeting decides which of the two corrections a computation is entitled to ignore: the geodetic latitude at 490 km, the rectifying latitude at 247 km, the conformal latitude at 23.8 km, the geocentric latitude at 17.6 km. What is taught wrongly

The correction is the smaller of the two corrections

Legendre's theorem subtracts a third of the spherical excess from each angle of a triangle and solves the rest as a plane figure. It is the correction every nineteenth-century computer applied and the one every textbook explains. Beside the error the choice of substituted sphere leaves in the same triangle it is small: on the thirty-kilometre triangle a first-order chain is made of, omitting it costs 166 millimetres in a computed side and using the geodetic-latitude sphere costs 43 metres. The two terms scale differently, so each sphere has a size at which the correction overtakes it — and for the two spheres a careful computer would have chosen, that size falls inside a real chain.

 A sea with two islands in it, and the three circles it already is. A stated sea 60° across with two islands 12° across, their centres 12° from its own, drawn on the globe and drawn again under the stereographic projection centred on the sea. Stereographic projection carries every circle on the sphere to a circle in the plane, so the flat picture is bounded by three exact circles with no fitting and no iteration: the sea's edge at radius 0.5359, and the two islands at radius 0.1060 with their centres 0.2108 from the origin. Normalised, the region's three conformal invariants are 0.2361, 0.6848, 0.1221 — three numbers, which is what a region with two holes carries and no angle-keeping map can change. The impossibility

Two islands are three numbers, and the proof does not survive them

A sea with one island at its centre has one conformal invariant and a forced best map. Put two islands in it and there are three invariants, readable in closed form because stereographic projection carries circles to circles — and the forcing is gone. The theorem that picked the map needed the region to be unchanged by rotation, not to have one hole, and a single harmonic term two hundredths in size beats the stereographic by 0.19 per cent the moment an island moves off centre.

 The seam corner falls faster than the paper and crosses it at 720 faces. For an icosahedron globe of 100 mm radius with every face divided ν ways, the gnomonic face map's worst seam corner against what a hand-folded paper net is accurate to. On twenty faces the corner is 9.19°, 12.8 times the whole budget of 0.716°. The corner falls as ν to the power -1.97 and the fold's error as ν to the -0.97, so the two lines converge; the budget stops falling once the crease, a fixed quarter-millimetre radius of paper, is larger than the fold. The corner passes under it at ν = 6, 720 faces, where it is 0.315° against 0.321°. The families

The seam falls faster than the paper

Divide each face of an icosahedron ν ways and two things shrink at different rates. The fold's error falls as 1/ν, because it is the face's own radius times a degree. The gnomonic seam corner falls as 1/ν², because it needs both a tilt between faces and a distance along their edge. So the gnomonic, 12.8 times the whole assembly budget on twenty faces, goes under it at 720 faces on a 100 mm globe. No globe of any size needs more than 5,120, and what it costs is paper folded out of sight: 5.8 per cent of the sheet at that crossing.

 The second clause moves the map without moving what the first clause measures. The plateau the tie-break changes most, drawn before and after. The contours are where the scale departs from true by the stated half a per cent, so the ground between them is the ground the specification asks to be kept. The rule leaves that ground alone — 52.23 per cent of the square before and 52.54 after — and spends its freedom outside it, where the worst departure falls from 27.16 per cent to 23.77. Of the 42 plateaux this is the largest change any of them shows. What each projection optimises

A second clause decides the half that was already decided

A specification written as a band admits forty-two maps keeping between 37.0 and 67.5 per cent of a region. Adding a tie-break — among those, take the one whose worst departure is least — is a linear programme, and it works: it makes each of the forty-two definite. It also merges three neighbours at the bottom into one, leaves 29.1 of the 30.5 points of spread in area kept, and names a single map only at the tolerance where every map keeps everything and the first clause has stopped asking for anything at all.

