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

Joined where the parallels agree, and the meridians turn a corner took Goode’s homolosine apart at its seam. Four quantities could be continuous at the join latitude and three are — the length of the parallel, the areal factor, the spacing of the parallels on the central meridian — and the fourth is not. Every meridian but the central one arrives along one slope and leaves along another, turning up to 13.3° at 110° east. Moving the join does not shrink the corner; it only moves where it is paid.

That measurement ended by naming the obvious repair and doubting it. A third map could be inserted — a strip of some width of latitude, equal-area, matching the sinusoidal at its lower edge and the Mollweide at its upper one in position and in slope. Four conditions where the plain join satisfies two. Whether such a strip exists, how wide it has to be, and how much angular deformation it carries compared with the maps it splices, were the questions.

It exists, it is unique, it can be one degree wide or thirty, and it carries less angular deformation than the maps it replaces.

Four conditions, not six #

The count is where the doubt was, and the count is smaller than it looks.

A pseudocylindrical is two functions of latitude: the half-length of the parallel $a(\varphi)$, so that $x = \lambda\,a(\varphi)$, and the height of the parallel $y(\varphi)$. Equal area forces

$$a(\varphi)\,y'(\varphi) = \cos\varphi$$

so $y$ is not free at all — it is the integral of $\cos\varphi / a$, determined by $a$ up to one constant, and that constant is a translation the composite may choose at each edge independently.

That is the count three conditions are one too many is about, arriving with the opposite sign. There a construction with two free numbers was asked for three exact conditions and the third had no freedom left to be satisfied with; here a construction is asked for six and two of them turn out to be free, because equal area has already spent them. A condition that is implied is not a condition, and counting the implied ones is how a repair comes to look impossible when it is not.

So the two conditions on $y$ vanish. What is left is four conditions on $a$: its value and its derivative at each edge.

And matching $a'$ at an edge matches the meridian’s slope there for every longitude at once. The meridian at longitude $\lambda$ is the curve $(\lambda a(\varphi),\, y(\varphi))$, so its tangent is $(\lambda a', y')$; if $a$ and $a'$ both match then $y'$ matches too, and the tangent direction matches for every $\lambda$. A corner that looked like thirty-six separate failures, one per drawn meridian, is one condition.

Four conditions on a cubic is a unique answer. There is no existence question and no search.

That is worth dwelling on because it is unusual in this subject. The nearest equal-area map to an impossible request and the fits beside it are searches over a truncated series with a residual to report; the maps with no family are simply better measures a function space with a dimension. Here there is no residual and no dimension to count: four numbers, four conditions, one cubic, exact. What remains to be measured is not whether the strip works but what it does to the map, and those are different questions.

Drawn as one function the whole construction is obvious, and so is what the corner was. Two curves cross at Goode’s latitude at an angle; the composite follows one and then the other; the kink in $a$ is the kink in every meridian. The strip replaces the crossing with a piece of cubic that leaves along one tangent and arrives along the other.

Why the width is free, and what that means #

The plain join has no width to choose: it happens at one latitude, the one where the two parallel lengths agree, and moving it costs a rescaling of the cap. The strip has a width and the width is genuinely free — any strip from a fraction of a degree to thirty degrees satisfies all four conditions exactly, with the Mollweide left unscaled in every case.

That is a change in the kind of object the composite is. Goode’s map is determined: two named maps and the one latitude at which they meet without adjustment. A spliced composite is a one-parameter family, and somebody has to pick the parameter.

The freedom comes from where the conditions are imposed. Four conditions at two edges determine the cubic given the edges; the edges themselves are two more numbers, and only their difference matters once the centre is fixed at Goode’s latitude. So the strip has one parameter left over, and the rest of this is about what that parameter buys.

The price that does not arrive #

The expected trade is that a strip bending hard between two stated slopes will be a bad map inside its own width — shear where there was none, bought to remove a kink.

It does not happen. The spliced composite’s worst angular deformation across the seam is 34.70° against the plain join’s 37.73° at sixty degrees of longitude — three degrees better, not worse.

