One curve beats the three maps it would replace

Only Eckert II’s family can be the strip, and its meridians are straight finishes a run of three essays that take Goode’s homolosine apart at its join, find a corner there, and remove it by splicing a third map between the two. The composite that results is three maps rather than two, and that essay’s closing question is whether it needed to be any maps at all — whether a single spacing curve, flat near the equator like the sinusoidal’s, turning over in the middle latitudes and closing towards the pole like the Mollweide’s, could run the whole way with no join in it.

The question has a clean form because an equal-area pseudocylindrical is one function. Fix the spacing of the parallels along the central meridian and the half-length of each parallel follows: their product has to be the cosine of the latitude, which is what makes the areal factor one everywhere. So a curve is a map, and the comparison this essay makes is between curves.

The answer is yes, easily, and the reasons it is easy are more interesting than the answer.

A curve is a map, and a curve has a free constant #

Two things have to be settled before any number can mean anything.

The first is what the curves are drawn from. Writing the spacing as the exponential of a cosine series in latitude keeps it positive, which it must be, and makes it smooth by construction — there is no spline and so nowhere for a join to hide. A curve with no coefficients at all is the sinusoidal; each coefficient added makes it more flexible, and a family is a function, not a list is the essay that makes the same move on the same family and finds two numbers enough to beat every named member of it.

The second is subtler and it changes the numbers a great deal. Equal area is one equation on two functions, so it leaves a whole function free — that is the subject of every equal-area map is every other one — but it also leaves something smaller and easier to miss. Divide the spacing by a constant and multiply every parallel’s half-length by the same constant, and the product is unchanged: the map is still exactly equal-area, and it is a different shape on the page. That constant is the map’s proportions, and it is free.

Every number below is quoted with that constant optimised for each map separately. A comparison that fixed it — at the value the Mollweide’s own construction gives, say — would be reporting the constant rather than the curves, and the differences it would report are larger than the differences between the curves themselves.

One coefficient beats the composite #

The first result is about the composite rather than about the curve, and it is the one worth stating first: joining the sinusoidal to the Mollweide produces a map worse than the Mollweide by itself. The composite carries 35.39 degrees of angular deformation on average over a hemisphere and the Mollweide carries 31.98. A cartographer who wanted an equal-area world map and had both in front of them would be better off with the Mollweide, and better off again with Eckert IV at 27.70.

That is not a criticism of Goode’s map and it is important to say why. The homolosine is not used uninterrupted. Its whole design is to be cut into lobes, so that no piece of land is far from a central meridian and the sinusoidal’s fast-growing deformation never gets the chance to grow. Giving up continuity prices what interruption buys and what a cut buys measures the currency. Read as one sheet, which is how every other map in this comparison is read, the composite is simply not a good map, and nothing in the three essays about its seam had asked.

The second result answers the question as asked. A curve with a single coefficient is already 6.83 degrees better than the three-piece composite, and it has no join anywhere, so the joins are certainly not doing work that no single curve can do. Two coefficients beat every named map in the library. Five reach 25.76 degrees and the fitting has nearly stopped improving, so the whole of what a smooth curve can buy here costs about two numbers.

That the improvement saturates at two or three coefficients is worth noticing on its own. It says the objective is not asking for a wiggly curve: it is asking for a shape, and the shape is describable in about as many numbers as a named projection already carries.

The one-term curve deserves a line to itself, because it is the kind of object this subject usually gives a name to. Its spacing is exp(0.294 cos 2φ) up to the equator’s value, which is one expression, has a closed-form derivative, and reaches every latitude without a case distinction. A cartographer who wanted a better equal-area world map than Goode’s composite and was unwilling to carry a table could use it as written. Nobody has, as far as the library here records, and the reason is presumably that the family has been explored by writing down constructions rather than by fitting curves — which is exactly the argument a family is a function, not a list makes about this same family one level down.

