The land puts the homolosine's join eleven degrees higher

On the land it was drawn for, the homolosine wins by its meridians finally gave Goode’s homolosine the objective its design answers to: least mean angular deformation over the land outside Antarctica, each point read from the central meridian of the lobe Goode put it in. On that objective the homolosine beat every named equal-area pseudocylindrical in the library, and the win came from above its join — every band of land north of 40.7° favoured it over the plain sinusoidal, every band south of the join favoured the sinusoidal.

That was suggestive about the join itself. Joined where the parallels agree, and the meridians turn a corner found where Goode’s join comes from: 40° 44′ 11.98″ is the one latitude at which the sinusoidal’s parallels and the Mollweide’s have the same length when the two are drawn to the same area, so the join is the only place the two maps can be glued without a step in any parallel. That is a fact about two named projections. It says nothing about whether 40.7° is where the land would want the switch.

Now that there is an objective, the join is a free parameter. The two maps can be joined at any latitude if the Mollweide cap is rescaled so each parallel stays continuous — the composite is then still equal-area and still a pseudocylindrical, only no longer made of the Mollweide at its own scale above the join. This essay asks where the land would put the join, how much it matters, and whether a map with no join at all does better.

The land’s best join is at fifty-two degrees #

Swept from a join at 20° to one at the pole, the land score has a single broad minimum. It is 17.34° with the join at 20°, where the composite is nearly all Mollweide; it falls through Goode’s 16.51° at 40.7° to its least, 16.29°, at a join of 51.7°; and it rises again to the plain sinusoidal’s 17.34° as the join reaches the pole and the cap vanishes.

The minimum is flat. Every join sampled from 50° to 60° is within a tenth of a degree of the best, and a join at 46° is too, at 16.38°; the whole of that plateau lies above Goode’s join, which scores 0.22° worse than the land’s optimum. So Goode’s latitude is not on the plateau — it is on the slope below it, about five degrees short of where the plateau begins.

The two sides of the minimum are not alike. Below it the score climbs slowly — 16.57° with the join at 40°, 16.81 at 35°, 17.02 at 30° — because moving the join down hands a little more mid-latitude land to the cap, which serves it slightly worse than the sinusoidal would. Above 60° it climbs steeply: 16.68° at 65°, 17.10 at 70°. A join that high leaves Scandinavia, northern Russia and the Canadian north on the sinusoidal, whose shear there, far from a meridian in Goode’s two wide northern lobes, is exactly what the Mollweide cap exists to relieve. So a cartographer unsure how much each continent should count does better to err low than high, and Goode erred low.

The two other curves show how particular this objective is. Over the whole sphere by area, the score falls monotonically all the way to the pole: the area-weighted objective wants no Mollweide cap at all, which is the earlier finding that the plain interrupted sinusoidal beats the homolosine on the sphere, seen here as a statement about the join. With Antarctica counted as land, the land score does the same — the one continent entirely poleward of any join is the one the cap serves worst, and counting it removes the minimum. It falls from 19.63° at Goode’s join to 17.91 at the pole, where it meets the plain sinusoidal. Only the land outside Antarctica has an interior optimum, and it is at 51.7°.

That is the same verdict scored lobe by lobe, the homolosine loses to its own lower half reached for the sphere, read as a statement about where the cap should start: on the sphere, nowhere. The cap is a device for land, and specifically for northern land near a lobe’s meridian; any objective that weights the far south or the open ocean heavily enough pushes it off the map.

A curve with no join does it better #

A composite is a sinusoidal below its join and a scaled Mollweide above it, and the join is the only thing about it that can move. A smooth spacing curve is freer. One curve beats the three maps it would replace wrote every equal-area pseudocylindrical as the exponential of a cosine series in latitude and fitted the series to the sphere; the same series fitted to the land, cut into Goode’s lobes and read from his meridians, is the land’s own answer with no named map in it.