 Cut into Goode's lobes, the ranking of uncut sheets turns over. Mean angular deformation over the sphere for eight equal-area pseudocylindricals, each read two ways: as one uncut sheet, and cut into Goode's lobes with every point measured from its own lobe's central meridian. Uncut, Eckert IV is the best named map at 27.70° and the one-term curve fitted to the uncut sheet beats Goode's composite by 6.83°. Cut, Eckert IV falls to 21.34°, second from last, the curve fitted uncut reads 20.77° against the composite's 18.48, and the plain sinusoidal beats the composite at 17.88°. A curve refitted for the cut sheet reaches 17.01°. The families

Scored lobe by lobe, the homolosine loses to its own lower half

Read as one uncut sheet, a single smooth curve beats Goode's composite by 9.6 degrees. But the homolosine is used cut into six lobes, so here it is scored cut, with each point measured from its own lobe's meridian, and the ranking turns over. Eckert IV, the best named map uncut, falls to second from last. The curve that beat the composite is now 2.3° worse than it. The plain sinusoidal, the composite's own lower half, wins at 17.88° against 18.48. The best smooth curve for a cut map is the sinusoidal with a few per cent of correction.

 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. Paths and directions

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.

 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. Paths and directions

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.

 Three exponents, three maps, and each is the only answer its specification has. The conformal maps of a 25° spherical square that minimise the mean of |ln k| raised to three exponents. The heavy contours are where the scale departs from true by half a per cent either way, so the ground between them is what a surveyor asking for that band would keep. At p = 1, the mean departure, the map keeps 52.8 per cent of the square inside the band; at p = 2, the mean square, 44.2; at p = 32, which is within a per cent of the least worst case, 32.7. Its worst departure falls the other way, from 3.25 per cent to 1.94 and 1.48. For every exponent from one upward the specification is convex and has one answer. What each projection optimises

One exponent names one map, and at tight bands a better one than the band usually finds

A specification written as a band keeps a count of area and throws the rest of the scale field away, which is why it had forty-two answers and why a tie-break could not name one. Written instead as the mean of |ln k| raised to a stated power, it is convex for every power from one up, and has one answer. At a band of a tenth of a per cent that single answer keeps more of the region than forty-one of the band's forty-two, and at half a per cent fewer than nine of them: the band's best answer leads it by 14.7 points there, and the band's own search started from it reaches within two points of that best. Only as the band approaches the least worst case does a large power take over.

 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. The families

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.

 The same two islands, and the cut that buys most is the central one's. A sea 60° across with a large island A, 12° across, and a small one B, 6° across, on opposite sides of its centre, drawn in the plane the stereographic makes of it. Each dashed line is one way of spending a single cut — from A to the open sea, from B to the open sea, or from A to B — and the solid line is the one that buys most, as a share of what two cuts buy. Left, A 18° from the centre and B 6°: A 9% · B 78% · between 21%. Right, A 6° and B 18°: A 81% · B 12% · between 11%. The better cut is the central island's in both, whichever island is large, and a cut joining the islands buys 21% and 11%. The impossibility

One cut belongs to the island that would cost most alone

A sea with two islands pays a few per cent for being drawn whole by an angle-keeping map, and one cut buys back part of it. Which island to cut is a design choice that leaves the ground exactly where it was, and it is not decided by size: across twenty arrangements of a large island and a small one, the cut worth most goes to the island that would cost the sea most if it were alone in it — in eighteen of twenty, the other two near-ties — which is usually the more central one. A cut joining the two islands buys almost nothing, and the better cut is nearly always the longer.

