The second derivative has its own ranking

This site refuses to call a projection best without naming the purpose, and which projection is best is the essay that does the refusing. The argument there is that every published ranking is a ranking against a criterion, that the criteria disagree, and that the disagreement is the content.

Every criterion in that argument was built from first derivatives. There are now two more, and they change the answer.

The two orderings, side by side #

The two agree on the winner and on the loser and on almost nothing else. Mercator rises three places, Mollweide falls two, Eckert IV falls two, Hammer rises two.

Mercator’s rise is the one worth understanding, because it is the projection this site’s headline essay is about and the one most often called the worst map in the world. What the second-order score says is narrow and defensible: a Mercator world map bends and lopsides shapes less than most of its competitors, and inflates their areas more than any of them. Those are different failures, they are measured by different derivatives, and a ranking that reports only one of them is answering a question its reader may not have asked.

Why the areas do not appear #

An areal factor is $ab$, a product of two singular values of the Jacobian, and so is first order. Nothing in flexion or skewness knows anything about it.

That is not an oversight in the criterion; it is the criterion. Gott and his co-authors proposed the pair precisely because the standard measures penalise a map for stretching a continent while saying nothing about bending it, and the eye’s complaint about a world map is often about shape rather than size. The reply this site has to make is that both are choices, and neither is the objective.

The weighting, which is arbitrary and matters more than the derivative #

The score plotted above is the root mean square of the two quantities. That combination says one radian of turning per radian of arc is exactly as bad as a factor of $e$ of scale change per radian of arc. Nothing justifies it. The two are dimensionally comparable — both are per radian — and there the case ends.

So the honest thing is to measure how much the choice matters, by ranking on each half alone:

Spearman’s correlation between those two orderings is 0.45. The correlation between the combined second-order score and the whole first-order criterion is 0.69.

That is the finding of this rung, and it is not the one the ladder was laid out to expect: the disagreement inside the second-order criterion is larger than the disagreement between the second-order criterion and the first-order one. Choosing how to weigh bending against stretching changes the answer more than choosing which derivative to measure at all.

The equirectangular is the clearest case. It is first on bending and sixth on stretching, which averages to fourth overall — and either of the two extremes could be quoted, truthfully, by somebody with a projection to sell.

Change the region and the whole order inverts #

Distortion over a region established the first-order version of this: a ranking is a ranking over the ground the map has to cover, and the answer moves when the ground does. At second order the movement is more violent.

The world column and the Europe column are close to reversals of each other. The three equal-area pseudocylindricals — Hammer, sinusoidal, Mollweide — are the three worst globally and the three best over a mid-latitude patch, because a projection whose failure is concentrated in the outer thirds of a world map has none of that failure inside a region that never goes there.

Mercator goes the other way: third globally, last over Europe. That is the same projection with the same formula, ranked on the same criterion, by the same code.

The mechanism is not mysterious and it is worth stating, because it explains why the two columns are near-inversions rather than merely different. Mercator’s second-order amplitude is exactly tan φ, established in the first rung of this ladder and matched to five figures against the closed form. Over the whole sphere that function is averaged with most of its weight near the equator, where it is small, and the map’s very worst latitudes are outside the sampled band anyway. Over a cap centred at 45° north there is no small part of it left in the average: every sample sits where tan φ is of order one. A projection whose failure is a monotone function of latitude is flattered by a global average and punished by a regional one, and the equal-area pseudocylindricals — whose failure is a function of longitude from the central meridian — are treated in exactly the opposite way.

The first-order ranking is not more stable, it is differently unstable #

A reader might reasonably suspect the second-order score of being fragile. It is not more fragile than the one this site has been using.

Rank the same six projections over two regions by a first-order criterion and the order is not the same in the two columns: a projection that suits a wide shallow region does not suit a long narrow one. Instability under a change of region is a property of ranking projections at all, not a defect of the criterion being used.

What is different is where the two criteria are stable. Over the tropics, the second-order and first-order orderings agree at 0.88 — a band that no projection in the table treats badly, so both criteria mostly measure the same mild thing. Over the whole sphere they fall to 0.69. Over Europe they are at 0.74 with completely different winners.

