A smooth curve beats the water's best join from its first term

The water would join Goode’s map at seventy-five degrees set the latitude of Goode’s join free and scored the composite on the water: the sinusoidal below the join, a rescaled Mollweide cap above it, cut into two lobes north and four south on the water’s own layout. The water’s score fell steadily as the join rose, bottomed out at 75° with 12.27° of mean angular deformation against 13.31° at Goode’s 40° 44′, and stayed within three hundredths of that up to a join at 89°, where the cap is a sliver round the pole and the map is the plain sinusoidal cut into lobes.

That left the cap doing almost nothing, and the essay ended by asking whether the water wants a cap at all — whether a smooth curve, fitted to the water with no join in it, would beat the best composite by as much as it beat the composite on the land.

The comparison has to be made cut. On the uncut sphere one curve beats the three maps it would replace by a wide margin from a single term, but scored lobe by lobe, the homolosine loses to its own lower half found that cutting a map into lobes turns that ranking over: read from each lobe’s own meridian, the curve fitted to the uncut sphere fell behind the composite it had beaten. Every score here is a cut score, on the lobes each surface chooses for itself.

The land’s answer is the one to measure against. The land puts the homolosine’s join eleven degrees higher fitted a smooth curve to the land on Goode’s lobes and found it needed three terms of its series to beat the land’s best composite at all. One or two terms could not hold the spacing flat through the mid-latitudes and then close it, which is the shape the land wanted, and the composite’s single join number did nearly as well as three coefficients.

The water’s answer is different in kind, and the difference is in the shape the water wants.

One term beats the water’s best join #

The curves are the family one curve beats the three maps it would replace introduced: the spacing of the parallels, relative to the equator’s, written as the exponential of a cosine series in latitude. Every such curve is an exactly equal-area pseudocylindrical, its parallels straight and its meridians spaced evenly along each, and the series’ coefficients are the only freedom. Each curve here is fitted by descending on the coefficients from several starts — the plain sinusoidal, the composites projected onto the series, and the fit with one term fewer — with the map’s proportions searched again at every step, and scored exactly as the composites were: every half-degree cell of the surface, weighted by its area, read from the central meridian of the lobe the layout puts it in.

On the water one term is enough. The best one-term curve scores 12.03°, against 12.27° for the composite at the water’s own best join: 0.23° better with a single coefficient, and 1.28° better than Goode’s homolosine at its own join on the water’s lobes. A second term takes it to 11.89°, a third to 11.73°, and a fourth adds only three hundredths more, at 11.70°. The fitted curve with three terms beats the water’s best composite by 0.54° — half again what moving the join from Goode’s latitude to 75° bought.

On the land the earlier pattern holds on the land’s own lobes too. Fitted to the land outside Antarctica, cut into the layout the land chooses at its best join of 60°, one term scores 14.31° and two terms the same, both nearly half a degree worse than the composite’s 13.83°. Three terms reach 13.71°, beating the composite by 0.12°.

So the land’s composite was a remarkably good design and the water’s is not. A composite has one number, the join, and the land’s best curve has the shape a join makes — flat, then closing — so the join gets nearly all of what the curve can. Whatever the water’s curve is doing, a join cannot do it at all.

The water’s curve barely closes #

The shape is the explanation. A composite joined at 75° keeps the sinusoidal’s even spacing all the way to 75° and then closes its parallels steeply through a cap a few degrees wide; Goode’s own closes from 40° 44′. The land’s three-term curve does what a join does by other means, holding within a few per cent of flat through the mid-latitudes and then turning down.

The water’s curve does neither. Its parallels close a little from the equator on — to 96 per cent of the equator’s spacing by 20° and 95 by 30° — hold there through the middle latitudes, still 96 per cent at 60°, and close slowly above that, reaching 93 at 70°, 90 at 80° and 88 at the pole. It closes fastest at 73°, but that fastest is gentle: nowhere does it fall below seven eighths of the equator’s spacing, where the composite’s cap closes to half and Goode’s to a third. The one-term curve is simpler still, falling smoothly to 93 per cent at the pole with nothing in it but one cosine.

A map whose parallels stay that evenly spaced is, to within a tenth, the sinusoidal. That is what the composite at 75° was too, and the reason the join could rise so far: the water wanted the sinusoidal nearly to the pole. What the curve finds is that the water wants the sinusoidal reshaped a few per cent everywhere — spacing pinched slightly through the tropics, held through the middle latitudes, eased toward the poles — and that a join, which can only leave a stretch of latitudes exactly flat and then close the rest, has no way to make any of those small changes.

Where on the water the curve gains #

The gain is not where the earlier essay’s gain was. Moving the join from Goode’s latitude to seventy-five degrees gained everything in the polar water and gave some back just above Goode’s join. The curve, measured against that composite band by band, gains nothing at the poles. It gives back about a tenth of a degree poleward of seventy, where its parallels stay at nine tenths of the equator’s spacing and the composite’s cap closes them, and about as much again between forty and seventy degrees, north and south.

