Families — the series
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Cylinders, cones and planes
The standard taxonomy sorts projections by the shape of paper they were notionally rolled from. It is memorable, it is how everyone is taught, and it says almost nothing about the properties anyone actually chooses on.
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What a standard parallel buys
A standard parallel is a line where the projection is exact. Choosing one does not reduce the distortion — it decides where the distortion is zero and lets everything grow away from it.
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The aspect is a free choice
A projection's distortion pattern is fixed relative to its own axis, and where that axis points is entirely up to the cartographer. Rotating it is the cheapest available improvement and it is the one most often left unmade.
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The projections that gave up being one thing
The polyconic is built from a different cone for every parallel, which means it is built from no cone at all. It preserves nothing the usual tests look for, it has an exact property neither of them measures, and its sheets do not fit together — a defect discovered in the field rather than at the drawing board.
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The conic is the whole family
Cylindrical and azimuthal are usually presented as two of three families beside the conic. They are the two ends of it. One parameter runs from the cylinder to the plane, and both limits are exact rather than suggestive — which is measurable, and measured here.
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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.
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The azimuthal family is one function
Five azimuthal projections are taught as five viewpoints — from the centre, from the far pole, from infinity. They are one projection with five choices of how far out to draw a point at angular distance ρ, and each named property is a differential equation in that one function: f′ = f/sin ρ integrates to the stereographic projection and f f′ = sin ρ to Lambert's, from nothing but true scale at the centre.
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The condition does not always decide the map
Write a family as a shape with an unknown function in it and every classical property becomes a differential equation. In three families the equation has one solution and the named projection is what comes back. In the fourth it has a whole function of solutions, which is why that family has forty members and the others have three.
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The developable surface was never necessary
Eight essays on this ladder start from cylinders, cones and planes, and none of them has asked which named projection is actually a projection onto one. Fitting a one-parameter perspective construction to eleven of them: six come back exact, and the three landmarks are the gnomonic at h = 0, the stereographic at h = 1 and the orthographic at h = ∞.
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The family is a symmetry, not a shape
Rung seven dissolves the taxonomy: eleven named maps fitted to one perspective formula, six reproduced exactly and five refused, so the developable surface describes how the projections were discovered rather than what they are. What is left is the question it does not ask. If the shapes are not the classification, what is?
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A projection defined by a table has an interpolation in it
One member of this library has no formula: Robinson set nineteen pairs of numbers by eye and the table is the definition. Three published interpolations of it draw graticules that differ by four parts in a thousand of the map's span and report angular deformations that differ by 8.4 degrees.
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A family is not closed under averaging
A family is a set of maps with a parameter running through it, and the compromise projections here are averages of members. An average of two members is not a member: two conics far apart average to something 31 per cent of their own separation outside the family — and averaging within a family makes the map worse every time, while averaging across two makes it 16 per cent better.
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A family is a function, not a list
The equal-area pseudocylindricals are the solutions of one equation in two unknown functions, so one function is free and the named members are points in a space of them. Writing that function as six numbers and searching over them finds a map at 25.03° of mean angular deformation against Eckert IV's 28.08 — and two numbers are already enough to beat every map the family has a name for.
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The maps with no family are simply better
Seven projections have no continuous symmetry, and they are almost exactly the set anybody would choose for a world map. That coincidence came with a conjecture and a test attached: their advantage should collapse under a criterion that does not care where anything is. Run, it does the opposite — 1.343 times under a uniform weighting and 1.204 under a concentration. The conjecture is refuted.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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The ocean centres its lobes on the land's cuts, and cannot keep its own cuts short
Run on the water instead of the land, the same exact search gives an interrupted homolosine whose lobes sit on the Pacific, the Atlantic and the Indian Ocean — five of its six central meridians within twenty degrees of a cut the land's own layout chose — and it deforms the water by 13.31° against the 19.75° the land's layout gives it. The inversion stops at the cuts. Every meridian on Earth crosses at least 7,172 kilometres of water, so the ocean's map has no layout whose cuts stay under seven thousand, where the land's went below nine hundred. And no layout serves both: none comes within 2.57° of both optima at once, though a single wide Pacific lobe buys the water 2.8° for a tenth of a degree of land.
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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.
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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.