With three terms it scores 16.08°, better than the land’s best composite by 0.21° and better than Goode’s homolosine by 0.43°. What it looks like is the interesting part. It does not blend a sinusoidal into a Mollweide gradually. Its spacing wanders within three per cent of the equator’s all the way to 50° — slightly closer together through the tropics, slightly further apart near 40° — and then closes steeply: 0.93 of the equator’s spacing at 55°, 0.67 at 70°. That is the shape of a join near 50°, arrived at by a curve that was never told there should be one.

It also takes different proportions. An equal-area pseudocylindrical is free to trade its width against its height, the one constant every equal-area map is every other one shows equal area leaves open, and the two composites take nearly the same, a width-to-height factor of 0.990 for Goode’s and 0.994 for the rejoined map. The fitted curve takes 1.175: it is drawn about a sixth flatter, with its parallels squeezed together near the equator. That costs it some shape on the tropical land, which the composites draw nearly square, and it gains it back further north. A single number of proportion is doing some of the work the join does in a composite.

So the land does not merely prefer a composite’s join at 51.7° among composites. Asked without any prejudice about joins, it asks for a map that is flat to fifty degrees and closes above, which is a smooth version of the rejoined homolosine. Goode drew a map of the right kind — sinusoidal low, closing high — and drew its knee about ten degrees too low for the land he drew it for.

The fit needs three terms to beat the join #

The smooth curve’s advantage is not cheap. With one term the best curve on land scores 17.10° and with two, 16.93 — both worse than Goode’s homolosine, and the two-term curve barely better than Craster’s parabolic. A cosine series in latitude with one or two terms cannot hold its spacing flat for fifty degrees and then close sharply; it has to begin closing earlier, and the land between 30° and 50°, the most densely inhabited continental ground on the map, pays for that. The third term is what lets the curve stay flat and then turn.

That makes the rejoined composite a remarkably good design for its simplicity. It reaches within 0.21° of the three-term curve with a single number — the join — where the smooth curve needs three coefficients to beat it at all. Goode’s composite construction was the right idea; it is the choice of join that the geometry made for him that costs.

For anybody designing an interrupted equal-area map for the inhabited continents today, the measurement says three things in order of how much they are worth. Put the lobes’ meridians through the continents, which is worth a degree and a half. Use a map that is sinusoidal low and closes high, which is worth most of another degree against the plain sinusoidal. And start the closing near fifty degrees rather than forty, which is worth a fifth of a degree — small, but free, since a composite joined at 51.7° is as easy to construct as one joined at 40.7° and differs from it only above the lower join.

Where moving the join gains #

The gain is not where the sweep’s shape suggests. Moving the join from 40.7° to 51.7° changes the map in two places, and the first of them loses. Between the two latitudes the rejoined map is still the sinusoidal where Goode’s is already the Mollweide, and away from the lobes’ meridians the Mollweide there was doing the better job: at 45° and twenty degrees of longitude from a meridian, Goode’s cap gives 9.2° of angular deformation and the sinusoidal 14.0. The land in that band — the northern United States and southern Canada, Europe north of the Alps, the steppe and northern China — gives back 0.95° of the land mean.

The second place is above the higher join, and it gains more. Goode’s cap starts closing at 40.7°, and a Mollweide carries shape distortion on its own central meridian that grows the further it has closed, so by 55° Goode’s map already has 10.6° of deformation on each lobe’s meridian. The rescaled cap starting at 51.7° has only just begun to close there, and carries 2.6. At 60° the two are 16.3° and 8.3; at 70°, 31.5 and 23.7. Because Goode’s lobes put northern land near their meridians — Scandinavia and European Russia near 30° E, the Canadian prairies and the lands round Hudson Bay near 100° W — the land north of 50° is exactly where a cap with less distortion on its meridians pays, and it pays 1.07° of the land mean.

The map shows the trade geographically: a band of loss between the two parallels across North America, Europe and Asia, and gain above it from Alaska to Siberia. The southern continents move very little, and slightly in the rejoined map’s favour, because each composite takes the proportions that suit its whole and the rejoined one takes proportions a shade closer to the sinusoidal’s, which suit the tropics. By land area, 71 per cent of the continents outside Antarctica are drawn better by the rejoined map, most of them by very little.