 The level is flat through the family, climbs over its last twentieth, and jumps at the cylinder. Each member's pass depth averaged over its 36 regions. From m = 0.2 to 0.9 it stays between 0.0224 and 0.0450 with no trend. From 0.95 it rises at every step — 0.0436, 0.0844, 0.0919, 0.1082, 0.1235, 0.1388 — a factor of 3.2 to the sinusoidal. And the cylinder at m = 0 stands at 0.2506, 8.5 times its nearest neighbour: a cylindrical map's pattern slides unchanged when it is turned about the Earth's axis, and no other member's does. What each projection optimises

The sinusoidal is the edge of its family, not outside it

The sinusoidal is the last member of a family of equal-area maps that begins at Lambert's cylindrical, and it was thought to be the one projection whose pass depth has no curve. Along the family the curve comes and goes: present at m = 0 and 0.8, absent at 0.6 and 0.9. What does move smoothly is the pass depth's level. It is flat from m = 0.2 to 0.9, then rises at every step over the last twentieth, 3.2-fold, to the sinusoid — whose high ridges are the end of that rise, not an exception to the family — while the cylinder at m = 0 stands eight times above its neighbour.

 The floor against where the streak is, and the stretch of it a crossing sweeps. The diversion floor for the New York–London chain with a streak 200 km/h faster than a 100 km/h jet core, against where along the jet the streak's centre lies, computed exactly on a grid of positions 125 km apart and read between them. With the streak far off the floor is the plain jet's, 624.7 km. With it over the Goose Bay half it rises by up to 41.1; over the Narsarsuaq half it falls by up to 44.8. The bar is the stretch a streak starting at -1400 km sweeps in a four-hour crossing at 250 km/h; the floor to be met is the worst on it, 666.0 km. The shaded band either side is a forecast position error of ±200 km at one standard deviation. Paths and directions

A forecast's error in where the streak is becomes a margin

A jet streak drifting along a gap moves the diversion floor by up to forty kilometres during a crossing, and an operator knows where the streak is only from a forecast. Planned to the worst over the forecast's own crossing window, the floor needs a margin of about eight kilometres for every hundred of error in the streak's position, and stops paying for itself against simply ignoring the streak near an error of two hundred and fifty. Planned to the departure analysis alone, it is worse than ignoring the streak at any error at all.

 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 families

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.

 Four cities, met — the triangular, x first construction. Every cell drawn here holds the same area of ground and a different area of page: the map has been constructed so that its areal scale factor is the density it was handed, cell by cell, over a contrast of 25.99 to one. The residual against that target is 5.1e-6, measured from the map's own derivatives rather than from the construction. What it cost is the shape: the redistribution alone reaches 139.1° of angular deformation and averages 70.4°, on top of whatever the equal-area projection under it was already doing. Measuring distortion

A map drawn to a density it was handed

Two hundred and twenty-six essays measure distortion after the fact. This one specifies it: a density is handed to a map as a boundary condition, the areal scale factor comes out equal to it to five parts in a million, and every other invariant the site owns becomes the price.

 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. Paths and directions

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.

 Two lawful systems, one sea: the lines move water between regimes and the outer limit does not move. The stated archipelago with the lawful system enclosing the most water (solid, 186,617 km² inside) and the one enclosing the least (dashed, 159,032 km²). Round each, the limit two hundred nautical miles from its baselines and from every island's coast: the two outer limits lie on top of each other everywhere but a few kilometres off the widest gaps. A straight coast stands 700 km east of the centre; the line equidistant between it and the archipelago is drawn for both systems and they coincide. Inside plus outside, the two systems account for 1,344,857 km² and 1,343,864 km² of sea. Grids, and what a survey does

The archipelago rule moves water between regimes, not between states

The eighteen baseline systems the archipelago article allows on a stated archipelago differ by 27,585 square kilometres of water inside the lines. Measured outward, as the Convention measures every zone, the difference almost vanishes: whatever a system takes inside it gives up from the zone outside, and water inside plus zone outside differs by 993 square kilometres, seven hundredths of a per cent. Against a straight neighbouring coast on any side, the equidistance line does not move at all, because a baseline's nearest point to a straight coast is always an island. Only a neighbour facing the middle of a gap feels the choice — 2,112 square kilometres at thirty kilometres off, 688 at 250.