Its whole gain, and more, is in the tropics and subtropics: 0.79° between forty south and forty north, most of it between five and twenty-five degrees north. That is the water a join cannot reach at all. The composite at seventy-five degrees is the plain sinusoidal there, exactly — even spacing, the sinusoidal’s own shear at every longitude from a lobe’s meridian — and no choice of join changes a degree of latitude below the join. The curve pinches the tropics’ spacing by about five per cent and settles on slightly different proportions, and in the wide tropical lobes of the North Pacific and the Atlantic, where much of the water lies far from any lobe’s meridian, that reduces the shear more than it costs anywhere else.

Two narrow bands go the other way, and both are informative. The band just south of the equator gives back a tenth of a degree, because the equatorial water sits where the sinusoidal is already nearly perfect — every lobe’s meridian and the equator itself carry no shear — and a curve that departs from the sinusoidal there has nothing to gain. And the band from 75° to 80°N gives back another tenth: the Arctic basin, which the composite’s cap was the one thing serving, and which the shared curve, holding its parallels at nine tenths of the equator’s spacing to the pole, serves slightly worse. That is the first sign of the split below. The northern water and the southern want different things at high latitude, and a curve shared by both serves neither pole exactly.

The lobes stay where they were #

Every score above puts the curve on lobes that were chosen for a different map, the composite at 75°, and that could flatter the composite or the curve. So the layout was searched again under the three-term curve, exactly as it was searched under each composite — every edge on a five-degree grid, every pair of northern lobes and every four southern ones by exact dynamic programming, each lobe’s central meridian chosen within it — and the curve fitted again on whatever layout came back.

The layout came back unchanged: the edge at 115°E through Siberia and Western Australia, the northern lobes centred on the North Pacific at 175°W and the Atlantic at 30°W, the southern on 170°E, 110°W, 20°W and 70°E, with the southern cuts through the open Pacific, South America and southern Africa. Refitted on it, the curve scores what it scored, to a thousandth. The composite scores 12.27° on that layout because it is that layout’s own.

The earlier essay found the water’s lobes moving between two nearly equal arrangements as the join rose, the order between them decided by details as small as where the cap began. Between the composite at 75° and the curve the difference is larger than any of those — over half a degree — and the lobes do not move at all. A lobe is chosen by where the water lies in longitude; the curve changes how the map treats each latitude, and every latitude is treated the same way in every lobe. A change that is uniform across longitude has no reason to move a cut, and for the water it does not.

Each hemisphere wants a curve of its own #

The earlier essay gave each hemisphere a join of its own and found it worth three hundredths of a degree: the southern water was indifferent to every join from sixty degrees up, and the northern water’s best was not far from the shared one. A curve for each hemisphere is worth seven times as much. Fitted separately, three terms north and three south, sharing one proportion for the whole map and meeting along the equator with parallels of one length, the two curves score 11.47° against the shared curve’s 11.73° — a quarter of a degree, nearly half of everything the shared curve gained over the best composite.

The two curves are not variations on one shape; they are different shapes. The northern water’s curve pinches its spacing to 92 per cent of the equator’s at about thirty-six degrees north and then opens again, to 96 per cent at sixty and 97 at eighty. That is a map that squeezes the latitudes of the northern subtropical oceans and keeps the Arctic nearly as the sinusoidal would draw it. The southern water’s curve does the opposite: it stays within three per cent of flat to fifty degrees south and then closes, to 95 per cent at sixty and 87 at eighty, through the Southern Ocean toward the Antarctic coast.

The shapes follow where each hemisphere’s water is. The northern oceans are widest in the subtropics, narrow between the continents further north and end in one small polar basin; the southern ocean is a continuous ring from forty degrees south to the ice. The northern curve gives its subtropical oceans the compression the shared curve gave the whole tropics, and leaves the Arctic nearly as the sinusoidal draws it. The southern curve leaves its middle latitudes alone and closes through the latitudes the ring of the Southern Ocean occupies.

A join could not have expressed either. A join leaves the latitudes below it exactly as the sinusoidal draws them, so it had nothing to say about the northern subtropics; and the southern water’s preference for closing between fifty and eighty degrees was, for a join, indistinguishable from not closing at all, which is why every southern join from sixty up scored alike. Given a shape rather than a latitude, each hemisphere says what it wants, and what it wants is different.

The map the water wants, drawn whole #

Drawn whole, before any cut, the two maps look almost alike, and that is the point worth seeing. Both are the sinusoidal’s pointed outline, meridians curving to meet at the poles. The composite runs evenly spaced parallels to 75° and then crowds its last few into a narrow cap, so its polar meridians turn a corner at the join that joined where the parallels agree, and the meridians turn a corner described. The curve’s parallels are not evenly spaced anywhere, but the unevenness is a few per cent and the eye does not find it; its meridians have no corner, because the curve has no join.

