Where a pseudocylindrical puts its error

A pseudocylindrical projection draws the parallels as straight horizontal lines and the central meridian as a straight vertical one, and then has to decide what happens at the top.

The decision #

On the globe the pole is a point. Every meridian passes through it, so the parallel at 89.9° is a small circle a few hundred kilometres round and the parallel at 90° has no length at all.

A projection whose parallels are straight horizontal lines must give the pole a horizontal line too — of some length, possibly zero. There are two choices and no third.

A pole point. The top parallel has length zero, so the meridians converge to a single spot. The sinusoidal, Mollweide and Hammer projections do this.

A pole line. The top parallel has a finite length, typically about half the equator’s. Eckert IV and Robinson do this.

Neither is a compromise between the other two, and neither is obviously right. What can be done is to measure what each buys and what each costs, which turns a matter of taste into a trade with two numbers on it.

What a pole line buys #

Averaged over the whole sphere with the area element, the angular deformation of these four projections is

The two pole-line projections are the two best, and the ordering is clean: the worst pole-line projection beats the best pole-point one, which is what makes this a property of the treatment rather than of the individual maps.

The mechanism is easy to state. On a pole-point projection the meridians converge to a spot, so near that spot they cross the parallels at a very acute angle and the shear is extreme over a wide region — the whole high-latitude band, on both sides of the central meridian. A pole line spreads the same convergence over a finite width and the shear is milder everywhere except in one specific place.

What a pole line costs #

That place is the last parallel, and the cost there is severe.

Measured at 89.5° north, 60° from the central meridian, the parallel scale factor $k$ — how much the ground running east–west is stretched — comes out at

Fifty times, against one. A pole-line projection takes the last few kilometres of parallel near the pole and draws them across half the width of the sheet, because that is what representing a point by a line means: the line has to be filled by something, and the only thing available is the ground immediately around the pole.

That is the trade, and it is worth stating in the form a cartographer would use. A pole line makes the whole map better and the Arctic Ocean absurd. The sinusoidal projection, which is the worst map in the table on average, is the only one of the four that draws the region round the pole at anything like its true east–west scale.

Both directions, or it is not a trade #

The gate asserts both halves, and the second is the one that makes the first worth printing.

If only the mean mattered, every pseudocylindrical would have a pole line and the family would have one member. If only the last parallel mattered, none would. The assertion requires the family to split into the two groups on the measurement of the pole’s image rather than on the names — a measured pole width above 2% of the equator’s counts as a line — and then requires the two groups to be ordered oppositely on the two quantities, with the last-parallel ratio at least a factor of five.

Measured, that ratio is 48.95 against 4.62, or 10.6, comfortably clear of the floor. Without the second half a bug that reported every projection as slightly better on average would pass.

Why anyone accepts either #

The answer is purpose before property, and the two treatments suit different purposes so cleanly that the argument is nearly settled by naming the use.

A thematic map of population, agriculture or disease wants low mean distortion over the inhabited world and does not care what happens at 89.5° north, because almost nothing is there. That is the pole-line case, and it is why Robinson and Winkel tripel — both pole-line — became the standard for general-purpose world maps.

A map that has to be honest about the polar regions cannot use a pole line, because a fiftyfold stretch is not a distortion a reader can discount. That is the pole-point case, and it is why the sinusoidal survives in exactly the applications where high latitudes matter.

A map of the world’s land can avoid the choice entirely by cutting the sheet, which is the third option and is what an interrupted projection is for.

The interruption reframes the question #

An interrupted construction is worth reading as a comment on this essay’s trade rather than as a separate projection.

The reason a pole-point projection has high mean distortion is that its meridians converge over a wide region, and the reason they converge that much is that the sheet has to accommodate a full 360° of longitude at every latitude. Cut the sheet into lobes and each lobe carries only 60° or 90°, the convergence within a lobe is mild, and the mean distortion falls without the pole line. Goode’s homolosine goes one step further and changes projection at 40°44′, keeping the sinusoidal for the equatorial band and handing the polar sections to Mollweide — three decisions, one for each part of the sheet, none of which the uninterrupted family is allowed to make.