So the land does not want the cap moved up because the sinusoidal is better in the forties. It wants the cap moved up because a Mollweide’s own meridian is its weakness, and the later the cap starts, the less of that weakness reaches the high northern land that Goode’s lobes centre on their meridians.

Goode’s objective and Goode’s join #

It would be neat if Goode had found by eye the optimum of an objective nobody had written down, and it would be neat the other way if his join were arbitrary. Neither is quite what the land says.

The objective is the one his design answers to — the land outside Antarctica, each point read from the lobe he put it in — and on that objective the kind of map he built, sinusoidal low and closing high, is right: the smooth curve fitted with no instruction about joins takes that shape of its own accord. The join he chose was fixed by a different requirement, that the two maps meet with no step in any parallel and no rescaling of either, and that requirement lands at 40.7° because of the two maps’ own geometry, which knows nothing about continents. The land wants the same construction with the knee at about fifty-two degrees, at the latitude of London, Kyiv and Calgary rather than of Madrid, Beijing and Philadelphia.

The cost of the geometric join is 0.22° of mean deformation over the land, about a quarter of what the homolosine gains on the plain sinusoidal. It is not large. It is the price of drawing the map from two named projections at their natural scales rather than from an objective — and a projection defined by a table has an interpolation in it is the reminder that in 1923 a map was drawn from named projections and tables, not from a numerical optimisation over a coastline raster.

How the joins were checked #

Joined at Goode’s latitude, the composite must be the homolosine. Its land score must reproduce the homolosine’s from the land comparison exactly, since it is the same map; it does, 16.512° against 16.512.

Joined at the pole, it must be the sinusoidal. With no cap left the composite is the plain sinusoidal and must score as it, to a hundredth of a degree; it scores 17.341 against 17.341.

A fit of more terms must do no worse. Every series of n terms contains every series of n − 1, so the fitted land scores must not rise with the order: 17.10°, 16.93, 16.08.

The finding must be there to be found. The land’s best join must lie away from Goode’s by more than three degrees, or there is nothing to report; it lies at 51.7°, eleven degrees higher.

Where the join stops #

The rescaled cap is one construction of many. Rejoining at another latitude here rescales the Mollweide so the parallels stay continuous; at Goode’s own latitude no rescaling is needed, which is why only Eckert II’s family can be the strip could smooth his seam with straight meridians, and at 51.7° the meridians still turn a corner at the join, where the spacing’s slope jumps. A composite could instead be joined through a splice strip, as an equal-area strip removes the corner and charges nothing built for Goode’s own join, or with a different map above the knee. Each would put its best join somewhere, and nothing here says they agree.

One coastline, one lobe layout, one measure. The land is Natural Earth’s 1:110m raster; the lobes are Goode’s published ones; the measure is mean angular deformation. Moving the lobes changes which land is near a meridian, and a different layout would move the best join with it. A different measure — the worst deformation over the land rather than its mean — would weight the high northern land far more heavily, and would push the join the other way; the average was a choice of norm is the reminder that the mean is a choice as well.

The fitted curve is a local optimum of its descent. It is the best of five starts, including the homolosine’s own spacing, refined by coordinate descent. A fourth term would do at least as well; nothing here says how much better.

Antarctica decides. Counted as land, it removes the interior optimum and sends the best join to the pole, where there is no cap. Every number above is for the land outside it, which is the inhabited world Goode drew for, and it is a choice.

Still open: whether the land also wants different lobes #

The join is one of the homolosine’s free choices, and the land moved it eleven degrees. The other is the lobe layout — where the cuts go and where each lobe’s meridian sits — and the land’s score depends on it at least as much: the earlier essay found Goode’s meridians worth 1.63° to his own map, eight times what moving the join is worth.

A layout and a join chosen together, on the land’s objective, might land on something close to Goode’s or somewhere else entirely; the cuts could move to put Greenland or New Zealand on a lobe’s meridian, or a sixth lobe could split Asia. Whether the land’s best layout keeps Goode’s two northern lobes, whether the best join moves again once the lobes are free, and how much of the 1.63° his meridians earned he left on the table, are questions a sweep of one parameter cannot ask.