It is not the shape the question guessed #

The curve the question imagined was a composite’s shape without the joins — flat near the equator, turning over somewhere in the middle latitudes, closing towards the pole. The best curve is not that.

It is flat only through the first fifteen degrees, at 97.2 per cent of the equator’s spacing at 15°. Then it closes, monotonically, and faster than anything named: 54.3 per cent at 60° where the Mollweide is at 77.3, and 24.9 per cent at the pole where the composite is at 21.7 and Eckert IV at 1.7. It never turns over at all.

Set beside the named maps the difference is not subtle. At 60° of latitude the fitted curve’s parallels are at 54.3 per cent of the equator’s spacing, Eckert IV’s at 64.0, the Mollweide’s at 77.3 and the composite’s at 85.8. At 75° they are at 33.4, 39.4, 61.2 and 67.9. The fitted curve is closer to Eckert IV than to either half of the composite through most of the range, which is consistent with Eckert IV being the best of the named maps on this measure — but it parts company at the pole, where Eckert IV’s spacing collapses almost to nothing (1.7 per cent) and the fitted curve’s stops at a quarter.

So the guess got the ends approximately right and the middle wrong, and the reason is visible in where the deformation sits.

The fitted curve buys a band and sells both ends. Through the twenties, thirties and forties it is an order of magnitude better than anything else here, dipping to three and a half degrees at 30°, which is nearly conformal for an equal-area map. At the equator it is twice the composite’s and above 55° it is worse than everything.

The reason that wins is the weighting, and the weighting is not a choice: a hemisphere’s area goes as the cosine of the latitude, so the band from 15° to 50° carries about half of it and everything above 60° carries a tenth. A curve that is very good where half the area is and poor where a tenth is will beat a curve that is moderate everywhere, and the composite is the second kind. Its sinusoidal half runs from 16° at 10° of latitude to 50° at 40° — rising fastest exactly through the band that matters — and the join hands over to the Mollweide only after the damage is done.

That also explains why the guess was wrong. A curve shaped to carry the sinusoidal’s behaviour near the equator inherits the sinusoidal’s behaviour at 30° and 40° as well, and that is where the area is. The best curve gives up the equator, which costs almost nothing in weighted terms, in order not to inherit it.

Seen as pictures they are not subtly different maps. The fitted curve is visibly taller for its width, its parallels close up from the start, and its outer meridians bow much further. A reader shown the two would call them different projections rather than variants, which is the honest description: this is not a refinement of Goode’s construction, it is a different map that happens to beat it.

What the average hides #

The curve was fitted on the mean, and it pays for that at the corner. Its worst angular deformation is 160.4 degrees against the composite’s 144.4 and the sinusoidal’s 113.3 — and on a pseudocylindrical the worst is always at the same place, the corner where the last meridian meets the pole, which is a point the reader of a world map looks at rarely and which several maps in the library interrupt away entirely.

The average was a choice of norm is the warning this belongs under, and it applies here without qualification: ranking these maps at the worst case reverses most of the order above. A fit on the worst case would find a different curve, probably closer to the sinusoidal, and would report a different winner. The claim this essay makes is the one its measure supports — that on the area-weighted mean a single smooth curve beats the composite by nearly ten degrees — and not the stronger one it does not.

One comparison does survive the choice of norm, and it is the one the three seam essays were missing. On the mean, Goode’s composite is worse than the Mollweide by 3.41 degrees. At the corner it is better, by 2.8 — 144.4° against 147.2 — which is a smaller margin than the mean’s and in the opposite direction. So a cartographer choosing between the composite and the Mollweide for an uncut sheet has no clear reason to take the composite on either measure, and the join those three essays are about is one they would not be making.

What this does to the three essays about the seam #

It does not undo any of them, and it is worth being exact about why.