Three correlations, three regions, and no ordering that survives all of them. That is what a ranking of general-purpose world maps is worth, and it is worth saying at both derivatives rather than at one.

Eight is a small number #

Every correlation quoted so far is a rank correlation over eight objects, and eight is few enough that the arithmetic of chance deserves stating before any of them is read as evidence. There are 8! = 40,320 orderings of eight projections. Drawing one of them uniformly at random and correlating it against a fixed ordering gives a Spearman coefficient that is not concentrated near zero at all: its distribution over those 40,320 outcomes has a standard deviation of 0.38, and it takes only 85 distinct values, so the statistic is coarse as well as noisy.

Enumerating all 40,320 permutations rather than consulting a table gives the exact tail probabilities for the four numbers this essay has quoted:

The 0.45 is the one that changes meaning under this reading. Thirteen per cent of random orderings of eight things agree with a fixed ordering at least that well, so the flexion ranking and the skewness ranking are not merely in tension: on eight projections they are statistically indistinguishable from two orderings drawn out of a hat. The essay’s claim that the weighting matters more than the derivative is if anything understated by its own number. There is no evidence here that the two halves of the second-order criterion are measuring a common thing at all.

The other three survive, in descending order of comfort. The one-tailed critical value at five per cent for eight items is 0.643, so the world figure of 0.69 clears it and nothing more; at two and a half per cent the critical value is 0.738 and the world figure fails. Only the tropical 0.88 is past the one-per-cent value of 0.833.

None of this is a defect in the measurements, which are exact to the precision the generators compute them at. It is a limit on how much can be learned from ranking eight things, and it is the reason this essay reports its correlations beside the orderings themselves rather than in place of them. A coefficient of 0.69 over eight objects means the ends agree and the middle is unresolved — which is precisely what the two-column table at the top of the essay shows by eye, and the eye is not being fooled.

The remedy available to a reader who wants a stable answer is not a better criterion but a longer list. The correlation between two orderings of forty projections, at the same underlying agreement, would have a chance-alone standard deviation of 0.16 rather than 0.38, and the same 0.69 would then be past any conventional threshold with room to spare. The library this site draws from is large enough to do that. What it would not fix is the finding: adding thirty-two more projections to a comparison whose two halves disagree does not make them agree, it measures the disagreement more precisely.

What the winner is winning #

The Winkel tripel comes first over the whole sphere on both criteria, which is the one robust result in this essay and is why it is the projection the National Geographic Society adopted in 1998 and why Gott’s own ranking put it at the top.

That is worth stating precisely because “best” invites the reading that the projection is good. Its combined score over the sphere is 0.87 per radian, which is a route bending through half a radian over a leg of 3,000 kilometres. The winner of this competition still bends a transatlantic great circle visibly, and no entrant does otherwise.

The one projection with a zero, and why it is not on the list #

The gnomonic projection has flexion of exactly zero, everywhere. On a criterion built from flexion alone it wins by an infinite margin, and it is absent from every table above.

It is absent because it cannot show a hemisphere, so the sphere-wide average that every row of every table is built from does not exist for it. That is not a technicality to be worked around; it is the reason a ranking has to state its region before it states anything else, and the reason choosing for a line, not a region exists as a separate argument.

What is left when the rankings are set aside #

Four numbers about a projection at a point are now available, and each answers a different question:

• the areal factor — how much larger or smaller a patch is drawn;
• ω — how much a shape is sheared;
• flexion — how far a route bends away from the ruled line;
• skewness — how fast the scale changes along that route.

Two of the four are first order and two are second, and no combination of the first pair predicts either of the second — which is the whole content of this ladder and is now measured rather than argued: the sinusoidal projection has an areal factor of exactly one and the worst flexion in the table, and the Mercator has the worst areal factor in the table and the third-best bending.

A map user has one of those questions and rarely two. A thematic map shaded by density needs the first and can ignore the rest, which is what the projection that shows true size is about. A navigator plotting a course needs the third. A surveyor reducing a distance needs the fourth. Nobody needs the average.

