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

Three criteria, three maps, and the tolerance band each one encloses

Drawn here at the parameters it defaults to, with every essay that calls it.
Three criteria, three maps, and the tolerance band each one encloses. A spherical square of 25° circumradius under three conformal maps, with the two contours at ±0.5% of true scale drawn heavy and the region between them the area each map keeps inside that tolerance. Chebyshev's map holds 37.0 per cent of the area there, the mean-square map 45.4, and the map fitted to the tolerance itself 64.1. The faint lines are further multiples of the tolerance, and they say where each map spends what it is not counting: the third map's outermost contours are much further out than the other two's.

A spherical square of 25° circumradius under three conformal maps, with the two contours at ±0.5% of true scale drawn heavy and the region between them the area each map keeps inside that tolerance. Chebyshev's map holds 37.0 per cent of the area there, the mean-square map 45.4, and the map fitted to the tolerance itself 64.1. The faint lines are further multiples of the tolerance, and they say where each map spends what it is not counting: the third map's outermost contours are much further out than the other two's.

It is one of 4 figures drawn by the same construction, and the drawing above is this one with nothing changed.

8 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

Three criteria, three maps, and the tolerance band each one encloses. A spherical square of 25° circumradius under three conformal maps, with the two contours at ±0.5% of true scale drawn heavy and the region between them the area each map keeps inside that tolerance. Chebyshev's map holds 37.0 per cent of the area there, the mean-square map 45.4, and the map fitted to the tolerance itself 64.1. The faint lines are further multiples of the tolerance, and they say where each map spends what it is not counting: the third map's outermost contours are much further out than the other two's. What each projection optimises

The map that keeps the most ground inside a tolerance

A surveyor does not want the smallest worst case or the smallest mean square. They want as much of the territory as possible within a stated tolerance of true scale and are indifferent to how far outside the rest goes. That is a third criterion, it is answered by a third map, and on a 25° square at half a per cent it keeps 64.1 per cent of the area where the mean-square map keeps 45.4 and Chebyshev's 37.0 — bought by sending a tail out to 5.6 per cent where Chebyshev's worst is 1.5.

 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.

 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.

 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.

 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.

 Two solved maps and the line between them keep what the whole family of exponents keeps. Area inside a band, per cent, against the band as a fraction of each region's least worst departure, from 0.3 to 1. Heavy: the best map on the line between the p = 1 map and the least worst map, found by a search along the line alone. Thin: the best of twelve exponents from 1 to 32, each solved separately. Dashed: the rule taking p = 1 below 0.9 and the least worst case above it; shaded, what that rule gives up. On the square the line trails the best exponent by at most 0.00 points and beats it by up to 1.3; the rule gives up 0.9. On the ellipse the line trails by at most 1.1 and the rule gives up 5.9. What each projection optimises

Two maps and the line between them do the work of every exponent

A conformal map's log scale is linear in its coefficients, so every blend of the map that minimises the mean departure and the least worst map is itself a conformal map, available the moment the two are solved. Searched along that one line for each band, the blend keeps within 0.17 points of the best of twelve exponents on every compact region and beats all twelve by up to 1.35 on the square; it trails by 1.5 at most on the thinnest ellipse and 0.74 on four regions with a bay, where the rule of switching between the two ends gave up as much as 7.1. The family's middle members lie up to two thirds of their own departure off the line. The line does not reproduce them; it does their work.

 Three solved maps span a plane of maps, and the band's best map lies well outside their triangle. Every conformal map of a 25° spherical square whose ln k is the p = 1 map's plus s times its difference from the least worst map plus u times its difference from the mean-square map, for s and u from −4 to 4, contoured by the share of the square it keeps inside a band of ±0.74 per cent — 0.5 of the least worst departure. The three maps are the corners of the shaded triangle at (0, 0), (1, 0) and (0, 1); the line between the first two is its lower edge, and the exponents from 1 to 32, dashed, run from one corner to the other through the mean-square map and out past it. The best map of the twelve exponents keeps 66.4 per cent, the best on the line 66.5, the best inside the triangle 66.5. The best in the plane sits at s = 0.72, u = −2.81, outside every one of them, and keeps 75.8. What each projection optimises

Three maps make a plane, and the band's best map lies outside their triangle

Every conformal map whose ln k is a fixed blend of three solved maps — the p = 1 map, the least worst map and the mean-square map — is itself a conformal map, so three solved maps are a plane of them. Inside their triangle the third map recovers most of what the line between the first two gave up on thin regions: 0.63 points short of the best exponent on the thinnest ellipse, where the line was 1.48 short. But the band's best map in the plane is not between the three. On 172 of 182 regions and bands it lies outside the triangle, and it keeps up to twenty points more than any exponent; started from it, a search over all twenty-five coefficients adds about a point on four regions of five and seven on a hexagon. Its price is a worst departure up to nearly five times the least worst.

 Each cap on the worst departure is a long convex band of maps, and the best map under it rides the band's edge. The plane of conformal maps of a 25° spherical square whose ln k is the p = 1 map's plus s times its difference from the least worst map plus u times its difference from the mean-square map, for s and u from −4 to 4. Dashed: the edges of the maps whose worst departure is at most 1.2, 1.5, 2, 3 times the least worst — convex, nested, long and narrow, each running diagonally through the least worst corner. Thin contour: the maps keeping as much of the region inside a band of ±0.74 per cent as the best exponent's 66.4 per cent; heavy contour: three points under the uncapped best, 72.8 per cent. Dots: the map keeping the most inside the band under each cap — 1.2: 59.8 per cent at (−0.30, 1.56); 1.5: 61.3 per cent at (−0.17, 1.03); 2: 65.8 per cent at (0.50, −0.36); 2.5: 69.1 per cent at (0.51, −1.00); 3: 72.1 per cent at (0.55, −1.63); 4: 75.8 per cent at (0.72, −2.81); the large dot is the uncapped best, 75.8 per cent at a worst of 3.82 times the least worst. What each projection optimises

A cap on the worst departure keeps the thin regions' gain and gives back the square's

The conformal map that keeps the most of a region inside a band can sit far out in the plane of three solved maps and let its edges go to nearly five times the least worst departure. Capped at twice the least worst, the plane still matches or beats the best exponent on 69 region-and-band pairs of 91: on the thin ellipses it keeps the whole of its gain, because that gain sits in a small pocket just past the exponents' own worst departure and arrives all at once; on the square and the bays it gives most of it back, because there the gain is a ramp that runs out to four times the least worst. Under any one cap the plane still beats the exponents that cap admits, by up to eighteen points — and it can name maps that no weighting of area against worst departure would ever choose.

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