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

Sinusoidal — where it appears

The equal-area pseudocylindrical whose parallels are true to scale and whose meridians are sine curves. Every point's east–west scale is therefore exactly its ground value, and the shape error depends only on distance from the central meridian, which is why interruption helps this projection more than any other.

Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.

What a pole line buys, and what it costs. four pseudocylindrical projections placed by the two numbers the decision moves. Across the bottom, the angular deformation averaged over the whole sphere, where the pole-line projections — Eckert IV and Robinson — are the better maps. Up the side, on a log axis, the factor by which the last parallel is stretched, where they are worse by more than tenfold: 49× at best against 4.6× at worst for a pole drawn as a point. A pole line represents one point of the globe by a line of the map, and that is the price.

Where a pseudocylindrical puts its error

Every projection in this family has to decide what to do at the pole, which on the globe is a point where every meridian meets. Drawing it as a point and drawing it as a line are the two answers, and the trade between them is measurable in both directions.

families · Families What a cut buys. The mean angular deformation of the interrupted sinusoidal against the total length of cut the interruption spends, for lobe counts from one to twenty-four. Goode's interruption — the one actually printed — is the marked point: it spends 100 thousand kilometres and returns 18.0°, where the even-lobed curve returns 9.2° for the same length. It is not on the frontier and it was never trying to be: its cuts are placed to keep continents whole.

What a cut buys

Cuts are usually counted and never measured. A cut is a curve on the sphere with a length in kilometres, the shape distortion it removes is a falling function of that length, and the interruption everybody prints spends a hundred thousand kilometres to reach a figure the even-lobed curve reaches with sixty.

impossibility · Topology One map made of two, and the latitude where they meet. Goode's homolosine drawn without its interruptions: sinusoidal between 40°44′11.98″ north and south and Mollweide beyond, joined at the latitude where a parallel has the same length in both. Dashed: the join. Heavy: four meridians, drawn from 60° south to 60° north, which cross it. The central meridian crosses smoothly; the others turn a corner there, reaching 13.30° at 110° of longitude, marked. The map is equal-area everywhere, and every parallel keeps its length across the join.

Joined where the parallels agree, and the meridians turn a corner

Goode's homolosine is two projections joined at 40°44′11.98″, and three quantities match there: the length of the parallel, the areal factor, and the spacing of the parallels on the central meridian. The fourth does not. Every meridian but the central one turns a corner at the join, up to 13.30° at 110° of longitude, and the angular deformation drops 26° across a line of no width. Moving the join decides where that corner is paid, not whether.

families · Families The same composite with a strip spliced in, and the meridians no longer turn a corner. Goode's composite redrawn with an equal-area strip of 8° of latitude spliced between the sinusoidal and the Mollweide cap, shaded. The strip's parallel half-length is the unique cubic matching the sinusoidal's value and slope at 36.74° and the Mollweide's at 44.74°, so every meridian arrives and leaves along one direction and the 11.2° corner the plain join turns at 60° of longitude is gone. Nothing else about the map is changed: the cap is the unmodified Mollweide, and the composite is equal-area throughout.

An equal-area strip removes the corner and charges nothing for it

Goode's seam matches three quantities and breaks the fourth: every meridian but the central one turns a corner there. A third map spliced between the two removes it — four conditions on one function, satisfied uniquely by a cubic — and the expected price does not arrive. The spliced composite carries less angular deformation across the seam than the plain join, not more: 34.7° against 37.7° at sixty degrees of longitude. What it does leave is the same defect one derivative up, a jump in curvature falling as the reciprocal of the strip's width.

families · Families Goode's composite with a strip of a generalised Eckert II, whose meridians are straight. The sinusoidal to 32.28°, a strip of the generalised Eckert II projection with r = 0.8311 from there to 49.27°, and the unscaled Mollweide beyond, both hemispheres; the strip's edges are dashed. Inside the strip every meridian is a straight line, as Eckert II's are, and it leaves the sinusoidal's meridian and joins the Mollweide's along their own tangents, so the 11.9° corner Goode's join turns at 180° of longitude is gone. The composite is equal-area throughout, and its strip is 17.0° wide and centred at 40.78°, four hundredths of a degree from Goode's own join.