So the trade is not between two treatments of the pole; it is between two treatments of the pole given that the map is one connected sheet. Dropping that constraint changes the answer, and dropping it is exactly what the tear is.

That is a general shape worth noticing. Several of the sharpest trades in this subject dissolve when a constraint nobody stated is removed, and the constraint is usually continuity — the same move the interrupted projections make and the same one a zone system makes at a completely different scale.

Where the shear actually sits #

The mean is a summary and the two treatments differ in pattern as well as in size, which a field of principal directions shows and a scatter plot cannot.

The sinusoidal’s field is the pole-point signature and reads differently. Its strokes rotate steadily with longitude at every latitude rather than concentrating at the top, so the shear is distributed over the whole outer half of the sheet instead of being packed into the polar band.

So the two treatments do not simply differ in how much distortion they impose. They differ in where they put it, and a mean over the sphere is precisely the statistic that cannot tell those apart. That is the same objection the regional criterion raises against every scalar summary, arriving here as a question about one family’s internal choice.

The pole width is a measurement, not a label #

One methodological point, because it is what makes the two groups groups.

The pole width used above is measured from each projection’s own forward map: evaluate the position of the pole at the eastern and western edges of the sheet, take the distance between them, and divide by the length of the equator. Sinusoidal returns 0.0000, Mollweide 0.0007, Eckert IV 0.5000, Robinson 0.5322.

Mollweide’s 0.0007 is the interesting entry. Its pole is a point mathematically, and the measurement of it at 89.999° returns a tiny but non-zero width, which is the numerical residue of a meridian that is still converging. A classification by name would call Mollweide a pole-point projection and be right; a classification by measurement calls it one too, at a threshold of 2%, with three orders of magnitude of margin. The measurement is preferred because it would also correctly classify a projection nobody had named.

Robinson’s 0.5322 is a reminder that the pole line’s length is itself a design parameter — Robinson chose 0.5322 of the equator by eye, along with the rest of the table of numbers he adjusted until it looked right.

The third answer, and why it is not one #

Winkel tripel is the projection most often reached for when this trade comes up, and it is worth saying exactly what it does about it, because the answer is nothing.

Winkel’s construction is the arithmetic mean of the equirectangular projection and the Aitoff — coordinate by coordinate, averaging two maps rather than optimising anything. It has a pole line because the equirectangular half has one, and the length of that line is whatever the average produces. There is no decision about the pole in the construction at all.

That it lands in a sensible place is a consequence of averaging two things that were each sensible, and it is the reason the projection is hard to argue with and hard to justify. The measured result is a mean angular deformation among the best in the family and a polar behaviour among the worst, which is the pole-line bargain arrived at by accident.

What the mean hides #

The mean angular deformation over the sphere is an area-weighted average, so it weights the tropics heavily and the polar regions barely at all. That is a defensible weighting for a general-purpose map and it is exactly the weighting that makes the pole line look good.

Changing the weighting changes the answer, which is the point the regional essay makes at length. Weight by population and the pole line looks better still. Weight by land area and it looks slightly worse. Weight equally over the graticule rather than over the sphere — which is what a naive average of sampled points does — and the polar regions are massively over-represented and the pole-point projections win.

None of those is wrong. What would be wrong is quoting a mean without saying which one, and the measurement here uses the area element throughout with the choice stated.

What it means for the family’s future #

The two treatments exhaust the possibilities only because the parallels are required to be straight and horizontal. Relax that and the question changes shape.

The pseudocylindrical definition is a strong constraint and it was adopted for a practical reason that has expired: straight parallels make a map easy to construct with drawing instruments and easy to read a latitude off. Neither matters when the map is computed and the reader is looking at a screen.

What the constraint buys geometrically is that the projection is a one-dimensional problem — pick a function giving the half-width of the sheet at each latitude and another giving the height, and the projection is determined. The whole family, from the sinusoidal to Robinson, is a set of choices of two functions of one variable, and the pole treatment is the boundary condition at the top of the range.