Joined where the parallels agree, and the meridians turn a corner found the corner and priced it; the strip essay removed it; the Eckert II essay identified the only named family that could supply the strip exactly. Every one of those is a statement about what happens at a join, and joins are a general feature of interrupted and composite maps rather than a peculiarity of this one. The corner at a seam where two equal-area maps meet with matched parallel lengths is real wherever it occurs.

What this essay adds is that the particular composite those essays used as their worked example is not, considered as a whole map, a good one — so the reason to care about its seam is the interruption rather than the seam’s own arithmetic. That reverses the direction of the interest rather than removing it.

What each number was compared against #

A spacing curve read as a map has to agree with the map. The sinusoidal’s spacing and the Mollweide’s are run through the same arithmetic and their angular deformation compared with what the projection library computes for the projections themselves. They agree, which is the check that the curve-to-map arithmetic is not quietly a different map.

Every curve is equal-area by construction, and it is checked rather than assumed. The areal factor is the product of the spacing and the half-length over the cosine of the latitude, and it is required to be one to within $10^{-9}$ at every latitude on every curve drawn.

A fitted curve must beat the named maps it generalises, or the fit is finding nothing the library did not already have. At three terms it does; at zero terms it is exactly the sinusoidal, which is the control at the other end.

The fits are nested. Each order is seeded from the one below it, because a curve of n terms contains every curve of n − 1 and its answer cannot be worse. Fitted independently they were not monotone — a three-term fit came back above a two-term one — which is a statement about a coordinate descent rather than about the curves, and is fixed rather than reported as a finding.

And the descent tries several step sizes in each direction rather than halving one. The objective has a long shallow floor, and a descent that walks along it instead of across it left a two-term fit at 28.10 degrees when 26.67 was reachable from a start it had already been handed.

Where the comparison is narrower than it sounds #

One measure, one weighting. The mean is area-weighted over a hemisphere and the whole ranking depends on that. Which projection a weighting can make best shows how much of a verdict a weighting decides, and an area weighting is itself a claim about the reader rather than a neutral default.

Uninterrupted. Goode’s map is used interrupted and this comparison is not. What it establishes is that the composite construction is not what makes the homolosine good; what makes it good is the cutting, which this essay does not price and the essays above it do.

Equal-area only. Every curve here satisfies the areal condition exactly, so the comparison is inside one family rather than across the subject. Nothing above says whether a compromise projection would beat all of them, and compromise projections says it probably would.

And the curve has no name. It is a set of coefficients, exactly as a map with no formula records for the solved conformal maps, with the same consequence: no closed-form inverse, and a table or an iteration wherever one is wanted. The one-term member is the exception and is the one worth using: exp(c cos 2φ) integrates to a northing by quadrature and inverts by a single Newton step from any sensible guess, which is no worse than the Mollweide already asks for.

The search is a fit and not a proof. Nothing here establishes that 25.76 degrees is the least a smooth equal-area pseudocylindrical can carry; it is the least the descent found over five coefficients from seven starting points, and a longer series or a different basis could do better. Every claim above is of the form a single curve beats the composite, which needs only the curve exhibited, and none is of the form this is the best curve, which would need something the search does not supply.

Still open: whether the cut is what should be fitted #

Everything above fits a curve and leaves the interruption out, and the interruption is what Goode’s map is for.

That suggests the measurement this pair of results actually asks for. An interrupted map has lobes, and each lobe has a central meridian, and the deformation a reader meets depends on how far the land they are looking at is from one. So the objective the homolosine is really optimising is not the mean over a hemisphere at all: it is the mean over land, with each point measured from its own lobe’s meridian. Under that objective the sinusoidal’s fast growth away from the central meridian is bounded by the lobe width rather than by ninety degrees of longitude, and the whole ranking above could change.

Whether a single smooth curve still beats the composite once both are interrupted, how much of the curve’s advantage survives being cut into six lobes, and whether the best curve for an interrupted map is the same shape as the best curve for a whole one — are questions a comparison of uncut sheets raises and cannot settle.