Only Eckert II's family can be the strip, and its meridians are straight

The cubic that smooths Goode's seam is four conditions on the length of a parallel, and equal area turns them into four conditions on how the parallels are spaced. The sinusoidal's spacing is flat, so a strip can leave it only where its own spacing is momentarily flat — and of eleven named equal-area maps, only Eckert II's spacing ever turns over. One member of its family meets all four conditions exactly, over seventeen degrees centred four hundredths of a degree from Goode's join, with every meridian straight. The cubic over the same span is the same map to four parts in a hundred thousand.

families · Families The curve that does the whole job has no kink and does not turn over. The parallel spacing along the central meridian, as a share of the equator's, for the best single smooth curve found and for three maps it is being compared with. Goode's composite is flat to its join at 40.7° and then follows the Mollweide's, and the kink where the two meet is the join every essay before this one has been about. The fitted curve has no kink, and it is not the shape the question expected either: it is flat only through the first fifteen degrees (97.2 per cent at 15°), then closes monotonically and faster than the Mollweide — 54.3 against 77.3 at 60° — and it never turns over.

One curve beats the three maps it would replace

Goode's composite is a sinusoidal, a Mollweide and now a splice strip between them, and the question left open was whether one smooth spacing curve could do the whole job or whether the joins are doing work no single curve can. One term does it: a curve with a single coefficient carries 6.8 degrees less angular deformation than the composite, two beat every named equal-area pseudocylindrical in the library, and the composite turns out to be worse than the Mollweide it is half made of.

families · Families 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 level is flat through the family, climbs over its last twentieth, and jumps at the cylinder. Each member's pass depth averaged over its 36 regions. From m = 0.2 to 0.9 it stays between 0.0224 and 0.0450 with no trend. From 0.95 it rises at every step — 0.0436, 0.0844, 0.0919, 0.1082, 0.1235, 0.1388 — a factor of 3.2 to the sinusoidal. And the cylinder at m = 0 stands at 0.2506, 8.5 times its nearest neighbour: a cylindrical map's pattern slides unchanged when it is turned about the Earth's axis, and no other member's does.

The sinusoidal is the edge of its family, not outside it

The sinusoidal is the last member of a family of equal-area maps that begins at Lambert's cylindrical, and it was thought to be the one projection whose pass depth has no curve. Along the family the curve comes and goes: present at m = 0 and 0.8, absent at 0.6 and 0.9. What does move smoothly is the pass depth's level. It is flat from m = 0.2 to 0.9, then rises at every step over the last twentieth, 3.2-fold, to the sinusoid — whose high ridges are the end of that rise, not an exception to the family — while the cylinder at m = 0 stands eight times above its neighbour.

choosing · Choosing 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 Goode's lobes and the lobes the land would choose, at his own count of two north and four south. The land in two-degree cells, Antarctica pale because it is not scored, with Goode's cuts dashed and the land's own solid; a short tick at the equator marks each lobe's central meridian. Goode's: edge at 180°, northern cut at 40°W, southern cuts at 100°W, 20°W and 80°E, meridians 100°W and 30°E north and 160°W, 60°W, 20°E, 140°E south — 16.51° of mean angular deformation on the land outside Antarctica. The land's, with no cut crossing more land than Goode's worst: edge at 20°W, northern cut at 165°E, southern cuts at 70°E, 155°E, 175°W, meridians 65°E and 100°W north and 25°E, 135°E, 175°E, 60°W south — 14.48°.

The land would move Goode's lobes before his join

Goode chose his lobes by eye: where the cuts run and where each lobe's meridian sits. Scored on the land outside Antarctica and found exactly on a five-degree grid, the land's own layout at his count of two lobes north and four south deforms it by 14.48° against his 16.51° — two degrees, more than his meridians were worth in the first place — while its cuts cross less land than his: 2,057 kilometres against 2,391, the worst of them 1,056 against the 1,946 his northern cut spends on Greenland. The land wants the map's edge in the Atlantic and its northern cut in the Pacific. His fourth southern lobe is worth four hundredths of a degree, and once the lobes are the land's the best join climbs again, to sixty degrees.

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.

Equal-areaPseudocylindricalAngular deformationGoode homolosineVerificationInterruptionOptimisationPurposeContinuityLobeProjection familySeam

All concepts