Seen that way, the pole line is a boundary condition, and the reason it is either zero or roughly half the equator is that intermediate values were tried and looked wrong: Eckert’s own pairs bracket the choice deliberately. A modern optimisation over the two functions, with the pole width as a free parameter and a stated objective, would return some particular width — and would be the same activity as fitting a compromise projection, applied to the one parameter this family argues about most.

Drawn as shapes rather than as numbers the trade is the same one: on Eckert IV the ellipses along the central meridian and the equator are nearly round, and those nearest the pole line are long and tilted — ground being stretched sideways to fill a line that stands for a point.

What was computed here #

The pole width, the parallel scale at 89.5° and the angular deformation at a high-latitude corner were computed from each projection’s forward map by the site’s ordinary derivative machinery. The mean angular deformation is an area-weighted integral over the whole sphere at 576 sample points per projection, with the area element $\cos\varphi$.

Three claims are asserted: the family splits into at least two members of each kind on the measured pole width; every pole-line projection has a lower mean deformation than every pole-point one; and every pole-line projection stretches the last parallel more than every pole-point one, by at least a factor of five.

What the pictures cannot show #

The scatter plot places each projection at a point and says nothing about where on the map either quantity is attained. The mean is an integral over the sphere and the last-parallel figure is one sample at one place, and a reader wanting the distribution rather than the summary needs the graticule figure beside it.

Nor does the measurement address the question a reader of an atlas actually asks, which is whether a particular continent looks right. That is a judgement about shape at a specific place, and it is what the whole apparatus of measured distortion exists to replace with something checkable — at the cost of no longer answering the original question directly.

Who found it, and when #

The sinusoidal projection is the oldest in the family, in use by the sixteenth century and named after nobody in particular. Mollweide published his in 1805, explicitly as an equal-area alternative with a less extreme high-latitude shear.

Eckert’s 1906 paper introduced six projections at once, in pairs — three with pole points and three with pole lines — which is the earliest systematic treatment of exactly the trade measured here. That he presented them as pairs rather than as six separate projections says that the decision was already understood as the family’s defining axis; what the pairs lacked was numbers, and the numbers say the pole line is worth about 25% of the mean deformation and costs a factor of fifty at the top of the sheet.

The point about Eckert’s pairs is worth pressing, because it describes what a measurement adds to an understanding that was already correct.

He knew what the axis was. Presenting six projections as three pairs, each pair differing in the pole treatment and in nothing else, is a statement that the pole decision is the family’s organising choice — and it is a better statement of that than any prose, because a reader can see the pairs.

What the presentation could not carry is the exchange rate. A pole line reduces the shear at the top and changes the shape elsewhere is qualitative, and every reader who saw the pairs already believed it. Whether the trade is worth taking depends on how much is bought and how much is paid, and neither number was available.

The numbers turn a taxonomy into a decision. Twenty-five per cent of the mean deformation, against a factor of fifty at the top of the sheet, is enough for a designer to decide — and the two figures point in opposite directions, which is why the family has both kinds of member a century later rather than converging on one.

It also explains why the family did not converge. A trade whose two sides are both large and point in opposite directions has no dominant answer, so both branches survive and are chosen by purpose — a world map for reading takes the pole line, a map for measuring near the pole takes the point. A family that had converged on one would be evidence that one side of the trade was much smaller than the other, and the measurement says it is not.

And that is the ordinary relation between a good qualitative account and a measurement. The measurement rarely overturns the account; it supplies the magnitudes the account could not, and the magnitudes are what a choice actually turns on.

Which is a reasonable test to apply to any long-lived family: convergence is evidence about the size of a trade, and persistence is evidence that both sides of it are real.

Both readings follow from the same two numbers rather than from any further work.

Neither needed a new measurement.

Where this goes next #

Both fourth rungs on this ladder have been about the shape of the graticule. The next question is about the sheet’s orientation rather than its shape: given a projection and a region, which way should the axis point?