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

Scored lobe by lobe, the homolosine loses to its own lower half finally judged Goode’s homolosine the way it is used: cut into Goode’s six lobes, every point measured from its own lobe’s central meridian. On that footing the ranking of equal-area maps turned almost completely over, and the plain sinusoidal, the homolosine’s own lower half, came out ahead of the homolosine by 0.60° of mean angular deformation over the sphere.

It then read the difference latitude by latitude. The homolosine wins from 32° to 61°, a band holding 35 per cent of the sphere’s area, and loses everywhere else. For it to draw level, a score would have to count that band 1.41 times as heavily as its area. A score weighted by land was the obvious candidate, since the homolosine was drawn for continents and the northern mid-latitudes are where the most land lies — but it needs a coastline, and the essay stopped there.

This is that measurement. It needs a decision made once, which is which coastline: Natural Earth’s land polygons at 1:110m, the coarsest of that public-domain series and the right one for a quantity summed over continents, rasterised at half a degree. The raster covers 28.9 per cent of the sphere, against about 29.2 for the Earth’s land including Antarctica.

The band holds its share of land and no more #

The first result is a disappointment for the band argument. The land is not concentrated in the northern mid-latitudes as heavily as a glance at a world map suggests. Without Antarctica, the band from 32° to 61° holds 35.8 per cent of the land against 34.9 per cent of the area, a factor of 1.03. With Antarctica — almost all of it south of 61° — the band holds 32.9 per cent, a factor of 0.94, less than its share.

The picture a map gives of northern land is itself the reason for the surprise. Every map in common use enlarges the high latitudes relative to the tropics or at least does not shrink them, and the Sahara, Arabia, India, Brazil and most of Africa are low-latitude land that equal-area maps draw at its true size and others do not. Counted by area on the sphere, the tropics hold as much land per band as the middle latitudes.

So a land-weighted score that looked only at latitude would leave the homolosine exactly where the area score left it, behind.

That is worth pausing on, because the homolosine’s design has always been explained by latitude. Joined where the parallels agree, and the meridians turn a corner found the join at 40° 44′ fixed by the one latitude where the sinusoidal’s and the Mollweide’s parallels have the same length, and the usual account says Goode used the sinusoidal where it is good, in the tropics, and the Mollweide where it is better, towards the poles. On land weighted by latitude, that account does not produce a winner. Whatever the homolosine has, it is not simply a better use of the land’s latitudes than the sinusoidal or Craster’s parabolic. On latitude alone — each row of the raster keeping its land weight, but the land in it spread evenly across the lobes as the area score spreads the sphere — the homolosine scores 18.14°, third, behind Craster’s parabolic at 17.85° and the Mollweide at 18.05°, and ahead of the sinusoidal at 18.52°.

Weighted by land where it lies, the homolosine wins #

The land is not spread evenly across the lobes, though, and that is the point of Goode’s lobes. Every land cell is read from the central meridian of the lobe it actually lies in — north of the equator, meridians at 100° W and 30° E with the cut at 40° W; south of it, meridians at 160° W, 60° W, 20° E and 140° E with cuts at 100° W, 20° W and 80° E — and every map is scored at its own best proportions for that weighting.

Without Antarctica, the homolosine is the best of the seven: 16.51° against Craster’s 17.00, the sinusoidal’s 17.34, the Mollweide’s 17.56 and Eckert II’s 17.67. Eckert IV, the best named map on an uncut sheet, is sixth at 19.34, and the one-term curve fitted to the uncut sheet is last at 21.33, as it was when the sphere was cut. The homolosine’s margin over the sinusoidal, 0.83°, is larger than the sinusoidal’s margin over it on the sphere, 0.60°.

Uncut, the land changes nothing that the sphere had not already shown. Read as one sheet centred on Greenwich, with Antarctica left out, Eckert IV is best on land at 24.76°, the one-term curve second at 27.64, the Mollweide third at 29.16, and the homolosine fifth at 31.48 — much the order one curve beats the three maps it would replace found on the sphere, with Eckert IV ahead of the one-term curve and the homolosine behind the Mollweide it is half made of. Cutting is worth fifteen degrees to the homolosine on land and five and a half to Eckert IV. The homolosine’s case has always been its lobes, and on land it is still its lobes.

