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

An equal-area strip removes the corner and charges nothing for it spliced a third map between the sinusoidal and the Mollweide at Goode’s seam. Its parallel’s half-length was a cubic, the smallest thing that could meet the four conditions — the half-length and its slope at each edge — and the corner every meridian turned at the plain join was gone. It ended by doubting its own construction. The cubic is not a piece of any projection anybody has named, so the composite is three maps rather than two, and the question it left was whether a named family could supply the strip instead: which come close, how far the nearest gets, and whether any meets all four conditions.

One does. Of eleven named equal-area pseudocylindricals only one could, and the reason is a single property that every named map either has or lacks and that can be read off a picture of its central meridian.

The four conditions are about spacing #

A pseudocylindrical is a parallel half-length $a(\varphi)$ and a parallel height $y(\varphi)$. Equal area forces $a\,y' = \cos\varphi$, so the length of each parallel and the spacing of the parallels along the central meridian are the same information: fix one and the other follows.

Differentiate that relation and the slope of the parallel’s length comes out in terms of the spacing:

$$\frac{a'}{a} = -\tan\varphi - \frac{y''}{y'}.$$

The sinusoidal is the map whose parallels are evenly spaced — $y = \varphi$, $y'' = 0$ — and for it $a'/a = -\tan\varphi$. A strip leaves the sinusoidal along the sinusoidal’s own tangent exactly when its $a$ and $a'$ agree with the sinusoidal’s there, and since the ratio already carries the whole of the condition on the slope, that happens exactly where the strip’s own $y''$ is zero: where the strip’s parallels are, for an instant, evenly spaced.

The other edge reads the same way. The strip meets the Mollweide smoothly where its spacing and the rate its spacing changes both match the Mollweide’s. So the four conditions the cubic satisfied are two conditions on the spacing of the parallels at each edge — its value and its slope — and a named map can supply the strip only if its spacing curve can be scaled and cut to fit them.

The scale is free: stretching a pseudocylindrical along the meridian by some factor and shrinking its parallels by the same factor keeps it equal-area, and changes the spacing’s value without changing where its slope is zero. So matching the length at one edge costs nothing. The slope condition at the sinusoidal’s edge does not come for free. It asks the named map for a latitude where its spacing turns over.

Eleven named maps, and one whose spacing turns over #

The eleven are the equal-area pseudocylindricals with closed forms that the textbooks tabulate. Their spacing curves split cleanly.

Eight close up throughout. From the equator to the pole their parallels crowd together steadily — the Mollweide’s at sixty degrees are 77 per cent as far apart as at the equator, Eckert IV’s 64 per cent. This is the design every one of them was drawn to: a pole shown as a point or a short line, with the high latitudes compressed north–south to pay for the length their parallels keep east–west. None of them has a latitude away from the equator where its spacing stops changing, and so none can leave the sinusoidal smoothly.

Collignon’s opens out. Édouard Collignon’s map of 1865 is the rhombus with straight meridians meeting at a point, and its parallels spread further apart all the way up — 137 per cent at sixty degrees. It fails for the mirror reason.

The sinusoidal’s is flat everywhere, which is what makes it the map being left.

Eckert II’s turns over. Max Eckert published six pseudocylindricals in 1906, and the second is equal-area with straight meridians broken at the equator and a pole half as long as the equator. Its spacing rises from the equator, peaks at 26.83 degrees, and falls to the pole. At that one latitude a scaled piece of Eckert II leaves the sinusoidal along its tangent. No other map of the eleven has such a latitude.

Four families, and the corners they leave #

A single named map has one latitude to offer and no shape to adjust, so it cannot also meet the Mollweide smoothly except by coincidence. A one-parameter family can. Each of four families of named maps has a shape parameter and a free scale, and with the two edges that is four unknowns against four conditions — a count that can close exactly. The doubt that closed the cubic’s essay assumed two parameters against four conditions, and forgot that the strip’s two edges are free as well.

The families are the ones the named maps sit in. The sine family holds the sinusoidal itself and Wagner I; the Mollweide family, where the auxiliary angle solves $2\theta + \sin 2\theta = p\sin\varphi$, holds the Mollweide and Wagner IV; the Eckert VI and Eckert IV families each generalise the constant in their defining equation. Every member of each is equal-area by construction, whatever its parameter.

The spacing argument says in advance what the search finds. The Mollweide, Eckert VI and Eckert IV families close up everywhere at every parameter, so their members cannot be tangent to the sinusoidal off the equator, and the best they manage is a strip that meets the sinusoidal at a corner nearly as large as Goode’s own: 11.9 degrees for the Mollweide family, whose best member is simply the Mollweide itself meeting at Goode’s latitude, and 10.0 for the Eckert VI family. No member of the Eckert IV family, scaled to the sinusoidal’s parallel, meets the Mollweide’s parallel again anywhere inside the window.