 Goode's lobes and the lobes the land would choose, at his own count of two north and four south. The land in two-degree cells, Antarctica pale because it is not scored, with Goode's cuts dashed and the land's own solid; a short tick at the equator marks each lobe's central meridian. Goode's: edge at 180°, northern cut at 40°W, southern cuts at 100°W, 20°W and 80°E, meridians 100°W and 30°E north and 160°W, 60°W, 20°E, 140°E south — 16.51° of mean angular deformation on the land outside Antarctica. The land's, with no cut crossing more land than Goode's worst: edge at 20°W, northern cut at 165°E, southern cuts at 70°E, 155°E, 175°W, meridians 65°E and 100°W north and 25°E, 135°E, 175°E, 60°W south — 14.48°. The families

The land would move Goode's lobes before his join

Goode chose his lobes by eye: where the cuts run and where each lobe's meridian sits. Scored on the land outside Antarctica and found exactly on a five-degree grid, the land's own layout at his count of two lobes north and four south deforms it by 14.48° against his 16.51° — two degrees, more than his meridians were worth in the first place — while its cuts cross less land than his: 2,057 kilometres against 2,391, the worst of them 1,056 against the 1,946 his northern cut spends on Greenland. The land wants the map's edge in the Atlantic and its northern cut in the Pacific. His fourth southern lobe is worth four hundredths of a degree, and once the lobes are the land's the best join climbs again, to sixty degrees.

 The two islands dearest alone are the wrong pair to cut when they sit on the same side. One of forty three-island seas, 60° across, drawn in the plane the stereographic makes of it; beside each island, what it would cost the sea if it were alone in it — P 0.97%, Q 0.72%, R 1.68%. Left: the two dearest islands, P and R, cut to the open sea, which buys 75% of what three cuts buy. Right: the best pair, Q and R, which buys 97%. The dearest islands are 69° apart as seen from the centre, and their single cuts overlap by 40 points. The impossibility

The second cut goes across the sea

With two islands and one cut, the island that would cost the sea most alone names the cut. With three islands and two cuts, the obvious rule — cut the two dearest — names the best pair in only 23 of 40 seas, and loses up to 23 points of what three cuts would buy. It fails when the two dearest sit on the same side of the sea, where their cuts do each other's work. The best pair keeps the dearest island in 37 of 40; its partner is the island across the centre from it, which is right in 32 of 40 and never costs more than 12 points. A cut joining two islands is never worth a second cut to the sea.

 A square switches from p = 1 to the worst case at one band; a thin ellipse passes through every exponent between. Area kept inside a band, per cent, against the band as a fraction of each region's least worst departure. Solid: the map minimising the mean of |ln k|, p = 1. Dotted: the least worst case, p = 32. Thin line: the best of twelve exponents from 1 to 32. Shaded: what a rule taking p = 1 below 0.9 and the worst case above it gives up against the best exponent. On the 25° square the best exponent is p = 1 up to 0.85 and the rule gives up at most 0.9 points. On an ellipse with axes 1 : 0.2 an intermediate exponent is best from 0.60 and the rule gives up 5.9. What each projection optimises

The band names the exponent for a compact region, and not for a thin one

A tolerance band is best served by the mean departure when it is tight and by the least worst case when it is wide. Read as a fraction of the region's own least worst departure, the band decides which, and the region's size drops out: a square, a triangle or a hexagon at 10°, 25° or 40° switches from one to the other at nine tenths, and a surveyor told only that loses at most two points of area on every compact region but the degenerate cap. A thin ellipse does not switch. From six tenths upward it is served best by exponents in between, and any rule with two exponents in it gives up five to seven points there, wherever its switch is put.

 Four cartograms of one density. The same stated density — four cities — met four different ways, drawn on the same cells. Every panel is a correct cartogram of the same numbers: a region's page area is proportional to its mass in all four. They do not look alike, because the areal scale factor fixes one number per point and a map has four derivatives, so three degrees of freedom per point are left over and each construction spends them differently. Measuring distortion

Every density can be met and none is free

Four maps of the same data, all of them correct, charging between 57.6° and 104.4° of angular deformation for it. There is no such thing as the cartogram of a density — there is an infinite family, and somebody picked a member of it without saying so.

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