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

Twelve earlier essays have asked what a family of projections is, and every answer has been a statement about one map at a time: a construction, a condition, a symmetry, a function space. The maps with no family are simply better ended by naming what none of them had touched — what happens when the thing being mapped is not one sphere drawn all at once, but one region joined to another.

That map exists and is in every school atlas. Goode’s homolosine is the sinusoidal projection between about 40°44′ north and south and the Mollweide projection beyond, and it is equal-area throughout because both pieces are. Its cuts through the oceans are a separate matter, priced in giving up continuity; this is about the seam that is there whether or not the map is cut, running right round the map at one latitude, through the middle of the Atlantic and across the Sahara.

Three quantities match at the join #

The join latitude is not chosen for the look of it. The sinusoidal draws the parallel at latitude $\varphi$ with length proportional to $\cos\varphi$, and the Mollweide draws it with length proportional to $(2\sqrt{2}/\pi)\cos\theta$, where $\theta$ is Mollweide’s own auxiliary angle. Those two are equal at exactly one latitude in each hemisphere, and solving for it gives 40°44′11.98″. At that latitude the two pieces can be laid edge to edge with no gap and no overlap, and every parallel crosses the seam without changing length.

Two further continuities come free, and both are worth stating because they are what makes the seam hard to see.

The areal factor is one on both sides. Each piece is equal-area pointwise, so the composite is too — exactly, everywhere, including along the join, where the measured areal factors are 1.000000000000 on the sinusoidal side and 1.000000000000 on the Mollweide side. Nothing about area changes at the seam.

The parallels on the central meridian are as far apart just above the join as just below it. That is not a third coincidence: it follows from the other two. An equal-area map’s meridian scale is fixed once its parallel scale is, because the product of the two is one; equal parallel lengths and equal areas therefore force equal spacing. The measured spacing is 1.000000 on each side.

So a reader running a finger up the central meridian crosses the join without feeling anything, and a reader measuring any country’s area crosses it without finding anything either. A family is not closed under averaging found that averaging two members of a family leaves the family; joining two members, this says, keeps every pointwise property both of them have.

The fourth quantity does not #

What a composite cannot inherit is a property of derivatives across the seam, and the direction of a meridian is one.

A meridian arrives at the join from below with one slope and leaves with another. On the central meridian the two slopes agree — the meridian is vertical on both sides — which is the continuity of spacing seen again. Away from it they do not, because the two pieces reach the join with different amounts of lean: the sinusoidal’s meridian leans by $\lambda\sin\varphi$ and the Mollweide’s by a different expression in $\theta$, and at the join those differ by about a third.

The corner grows to 13.02 degrees at 90 degrees of longitude, peaks at 13.30 degrees at 110 degrees, and falls back to 11.92 at the edge of the map. That it peaks inside rather than at the edge is worth a sentence: the corner is an angle between two directions, and as the longitude grows both directions turn towards the horizontal, so their difference stops growing and eventually shrinks.

Thirteen degrees is not a subtlety. It is a visible kink in a line a reader follows with a finger, and it is at the same latitude all the way round the map. Giving up continuity notes that it is visible in any Goode homolosine once it is looked for, and a projection defined by a table has an interpolation in it records that the homolosine is joined so that the positions agree and the derivatives do not. Neither says how large the corner is, and it is worth having the number, because a kink of a degree would be a curiosity and one of thirteen is a feature of the map.

Why the corner is thirteen degrees, and why moving the join cannot help #

The corner has a closed form, and it is what explains the flat curve in the sweep below.

At the join the two pieces draw a meridian with leans in a fixed ratio. The sinusoidal’s meridian at longitude $\lambda$ leans by $\lambda\sin\varphi$ for every unit of northing; the Mollweide’s leans by its own expression in $\theta$, and at the join the second is $r$ times the first, with

$$r = \frac{2}{\pi}\,s^{2}\,\frac{\tan\theta_j}{\sin\varphi_j} ,$$

where $s$ is whatever scaling the polar piece needed. At Goode’s join, where $s$ is one, $r$ is 0.6260: every meridian arrives leaning by one amount and leaves leaning by five-eighths of it.