The rankings in this essay are therefore offered as evidence about rankings rather than as advice. The evidence is: two derivatives, three regions, two weightings, and eight projections produce eight orderings with one stable winner and no stable middle.

Where the model stops #

The score is a root mean square over directions and over area, so it is dominated by the worst regions of a map and is insensitive to what happens in the best ones. A maximum-based score would rank differently again; the site’s own where the worst point is shows how far a maximum and a mean can diverge for the first-order quantities, and there is no reason to expect otherwise here.

The average is taken over a lattice of sample points and weighted by area, which is the right weighting and is not a free choice: an unweighted mean over a longitude–latitude grid counts a square degree at 80° north as heavily as one at the equator, and the poles are exactly where every cylindrical projection’s second-order failure lives. What the weighting cannot repair is that the quantity being averaged is unbounded. Mercator’s amplitude is tan φ, so the integral of its square against the area element is the integral of sin²φ / cos φ, which diverges logarithmically at the pole. The sphere-wide score for a cylindrical projection is therefore not a number the sampling converges to; it is a number that creeps upward as the lattice reaches closer to the poles, and every world-column figure in this essay should be read with that attached.

The creep is slow, which is why it is easy to miss and worth measuring. Refining the lattice from 36 points to 2,304 — the highest sampled latitude moving from 70° to 82° — takes Mercator’s combined score from 1.256 to 1.397 and the Winkel tripel’s from 1.082 to 1.339. Both rise, so the ordering between them survives; what does not survive is the margin. Mercator and Miller are separated by 0.056 on the coarse lattice and by 0.006 on the fine one, a gap that has fallen by a factor of nine and shows no sign of stopping. Two of the eight rows in the first table of this essay are, at sufficient resolution, a tie.

Two projections in the tables are also not twice differentiable everywhere. Robinson is excluded for that reason — its interpolated table has no second derivative at an entry — and the polyconic, which has a kink of its own along the central meridian, is not in the list either. A criterion that cannot be evaluated on a projection cannot rank it, and quietly ranking it anyway on values computed a degree away is exactly the sort of thing this site exists to refuse.

The score is also computed on the sphere. Recomputing it on the ellipsoid would move every number by of order the flattening, which is a third of a per cent and smaller than the gap between any two adjacent rows in any table here — the same conclusion datum shifts dwarf projection errors reaches about the first-order quantities, and for the same reason.

Finally, the whole apparatus assumes the reader is drawing straight lines between points. A reader who drapes a great-circle route onto the map, as any competent atlas does, meets none of this — the flexion is then the cartographer’s problem and not the reader’s, and the map’s second-order failure shows up as ink rather than as error.

Who found it, and when #

Kavrayskiy’s criterion dates from the 1930s and Airy’s from 1861, and both are integrals of first-order quantities. The second-order pair is from 2007. That is a hundred and forty-six years between the two derivatives, on a subject whose central question — which map should be used — was being answered in print the whole time.

The Winkel tripel’s position at the top of both is a genuine convergence of independent criteria and is the strongest argument for a compromise projection this site has been able to measure. What the measurement does not support is the further step, taken often, of describing the winner as accurate: it is the least bad of eight bad options on two arbitrary scores over one region, and its own numbers say so.

And the instability is not noise to be averaged away. Each ranking is exact: given a region and a criterion, the order it produces is the order, computed to the arithmetic’s floor and reproducible. What moves is the question, not the answer, so a ranking averaged over regions is not a better estimate of anything — it is a fifth ranking, of a fifth question, with no more claim on a reader than the four it was made from.

Where the ladder goes next #

The flexion anchor now carries a definition, a coupling theorem and a ranking, and every one of them is about a projection that is differentiable twice. The obvious fourth rung is the one not written here: what the second-order quantities do at the seams — at an interruption, at a table entry, along the polyconic’s central meridian — where the first derivative exists and the second does not.

That is a question about where a projection stops being a function, and giving up continuity is the essay that argues an interruption is a choice rather than an accident. Measuring what it costs at second order needs a definition of flexion at a point where the limit does not exist, and no definition of that kind exists here yet.