Concept

Lobe — where it appears

One piece of an interrupted map: a strip of the sphere drawn about its own central meridian and separated from its neighbours by a cut. Every point in a lobe is measured from that meridian rather than from the sheet's, which is the whole of what interruption changes.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Sinusoidal, cut into 6 lobes. six lobes, cut through the oceans so each continent stays whole. Each lobe is the same projection about its own central meridian, so each point is near a line where the shape distortion vanishes: the mean angular deformation over the mapped world falls from 38.6° uninterrupted to 17.8° here. What is given up is that the map is no longer one surface — the graticule stops at every cut because there is nothing on the other side of it.

Giving up continuity

Cutting a map into lobes really does reduce the distortion, by a factor that can be measured. What is paid is that the map stops being one surface — and the size of the tear is the number that pictures of interrupted maps never carry.

choosing · Choosing

Cut into Goode's lobes, the ranking of uncut sheets turns over. Mean angular deformation over the sphere for eight equal-area pseudocylindricals, each read two ways: as one uncut sheet, and cut into Goode's lobes with every point measured from its own lobe's central meridian. Uncut, Eckert IV is the best named map at 27.70° and the one-term curve fitted to the uncut sheet beats Goode's composite by 6.83°. Cut, Eckert IV falls to 21.34°, second from last, the curve fitted uncut reads 20.77° against the composite's 18.48, and the plain sinusoidal beats the composite at 17.88°. A curve refitted for the cut sheet reaches 17.01°.

Scored lobe by lobe, the homolosine loses to its own lower half

Read as one uncut sheet, a single smooth curve beats Goode's composite by 9.6 degrees. But the homolosine is used cut into six lobes, so here it is scored cut, with each point measured from its own lobe's meridian, and the ranking turns over. Eckert IV, the best named map uncut, falls to second from last. The curve that beat the composite is now 2.3° worse than it. The plain sinusoidal, the composite's own lower half, wins at 17.88° against 18.48. The best smooth curve for a cut map is the sinusoidal with a few per cent of correction.

families · Families

Where on the land the homolosine beats the sinusoidal, read from Goode's own meridians. The land of the world outside Antarctica on a plain latitude–longitude grid, in two-degree cells, each shaded by how much less angular deformation Goode's homolosine gives it than the plain sinusoidal does, both cut into Goode's lobes and each point read from its own lobe's central meridian (dashed); solid lines are the cuts, and the dotted parallels are Goode's join at 40.7°. The key names the two inks: one where the homolosine wins, the other where the sinusoidal does, deeper for a larger difference. Weighted by land, the homolosine averages 16.51° and the sinusoidal 17.34°. Antarctica, drawn pale, is not in this score.

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

Scored over the whole sphere and cut into Goode's lobes, the homolosine loses to the plain sinusoidal; for it to win on latitude alone, a reader would have to care about the band from 32° to 61° forty per cent more than its area. The land does not do that — the band holds 1.03 times its share of the continents. Weighted by land anyway, and each point read from the meridian of the lobe it lies in, the homolosine wins: 16.51° against Craster's 17.00 and the sinusoidal's 17.34. The win is in where Goode put his meridians, which lift his map from third to first; count Antarctica as land and it falls to fourth.

families · Families

The land wants the join at fifty-two degrees; Goode put it at forty-one. The homolosine rejoined at every latitude from 20° to the pole, the Mollweide cap rescaled so each parallel stays continuous, each composite cut into Goode's lobes at its own best proportions. Solid: mean angular deformation over the land outside Antarctica. It falls from 17.33° at 20° to its least, 16.29°, at a join of 51.7°, and rises again to the sinusoidal's 17.34° at the pole; Goode's join at 40.7° scores 16.51°, and every join sampled from 50° to 60° is within a tenth of a degree of the best. Dashed: with Antarctica counted as land, and dotted: over the whole sphere by area — both fall all the way to the pole, where there is no cap at all.

The land puts the homolosine's join eleven degrees higher

Goode joined the sinusoidal to the Mollweide at 40° 44′ because that is where their parallels have the same length — a fact of geometry, not of any objective. On the land his map was drawn for, read from his own lobes, the join has an objective at last, and its best value is 51.7°: a composite joined there gains a fifth of a degree over Goode's, and every join from about 46° to 60° does nearly as well. A smooth curve fitted to the land with no join in it at all stays within three per cent of flat to 50° and then closes, gaining twice as much again. Goode found the right shape and put its knee too low; with Antarctica counted, the land wants no knee at all.

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

One smooth term beats the water's best join; the land needs three. Equal-area spacing curves with no join — the exponential of a cosine series in latitude — fitted to each surface on that surface's own lobes at its own best join, two lobes north and four south, each map at its own best proportions; the gain is over Goode's composite rejoined at the surface's best latitude (zero line). On the water to 88°, against the composite joined at 75° (12.27°): 1 term 12.03°, 2 terms 11.89°, 3 terms 11.73°, 4 terms 11.70°. On the land outside Antarctica, against the composite joined at 60° (13.83°): 1 term 14.31°, 2 terms 14.31°, 3 terms 13.71°.

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

On the land, a smooth equal-area curve with no join needed three terms before it beat Goode's composite rejoined at the land's best latitude, and on the land's own lobes it needs three again. On the water a single term beats the water's best composite, a second and third add more, and the best curve is 0.54 degrees of mean angular deformation better than the composite joined at seventy-five and 1.6 better than Goode's. It is nothing like the composite in shape: its parallels never close to less than seven eighths of the equator's spacing, so the map the water wants is the sinusoidal reshaped by a few per cent at every latitude, not the sinusoidal with a cap. The lobes do not move under it. And where a separate join for each hemisphere was worth three hundredths of a degree, a separate curve for each is worth a quarter of a degree.

families · Families

Named alongside it

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

Angular deformation Equal-area Interruption Goode homolosine Pseudocylindrical Sinusoidal Optimisation Purpose Trade-off Verification Homolosine Weighting

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