A family is not closed under averaging is the standing warning against expecting the thing between two maps to behave like either of them, and it applies here in the reader’s favour rather than against it. The reason is visible once the two curves are beside each other. Just above Goode’s join the Mollweide’s own angular deformation is higher than the sinusoidal’s just below it: the corner in the meridian is exactly a jump in the local shear, and the map on the poleward side of the seam starts worse. The strip does not interpolate between two good maps; it interpolates between a good one and a worse one, and it spends its width in the region where the worse one was.

There is a best width and it is about eight degrees, and the shape of the curve says why. A one-degree strip has too little room to change anything and recovers a third of a degree. A thirty-degree strip is so wide that the worst point in the window is out beyond its edges, where the ordinary Mollweide is doing what it always did, and the gain is given back.

Where the seam’s own quality sits among the map’s #

One more comparison places the result. Across the whole composite the worst angular deformation is at 180° of longitude near the equator, where a pseudocylindrical’s shear approaches a right angle and nothing about the seam is relevant. What the strip changes is a band a few degrees wide in the middle latitudes — precisely where a world map carries most of the land a reader is looking at.

So the three degrees recovered at sixty degrees of longitude are not three degrees off the map’s worst case. They are three degrees off a region that was locally worse than its neighbours for a reason — the corner — and the repair removes the reason and the symptom together. What a cut buys makes the general point that a seam has a length and a price measured in the quantity it was introduced to save; this is the same accounting for a seam that is not cut at all.

What it does cost #

Something has to be left broken, and it is.

A cubic pinned by four numbers has no freedom left for its second derivative, so $a''$ jumps at both edges. The meridian is smooth — it arrives and leaves along one direction — and its curvature is not: the line bends at one rate up to the strip’s edge and at another just inside it.

That is the same defect the plain join had, moved one derivative up, and the width sets its size. A one-degree strip trades an 11.2° corner for a curvature jump of twenty-two-fold; an eight-degree strip trades it for 2.8; a thirty-degree strip for 0.78, which is less than the curvature already present.

Neither quantity is an objective anybody has written down for a composite. Every projection minimises something catalogues what cartographers have chosen to minimise, and a seam’s curvature jump is not on the list — which is the ordinary situation for a defect that has only just been given a number.

So the two costs point the same way up to about eight degrees and then part. Below that, a wider strip is better on both counts and there is nothing to decide. Above it, a wider strip goes on smoothing the curvature and gives back the deformation it had gained, which is a trade — and a mild one, since the deformation returns only to the value the plain join already had.

Whether a reader would see the curvature jump at all is another matter. A twenty-two-fold change in the rate at which a meridian bends is visible on a drawn map at a one-degree strip; a factor of 2.8 at eight degrees is not obviously so, and nothing here measures what a reader notices.

The corner and the curvature are the same defect, counted twice #

It is worth naming what has actually happened, because “the corner is removed and a curvature jump appears” can sound like a shell game.

A join between two maps is a matching problem in derivatives. Match nothing and the map tears. Match the value of $a$ and the map is continuous but kinked — which is what a naive join at an arbitrary latitude gives. Match the value and the slope and the map is smooth with a curvature jump, which is the strip. Match value, slope and curvature and it is smooth with a jump one further up, and so on.

Goode’s join is not the first of those but a special case of the second-from-worst: it matches $a$ at the one latitude where the two maps’ values happen to agree, which is why it needs no rescaling there, and it cannot match $a'$ because it has no freedom to. The strip supplies the freedom.

The exact map says the seam is smooth meets the same sequence of derivatives on a polyhedral net, where a seam that looked like a corner turned out to be smooth and the measured corner was an artefact of a finite span. The difference is instructive: there the smoothness was a property of the exact map and the corner was the instrument’s; here the corner is real and the smoothness is bought.

What a printed atlas would notice #

Goode’s homolosine is a printed map with a long history of being printed, so it is fair to ask whether any of this would show.

The corner is 11.2° at sixty degrees of longitude and 13.3° at its worst. On a page where a meridian is drawn as a line a fraction of a millimetre wide, a 13° change of direction at a point is visible if one looks for it and easy to miss if one does not — which is presumably why the seam is usually described by the quantity that is continuous there rather than the one that is not.