With Antarctica counted as land, the homolosine is fourth, at 19.63° against 17.91 for both the sinusoidal and Craster’s parabolic. Antarctica is a continent of fourteen million square kilometres lying almost entirely beyond 61° south, which is exactly where the homolosine’s Mollweide cap is worst — the earlier essay found it at 66.7° of deformation at 80° against the sinusoidal’s 33.8 — and a score that counts it as land counts the homolosine’s weakest ground as among the most important on the map. Goode’s own map gave Antarctica no lobe of its own and split it across four.

The meridians did it #

The difference between the two scores — latitudes only, and the land where it lies — is what Goode’s choice of central meridians is worth, because the only thing it adds is the land’s distance from those meridians. It is worth something to every map: land sits nearer the lobes’ meridians than an even spread would put it, which is what the lobes were drawn to do. But it is worth different amounts to different maps, and the differences decide the ranking.

How much nearer the meridians the land lies is easy to state. Spread evenly across Goode’s two northern lobes, a point would lie on average 47.2° of longitude from its lobe’s meridian; the northern land lies 38.0° from it. In the south, where there are four lobes and each is narrower, an even spread would lie 22.8° out, and the southern continents lie 9.6° from their meridians — Goode centred his southern lobes on South America, Africa and Australia almost exactly. Three quarters of the land outside Antarctica is in the north, where the lobes are wide and the gain is smaller, and a quarter in the south, where the lobes fit the continents closely.

To the homolosine it is worth 1.63°, enough to lift it from third to first. To the sinusoidal, 1.18. To Craster’s parabolic, which led on latitudes alone, 0.85, and to the Mollweide only 0.49. Eckert II gains most of all, 3.31°, because its deformation grows fastest with distance from the meridian, but it starts too far back to catch up.

The homolosine and the sinusoidal are the same map below Goode’s join, so the half-degree more that the meridians are worth to the homolosine is earned above it, on the land its Mollweide cap covers — northern North America under the 100° W meridian, Europe and Russia under the 30° E one. The measurement records that the cap gains more than the sinusoidal would from having that land where it is rather than spread across the lobe; it does not separate how much of the gain is the land’s nearness to the meridians and how much its particular latitudes, which are the two things the latitude-only score throws away together.

Where the lead is won #

Split by latitude, the homolosine’s lead of 0.83° on land is won entirely above its join and partly given back below it. Every band north of 40.7° favours the homolosine, together by 1.54°. Every band south of the join, where both maps are the sinusoidal and differ only in the proportions each takes to suit its whole, favours the plain sinusoidal, together by 0.71°: the homolosine’s best proportions are set for its cap and cost it a little on the continents’ low-latitude land.

The map shows the same thing geographically. North of the join nearly all the land is better on the homolosine — Canada, the northern United States, Europe, Russia and northern China. South of it the plain sinusoidal is slightly better almost everywhere, on Africa, South America, India and Australia, by margins too small to see on the grid, and the homolosine’s lead is the northern gain less that southern loss. The ground the homolosine wins is the ground Goode chose to fix with his cap; the earlier essay called it “the continents it most needed to fix”, and on the land itself that turns out to be true.

Goode’s objective, written down #

The earlier essay stated the homolosine’s case as a condition on latitude: it beats the plain sinusoidal if and only if its reader weights the band from 32° to 61° about forty per cent above its area. That is exact for any weighting that depends on latitude alone, and it is the wrong family of weightings for a map designed around where the continents are. Land is a weighting in latitude and longitude, and Goode placed his meridians in longitude.

So the objective his design answers to can now be stated, which he never did. It is: least mean angular deformation over the land outside Antarctica, each point read from the meridian of its own lobe, with those meridians where Goode put them. On that objective the homolosine beats every named equal-area pseudocylindrical in the library, and the smooth curve that beat it on the uncut sphere falls to last. The area weighting was a readership all along found the same shape of result for projection comparisons generally: the verdict belongs to the weighting, and here the weighting that crowns the homolosine is the one its designer had in mind, provided one continent is left out.