The curve is also taller for its width. Drawn whole at the proportions that serve the water best, its height is 0.504 of its width, against 0.481 for the composite at 75° and 0.444 for Goode’s own — and 0.500 for the plain sinusoidal at its natural proportions. The composite is squatter almost entirely because of its cap, which packs the last fifteen degrees of latitude into the space of eleven; below the cap it is the sinusoidal at very nearly its own proportions. The curve keeps its polar parallels wider apart and pays for it with nothing visible, and its outline comes out within a per cent of the sinusoid’s own. What the water wants is not a different outline from the sinusoidal’s but a different distribution of the parallels inside it.

What an ocean map can take from this #

Goode’s design was a composite because composites were what could be drawn. A projection in 1923 had to be computed from formulae a draughtsman could tabulate, and two named maps glued at a latitude where their parallels agree is a construction with a table for each half and nothing to compute at the join. A spacing curve fitted to an objective was not a construction anybody could have used, and Goode was choosing for the land, where a composite turns out to be very nearly as good as anything smooth.

For the water that is not so, and the reason is general. A join can do one thing: keep a range of latitudes exactly as the sinusoidal draws them and close the rest. That is the right thing to do when a surface’s area lies where the sinusoidal is good and the cap is needed only beyond it, which is the land’s case. The water’s area lies everywhere, with heavy weight in the Southern Ocean and the Arctic, and the sinusoidal is close to right at every latitude without being exactly right at any. The improvement available is a little everywhere, and only a curve has a little everywhere to give.

For a cartographer drawing an interrupted ocean map, the measurements on this ground now say three things. Cut where the water’s own lobes put the cuts, which the ocean essay found centring on the land’s own cuts, as the land would move Goode’s lobes before his join found the land doing for itself. Join, if a composite is wanted, near seventy-five degrees, which is nearly not joining. And if the projection can be anything equal-area, use a smooth curve fitted to the water, which buys half a degree more than any join with three numbers, and a quarter of a degree beyond that with a curve for each hemisphere — the cut at the equator already makes the two hemispheres separate maps, so the price of letting them differ is paid already.

How the curves were checked #

A curve of more terms contains every curve of fewer. The scores must not rise as terms are added, and each fit is started from the one below it with a zero appended; they fall from 12.03° to 11.70°.

The measure is the join search’s. The composite at 75°, scored by the same arithmetic the curves are scored by, on the layout the join search found, must reproduce that search’s score exactly. It does, to a billionth of a degree.

A layout searched for the curve must serve it no worse, and two curves no worse than one. The layout searched under the three-term curve must score it no worse than the composite’s layout, beyond a thousandth; it is the same layout. The curve for each hemisphere must score no worse than the shared curve it started from; it is 0.25° better.

The finding has to be there to fail. On the water one term must beat the water’s best composite, and on the land one term must not beat the land’s. The first is 12.03° against 12.27°; the second, 14.31° against 13.83°.

The descent must not stall in a valley. Fitted one coefficient at a time, the land’s two-term curve once stopped exactly where the one-term curve was. Every fit now also tries moving pairs of coefficients together when no single one improves, and the two-term fits start from a coarse grid of curves as well; the land’s two-term curve still finds nothing a second term can buy.

Where the curves stop #

Three terms, one family. The curves are cosine series of up to four terms in latitude. The fourth term adds three hundredths, so the series is near its limit for this objective, but a family with a different basis — one able to change faster near a pole — could find more, particularly in the Arctic basin, which the curve treats exactly as it treats every other water at its latitude.

Equal area throughout. Every curve here is an exactly equal-area pseudocylindrical. An ocean map that may trade a little area for less shear is a different problem, with different answers, and a different family of curves.

The water weighted by area to 88°. The Arctic and Southern Ocean count for their area; a map weighted by shipping, by fisheries, or with the ice-covered water left out would want a different curve and very likely a different hemisphere split, as the area weighting was a readership all along found for every weighting.

One proportion for both hemispheres. The curve for each hemisphere shares the map’s height-to-width proportion with the other, as the separate joins did; each hemisphere could be stretched differently too.

Still open: whether one curve can serve both surfaces #

The earlier essays found no single layout that serves the land and the water well at once: weighted between the two, the best layout jumps from a land layout to an ocean layout with nothing between, and with the join free the floor under any compromise rose from 2.57° to 3.30°. A free join helped each surface’s best more than it helped any compromise, because the land and the water wanted different joins.

A curve is freer than a join, and the two surfaces want curves of different kinds — the land’s flat and then closing, the water’s nearly even throughout. Whether a curve fitted to the blended score at each weight narrows the floor or widens it again, whether the jump between land layouts and ocean layouts survives a free curve, and whether the hemisphere split the water wants is wanted by the land as well, are questions a fit to one surface at a time cannot ask.