The sine family comes closest, and not by accident: it contains the sinusoidal. Its best strip, at m = 0.939 from 25.7 to 51.3 degrees, leaves corners of 4.3 degrees at its lower edge and 3.8 at its upper — a third of Goode’s corner, and still a corner, because every member of the family that is not the sinusoidal itself closes up everywhere and cannot leave the sinusoidal smoothly.

The family that meets all four #

Eckert II’s spacing is $y' \propto \cos\varphi / \sqrt{4 - 3\sin\varphi}$. The natural family around it replaces the 3/4 inside the root with a parameter:

$$y' \propto \frac{\cos\varphi}{\sqrt{1 - r\sin\varphi}}, \qquad a \propto \sqrt{1 - r\sin\varphi},$$

with Eckert II at r = 3/4. Every member is equal-area, and every member keeps Eckert II’s defining feature: $a$ is a linear function of $y$, so each meridian is a straight line.

The tangency condition at the sinusoidal’s edge has a closed form. The member with parameter r turns over, and so leaves the sinusoidal smoothly, at the latitude where

$$r = \frac{2\sin\varphi_1}{1 + \sin^2\varphi_1},$$

which puts Eckert II’s own turn at 26.83 degrees, as measured. The scale then fixes the half-length at that edge, and what remains are two conditions at the Mollweide’s edge — length and slope — in two unknowns: where the strip leaves the sinusoidal and where it meets the Mollweide.

Walking the family up the sinusoidal shows how the solution sits. A member leaving the sinusoidal below 32.28 degrees has parallels that stay longer than the Mollweide’s all the way to the pole: it never meets the cap at all. At 32.28 degrees, the member with r = 0.8311 comes down to the Mollweide’s parallel length at 49.27 degrees and touches it there, the two curves meeting with the same slope. That touching is the fourth condition. Above 32.28 degrees each member crosses the Mollweide’s curve instead of touching it, at a latitude that falls as the lower edge rises, and the corner at the crossing grows — until at Goode’s join the strip has shrunk to nothing and the corner is Goode’s own 11.9 degrees.

So the family does more than contain one smooth strip. It is a one-parameter path from Goode’s plain join, a strip of zero width with an 11.9-degree corner, to the smooth strip seventeen degrees wide, and the corner falls steadily along it.

The solution is exact: the four conditions hold to rounding, the strip’s parallels stay positive, it is equal-area to seven places, and the composite is continuous at both edges. And its centre, halfway between 32.28 and 49.27 degrees, is 40.78 degrees — four hundredths of a degree from 40.74, the latitude where Goode joined the two maps with no strip at all.

The cubic was already this map #

The cubic of the essay that built it has a free width, and at seventeen degrees centred on the same latitudes it meets the same four conditions as the Eckert II strip. Two functions meeting the same four conditions over the same interval need not agree in between. These do, almost exactly: their parallel lengths differ by at most 3.9 parts in a hundred thousand, at 41.2 degrees, and the cubic’s meridian at 180 degrees departs from a straight line by a third of a kilometre across a chord more than three and a half thousand kilometres long.

That is closer than it has any right to be, and it has a reason. A straight meridian means $a$ is linear in $y$, and over a strip the cubic interpolates a function whose ends are pinned in value and slope; if a function with straight meridians satisfies those same end conditions, the cubic is its Hermite interpolant and differs from it only at fourth order in the width. Seventeen degrees is under a third of a radian, and the fourth power of that is less than a hundredth, before the small derivative it multiplies. The cubic at this width was a piece of generalised Eckert II with a third of a kilometre of rounding on its meridians.

The map a reader would see agrees. Along the meridian at sixty degrees of longitude, the plain join’s worst angular deformation inside the strip’s latitudes is 37.59 degrees, the cubic’s 31.251 and the Eckert II strip’s 31.247. The refund the cubic was found to collect is the Eckert II strip’s refund, to four thousandths of a degree.

What naming it changes #

The composite with the cubic was three maps, one of them anonymous. The same composite at a width of seventeen degrees is the sinusoidal, a generalised Eckert II and the Mollweide: three maps with names, or two names and a parameter. That is a small change in bookkeeping and a larger one in description, and it answers the question in both of the forms it was asked.

Which named families come close. Of four one-parameter families around six named maps, the nearest leaves a 4.3-degree corner and the others ten or twelve. None of the four can do better, because every member except the sinusoidal closes up its parallels everywhere.