A corner between two directions whose tangents stand in a fixed ratio is $\arctan a - \arctan(ra)$, with $a = \lambda\sin\varphi_j$ growing along the join. That is zero at the central meridian, grows, and turns over at $a = 1/\sqrt{r}$, where it reaches

$$90° - 2\arctan\sqrt{r} .$$

For $r = 0.6260$ that is 13.297 degrees, at a longitude of 111 degrees — against a measured 13.297 at 111. The same arithmetic matches the measurement at every join latitude tried: 13.489 at a join of 30°, 13.202 at 45°, 12.775 at 60°, each agreeing to three decimal places with the sweep below, and each peaking where the formula says.

That is also the answer to why moving the join buys nothing. The ratio $r$ is 0.6190 at a join of 20° and 0.6378 at 60° — it barely moves, because raising the join raises $\tan\theta_j$ and $\sin\varphi_j$ together and the scaling $s^2$ pulls the other way. A quantity that depends on $r$ only through $\sqrt{r}$, inside an arctangent, cannot move much either. The corner is not a property of Goode’s choice; it is a property of these two families meeting at all.

One detail the formula supplies that the measurement alone would leave as a curiosity: at a join of 20° the peak is predicted at 213° of longitude, which is off the map. So the corner there is still climbing when the map runs out at 180°, and the largest corner the map actually shows, 13.43°, is at its edge rather than at a turning point.

The polyhedral seam, met again #

This is the same quantity four essays on polyhedral maps spent their length measuring. The gnomonic crosses a seam without a corner measures the corner a feature acquires crossing from one face of a solid to another, and the corner is not at the midpoint finds that its value depends on where the crossing is taken.

The parallel is close. A polyhedral map is a composite of face maps, joined along the edges of a solid; the homolosine is a composite of two pseudocylindricals, joined along a parallel. In both cases the pieces agree in position, each piece has its properties pointwise, and the derivative across the join is where the join shows. The difference is which crossings are smooth: on a polyhedral seam the corner vanishes at the edge’s midpoint, where the face’s own mirror symmetry forces it; here it vanishes on the central meridian, where the map’s mirror symmetry does the same. A seam is smooth exactly where a symmetry makes it so, and nowhere else.

Along the join, the shape improves by twenty-six degrees #

The corner is what a reader sees. What a measurement sees is a jump in distortion.

The sinusoidal’s shape distortion at 40°44′ grows to 91.4 degrees at the edge of the map, which is what an equal-area map with straight parallels and a pole at a point has to pay far from its centre. The Mollweide’s at the same place is 65.4. Crossing the join northwards, the angular deformation of the map falls by 26 degrees, instantly.

That drop is the reason the map exists. Goode’s assembly is not a compromise between two maps: it is a decision to use each one where it is better, and the join is placed as low as the parallels allow so that the Mollweide’s gentler shape covers as much of the high latitudes as possible. Compromise projections is about maps that give up an exact property to reduce distortion everywhere; this is the other way of spending the same wish — keep the exact property, change maps halfway.

Moving the join #

The join could be somewhere else. Putting it at another latitude means the two parallel lengths no longer agree there, so the polar piece has to be scaled to meet the equatorial one — and to stay equal-area it must be scaled the other way along the meridian, by the reciprocal.

The corner is unmoved by any of this. Across joins from 20 to 60 degrees the largest corner stays between 12.77 and 13.56 degrees — a spread of less than a degree on a quantity of thirteen — and what changes is where along the parallel it peaks, from the edge of the map at a low join to 82 degrees of longitude at a high one. Joining the two maps costs about thirteen degrees of corner wherever it is done.

What the join latitude does decide is the central meridian. At any latitude but Goode’s the scaling puts a step in the spacing of the parallels: 7.8 per cent at 20 degrees, 16.5 per cent at 60, and nothing at 40°44′11.98″.

So Goode’s latitude is not the latitude that minimises anything. It is the one latitude at which no scaling is needed, and therefore the only one where the central meridian — the line down the middle of every lobe, the one a reader’s eye follows first — crosses the seam without a visible step. Every other join trades that away and buys nothing back.

Smooth, and bending immediately after #

One thing remains to be said about the central meridian, because “crosses smoothly” is a claim about a first derivative and there is a second.

Below the join the sinusoidal’s parallels are evenly spaced: the spacing is one at every latitude, and its rate of change is zero. Above it the Mollweide’s parallels close up towards the pole, and at the join that closing has already begun at a rate of 0.3221 per radian. So the composite’s central meridian is continuous and has a continuous first derivative, and its second derivative jumps from nothing to something.