What a reader would notice more readily is the shape of the land near the seam, and that is the angular deformation rather than the meridian’s direction. Three degrees off 37.7 is an eight per cent improvement in the local shear, which is the kind of change that shows as a slightly less slanted coastline rather than as anything nameable.

So the honest summary of the repair is modest: it removes a defect that is real and rarely noticed, it improves a quantity that is noticed and by a small amount, and it charges a curvature jump nobody has a name for. That it charges nothing in the map’s own currency is the surprise; that the surprise is small is not a reason to misreport it.

What each number was checked against #

A strip between two copies of the same map must be that map. Given both edges on the sinusoidal, the cubic through its value and slope at two points must reproduce it; it departs by under 10⁻⁶ over the whole strip. A construction that bent there would be bending everywhere, and every number here would be measuring the interpolation rather than the join.

The strip must be equal-area everywhere inside it, which it is to 10⁻¹² — by construction, since $y$ is defined as the integral of $\cos\varphi/a$, but the identity is checked rather than assumed.

The meridian must actually be smooth. The strip’s value and slope must match the map on each side to rounding: they agree to 10⁻¹² in value and 10⁻⁹ in slope at both edges.

It must not fold. The parallel’s half-length must stay positive throughout, or the map covers part of itself; its smallest value is 0.73 at an eight-degree strip and 0.63 at thirty.

And the cost must rise as the strip narrows, or the width is free and there was no trade to find. The curvature jump goes 0.78, 2.8, 22.4 as the width goes 30°, 8°, 1°.

What one strip does not settle #

The strip is centred on Goode’s latitude. Every measurement takes the strip symmetric about the latitude the plain join uses. An asymmetric strip — more of it on the Mollweide side, say, where the deformation is worse — is a second parameter this does not explore, and the fact that widening helps up to eight degrees suggests it might not be neutral.

A cubic is a choice. Four conditions could be met by any function with four free parameters, and a cubic is the simplest. A quintic matching the second derivative too would remove the curvature jump and leave a jump in the third, at whatever cost in deformation its extra bending brings — which is not measured here and is the obvious next thing.

The comparison is against the plain join and not against the best available map. Three degrees better than Goode’s seam is not three degrees better than anything else: the maps with no family are simply better records that unfamilied compromise maps beat the named families on several scores, and none of that is disturbed here. This measures a repair to one map, not a claim about the field.

Angular deformation is measured along meridians at stated longitudes. The figures take 60° and 150°; the worst over the whole map is at 180°, where a pseudocylindrical’s shear approaches a right angle and every map in the comparison is equally bad. The strip’s advantage is real in the middle longitudes and vanishes at the edge, where nothing could help.

Only one join latitude was tried. The strip is centred on Goode’s, which is where the plain join must be if the cap is not to be rescaled. A strip centred elsewhere with the cap rescaled is the generalisation joined where the parallels agree swept for the plain join, and it is not swept here.

The cap is unscaled. Every measurement here keeps the Mollweide exactly as it is, which is possible because the strip absorbs the mismatch. A strip that also rescaled the cap has one more freedom and is a different construction.

The best width is a best for one longitude. Eight degrees is where the worst deformation across the seam is lowest at 60° of longitude. At 30° and at 120° the curve has the same shape and its minimum is not at exactly the same width, so “about eight degrees” is the honest statement and not “eight”.

And the composite is not interrupted. Goode’s map as printed is cut into lobes, and the interruptions are a separate matter from the seam. Splicing changes nothing about where the cuts go.

Still open: whether the strip should be a piece of a known map at all #

The strip here is a cubic in the parallel’s half-length, chosen because four conditions and four parameters is the smallest thing that works. It is not a piece of any named projection, and the composite is therefore three maps rather than two — which is a real change in what the object is, and not obviously an improvement on a map named for the two it is made of.

The alternative is to ask which named equal-area pseudocylindrical, if any, leaves the sinusoidal along its own tangent and arrives at the Mollweide along the Mollweide’s. Each such family has a parameter or two, and matching four conditions with two parameters is over-determined — so either the answer is a family nobody has written down, or it is none.

Which named families come close, how far from an exact match the nearest gets, and whether a two-parameter family exists that meets all four conditions at some width, are questions a cubic answers by making them unnecessary and does not answer.