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

Pareto front — where it appears

The members of a set that nothing beats on every criterion at once. It says what is not ruled out rather than what a stated preference selects, and a weighted sum of the criteria can only ever choose a vertex of the front's convex hull.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

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

A rule of thumb has been scored over thirty regions and then forty-five. The same populations answer a harder question nobody has 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 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 The shortest crossing that never goes more than 745 km from an airfield. New York to London, drawn on a conformal conic, with a circle of 745 km round each of twenty airfields on both sides of the North Atlantic. The dashed line is the great circle, 5,540 km. The solid line is the shortest route that stays inside the circles, 5,698 km — 158 km or 2.85 per cent longer — and it bends only where two circles cross: between Narsarsuaq and Keflavík; Keflavík and Vágar. Nowhere else can it bend, because anywhere else a straight cut stays inside a circle.

The rule that keeps a route near land is pinned at both ends

A crossing that may never be more than a stated distance from an airfield is refused below one radius and free above another, and each wall is set by a single feature of the ocean. Between them the price falls continuously, with corners and no jumps — and at the lower wall it falls as the square root of the allowance, so the first kilometre is worth twelve of route.

paths · Paths 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 lobes the ocean would choose, against the lobes the land chose. The water in two-degree cells, with the land's own layout dashed and the ocean's solid; a short tick at the equator marks each lobe's central meridian. The land's, at two lobes north and four south: edge at 20°W, northern cut at 165°E, southern cuts at 70°E, 135°E, 180° — 14.40° of mean angular deformation on the land outside Antarctica and 19.75° on the water. The ocean's: edge at 40°E, northern cut at 110°W, southern cuts at 135°E, 140°W, 60°W, meridians 175°E and 40°W north and 85°E, 180°, 105°W, 10°W south — 13.31° on the water and 26.18° on the land.

The ocean centres its lobes on the land's cuts, and cannot keep its own cuts short

Run on the water instead of the land, the same exact search gives an interrupted homolosine whose lobes sit on the Pacific, the Atlantic and the Indian Ocean — five of its six central meridians within twenty degrees of a cut the land's own layout chose — and it deforms the water by 13.31° against the 19.75° the land's layout gives it. The inversion stops at the cuts. Every meridian on Earth crosses at least 7,172 kilometres of water, so the ocean's map has no layout whose cuts stay under seven thousand, where the land's went below nine hundred. And no layout serves both: none comes within 2.57° of both optima at once, though a single wide Pacific lobe buys the water 2.8° for a tenth of a degree of land.

families · Families The water moves the join higher than the land does, to seventy-five degrees. Goode's composite joined at each latitude, two lobes north and four south. On the water to 88°, with the layout searched afresh at each join (solid): 14.10° at 25°, 13.87° at 30°, 13.60° at 35°, 13.37° at 40°, 13.31° at 40.7°, 13.14° at 45°, 12.89° at 50°, 12.66° at 55°, 12.52° at 60°, 12.40° at 65°, 12.29° at 70°, 12.27° at 75°, 12.27° at 80°, 12.29° at 85°, 12.30° at 89°; with the ocean's layout found at Goode's join held (dashed), within 0.05° of it everywhere. On the land outside Antarctica, the land's own layout searched at each join: best at 60° with 13.83° against 14.40° at Goode's; the water's best is 75° with 12.27° against 13.31°.

The water would join Goode's map at seventy-five degrees

Goode joined the sinusoidal to the Mollweide at 40° 44′, and the land, given its own lobes, moves that join up to sixty degrees. The water moves it further, to seventy-five, and gains twice as much by it — a full degree of mean angular deformation against the land's half — though less of the northern water than of the northern land lies above Goode's join. The gain is in the polar water, north and south, where Goode's Mollweide cap shears even the lobes' own meridians; the water just above his join, the northern edge of the Southern Ocean among it, pays part of it back. A separate join for each hemisphere is worth a few hundredths of a degree, and with the join free the one-layout-for-both question comes out the same way it did with the join fixed, only worse: no layout comes within 3.3 degrees of both surfaces' best at once.

families · Families 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.

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.

choosing · Condition

Named alongside it

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

Trade-offVerificationAngular deformationPurposeCentral meridianEqual-areaHomolosineInterruptionMinimaxObjective functionOptimisationProjection selection

All concepts