A reader cannot see that directly — nobody looks at the second derivative of a meridian — but it is visible in the aggregate, as the point where the parallels stop being evenly spaced and begin to crowd. On Goode’s map that happens exactly at the join, and it is the only sign the central meridian gives that anything happened there.

What interruption does to the corner #

The map measured here is drawn whole, and the map in the atlas is not. That difference turns out to matter for the corner, in the direction that flatters the real map.

The corner depends on longitude measured from the central meridian, and interrupting the map gives every lobe a central meridian of its own. A feature at 110° of longitude on the whole map sits at 110° from the one centre; on the interrupted map it sits inside some lobe, at most a lobe’s half-width from that lobe’s centre. Goode’s land lobes are roughly forty to eighty degrees wide, so a meridian in one of them is at most twenty to forty degrees from its own centre — where the corner is 4.72° and 8.57° rather than 13.30.

So the cuts buy smoothness as well as shape. Giving up continuity prices interruption in the currency of shape distortion and counts the tears it costs; this adds a second thing the same cuts are quietly buying, which is that no meridian on the finished map ever gets far enough from a centre to turn the full corner. The worst corner on a Goode map as printed is set by its widest lobe, and a map cut into narrower lobes is smoother at the seam as well as truer in shape — an argument for cutting that has nothing to do with the reason cutting is usually defended.

Who joined them, and when #

The two pieces are much older than the join. The sinusoidal is sixteenth-century and was in use before anybody wrote down what equal-area meant; Mollweide published his ellipse in 1805, and it carried the name homalographic — equal-writing — for a century. Goode put them together in 1923 and named the result for both: homolosine, from homalographic and sinusoidal.

What he chose was the latitude. The two maps were already equal-area, already pseudocylindrical, already drawn with straight parallels; the only decision the composite needed was where to change from one to the other, and the condition that fixes it — equal parallel lengths — is the one that makes the change invisible in everything except the derivative. That the meridians would kink was known to him and is mentioned in the descriptive literature ever since, always as a qualitative remark. The number it has is thirteen degrees.

What each number was checked against #

The two pieces are the library’s own projections. The composite’s equatorial half is required to agree with the sinusoidal projection, and its polar half with the Mollweide, to twelve decimal places at stated points — so every derivative measured here is a derivative of those maps rather than of a re-implementation.

The three continuities are required and measured. At the join the areal factors must both be one to nine decimal places, and the central meridian’s spacing must agree on the two sides to nine. They do.

The corner must be zero where the symmetry says and large elsewhere. On the central meridian it must be below $10^{-9}$ degrees, and somewhere else it must exceed ten degrees; it is $0$ and 13.30.

And the refusal is a join moved to 30°. It must keep the corner — within a degree and a half of Goode’s largest — and break the central meridian, whose spacing must then jump by more than four per cent. It keeps 13.49 degrees of corner and jumps 4.9 per cent. A sweep that removed the corner by moving the join would mean the corner was an artefact of Goode’s particular latitude rather than of joining two families at all.

What one composite cannot settle #

Two pieces, one seam, one pair of maps. Everything here is the sinusoidal against the Mollweide. Another pair — two conics, a cylindrical against an azimuthal — would have its own join condition and its own corner, and nothing measured here says how large.

The corner is measured across the join and not along a route. A feature crossing the seam at an angle acquires the corner as its own kink; a feature running along the parallel does not notice it at all. Where a pseudocylindrical puts its error prices what the family’s choices do to features generally, and this essay prices only what the seam does at the crossing.

And the interruptions are left out. The real homolosine is cut through the oceans, and those cuts are a second kind of discontinuity — a gap rather than a corner — with its own price, already measured. Drawing the map whole is what makes the seam the only thing in the picture.

Still open: a join that is smooth #

The corner is not a law. It is the consequence of joining two maps whose meridians happen to arrive at the seam with different slopes, and a third map could be inserted between them: a strip, of some small width of latitude, whose parallels match the sinusoidal’s at its lower edge and the Mollweide’s at its upper one, and whose meridian slopes match at both.

Such a strip would have to be equal-area to keep the composite’s one exact property, and matching a slope at each edge as well as a position is four conditions where the join above satisfies two. Whether an equal-area strip exists that meets all four, how wide it has to be — a degree of latitude, or ten — and how much angular deformation it carries compared with the two maps it is splicing, are questions a join with no thickness cannot ask.