Whether a family meets all four conditions. One does, around the one named map whose spacing turns over, and it meets them at a single member and a single width. The cubic’s width was free; this strip’s is not. Asking the strip to be a piece of a named family spends the freedom the cubic had and puts the strip at 32.28 to 49.27 degrees, centred almost exactly on Goode’s latitude.

Whether it matters. For the map, hardly: the cubic at that width is the same map to a third of a kilometre. For the argument it is decisive. The cubic looked like an interpolation invented to patch a seam, and it turns out to have been a piece of a projection with straight meridians, one parameter away from a map published in 1906.

A family is a function, not a list found the equal-area pseudocylindricals to be the solutions of one equation with a function free, and named maps as points in that space. The spacing curve is that free function, drawn. Where a pseudocylindrical puts its error measured the trade at the pole between a point and a line, and the spacing curve is where that trade is made: the eight maps that close up are the ones that paid for a short pole with crowded high latitudes. And three conditions are one too many is the counting lesson again from the other side: a construction with the right number of free parameters can still fail when a condition it cannot bend asks for a property it does not have, and the property here was a spacing curve that turns over.

The spacing curve is the map #

Everything above has treated a spacing curve as a way of describing a named map. It is more than a description: for an equal-area pseudocylindrical it is the whole map. Choose any positive spacing along the central meridian and equal area sets the length of every parallel, $a = \cos\varphi / y'$, with nothing left to decide.

That is the same shape of result the condition does not always decide the map found when it wrote each classical property as a differential equation: for three families the equation had one solution, and for the equal-area pseudocylindricals it had a whole function of them. The azimuthal family is one function found the same for azimuthal maps, where the one function is how far out a point at a given distance is drawn. Here the one function is how far apart the parallels are, and every named equal-area pseudocylindrical is a curve drawn in it.

It also makes the space closed under averaging, which is not true of every family. A family is not closed under averaging found that the average of two conics is not a conic. The average of two spacing curves is a spacing curve, so the average of two equal-area pseudocylindricals, taken that way, is another. And averaging can make a turn that neither parent has: an even blend of Collignon’s spacing, which opens out, and the Mollweide’s, which closes up, turns over at 41.1 degrees. A blend three quarters Mollweide turns at 20.0 degrees; three quarters Collignon at 62.0. Eckert II’s spacing is one natural shape in that space, and far from the only one that turns over.

How the claims were checked #

Only the sinusoidal’s spacing may be flat, and only Eckert II’s may turn over. Each of the eleven spacing curves is scanned for a change of sign in its slope between half a degree and eighty-nine and a half. Exactly one turns over, Eckert II’s, and exactly one is flat.

Eckert II must turn over where the closed form says. For r = 3/4 the formula puts the turn at 26.829°, and the scan finds it at the same latitude to a millionth of a degree.

The family containing the sinusoidal must be tangent to it. The sine family at m = 1 must have exactly the sinusoidal’s $a'/a$ at thirty degrees, and does to $10^{-12}$ — the control that the tangency test can say yes.

The Eckert II strip must meet all four conditions and be a map. Length and slope at both edges agree to $10^{-12}$; the strip is equal-area to $10^{-7}$ everywhere across it, its parallels stay positive, and the composite built from it has no gap at either edge.

Where the measurement stops #

Eleven maps are not all maps. They are the equal-area pseudocylindricals with simple closed forms. Others exist — Boggs’s eumorphic, defined as an average of two of these, and a long tail of interrupted and tabulated designs — and a map whose spacing turns over would give a second family to try.

Four conditions, not six. The strip is smooth in the meridian’s direction and not in its curvature: like the cubic, the Eckert II strip leaves a jump in the second derivative of the parallel’s length at each edge, of about the same size as the cubic’s at the same width. Straight meridians inside the strip mean the meridian’s curvature is zero there and not zero on either side.

The cap is the unscaled Mollweide. Every strip here meets the Mollweide as published. A cap rescaled by a few per cent would move every solution, and the plain join’s own latitude depends on exactly that scale.

Still open: whether the whole composite can be one family #

The strip is one member of a family chosen for one latitude band. The sinusoidal on one side is a member of the sine family, and the Mollweide on the other a member of the Mollweide family, and nothing so far has asked whether a single family could run all the way from the equator to the pole — a spacing curve flat near the equator like the sinusoidal’s, turning over in the middle latitudes like Eckert II’s, and closing towards the pole like the Mollweide’s — so that Goode’s composite is not three maps joined but one map with no join at all.

Such a curve would be one continuous function where the composite is three pieces, and the question is what it would cost. Equal area leaves the spacing free, so the curve can be drawn; whether any smooth spacing curve keeps the sinusoidal’s small angular deformation near the equator and the Mollweide’s near the poles, or whether the joins in Goode’s map are doing work that no single smooth curve can, is a question a strip that lives between two latitudes cannot ask.