That last proviso is not a small one. Antarctica is 8.3 per cent of the land on this raster, and it decides the ranking. Leaving it out of a world map’s objective is a choice, and it is the choice every map built for the inhabited continents makes; counting it is equally defensible for a map of land as land. The homolosine is the best map in the library for the first reader and fourth for the second, and no weighting between them can make it both, since the one continent the choice concerns lies wholly on the ground where the homolosine is worst.

The proportions each map takes move less than the ranking does. The homolosine’s best width-to-height ratio is 0.99 on land without Antarctica and 0.98 with it, against 0.982 on the sphere; the sinusoidal’s is 1.00 in all three. Every equal-area map is every other one is why that ratio is free for an equal-area map, and it is the one number of a map’s own that a weighting can move. Here it barely does, so what the weighting changes is almost entirely which map is best rather than how each should be drawn.

How the land was counted #

The raster must be the land. Natural Earth’s 1:110m polygons, each half-degree cell counted as land when its centre is inside a polygon by the even–odd rule over every ring. The area-weighted share must be within a point of the Earth’s 29.2 per cent, and is 28.9; the middles of six continents must be land and of four oceans sea, and are.

On all cells, the land score must be the area score. With every cell of a one-degree grid counted as land, the land-weighted score on Goode’s layout must reproduce the area score on Goode’s lobes, and on an uncut layout the uncut area score. They agree to 0.11° cut and 0.36° uncut, where the deformation at the sheet’s edge is largest and the two samplings place their points differently.

The finding must be there to be found. Without Antarctica the homolosine must beat the sinusoidal on land read from its own lobes, and with Antarctica it must not — it does, and it does not.

Every map is scored at its own best proportions. The constant each equal-area pseudocylindrical leaves free, the ratio of its width to its height, is searched for every map under every weighting separately, so no map is judged at proportions chosen for another objective. It matters here more than it might: which projection a weighting can make best found that a weighting cannot put every map first, and a map judged at the wrong width can look like one of those it cannot.

Where the land stops #

One coastline at one resolution. At 1:110m, small islands and narrow peninsulas are missing or merged, and a half-degree cell is land or not by its centre. Both change the land’s area by far less than a per cent and its distribution by latitude by less; the ranking’s margins, half a degree and more, are well above that.

Goode’s lobes, as published. The meridians and cuts are Goode’s 1923 layout, and the cuts are taken as free. Giving up continuity is the reminder that they are not: a reader crossing a cut pays in continuity, which no angular measure counts. And the homolosine’s own seam at the join is scored as it is, corner and all; an equal-area strip removes the corner and charges nothing found a splice that would take the corner away, and it is not in this comparison. A different layout — Goode himself published variants, including ones with lobes for the oceans — would reweight every map, and nothing here says whether his layout is the best one for his own map on this objective.

Angular deformation only. Every map compared is equal-area, so area is not at issue, and the score is the mean of the maximum angular deformation. Distance and the look of a coastline’s shape are not the same measure, and a reader who cares about the outline of continents cares about something close to, but not exactly, this.

Every point of land counts the same. A square kilometre of the Canadian Arctic weighs what a square kilometre of Bengal does. A map for people rather than for land would weight by population, and population is concentrated far more heavily in latitude than land is; the same score would read it, with a population raster in place of the coastline.

Still open: where Goode should have put the join #

The land’s contribution to the homolosine’s lead changes sign exactly at the join, and that is suggestive. Every band above 40.7° favours the Mollweide cap, and every band below favours the sinusoidal. A join placed lower would give the cap more of the land it wins on, and more of the land it loses on; a join placed higher, the reverse. On the sphere’s area the earlier essays found the join where the parallels agree, a fact of geometry; on the land, the join’s position is a free parameter with an objective now attached to it.

Whether a homolosine joined somewhere other than 40° 44′ does better on the land outside Antarctica, where the best join falls, and whether a smooth spacing curve fitted to the land-weighted, cut objective closes its parallels fastest near Goode’s latitude — which would mean he found by eye the optimum of an objective nobody had written down — are questions one join and one set of named maps cannot answer.