The average was a choice of norm

Eleven rungs of this ladder have taken apart the ways a comparison of projections can be made to say what its author wanted. The rule of thumb, scored tested the advice; which projection a weighting can make best measured how much freedom a weighted sum of criteria hands the person setting the weights; the ranking is not an order removed the weights entirely and found the majority relation running in circles.

Every one of those took the criteria as given. A criterion arrives as a name — Airy, Kavrayskiy, worst-case angular deformation — and the argument has always been about which of them to believe and how to combine them. Underneath the names is a step nobody has looked at, and it is the same step in all of them.

A projection’s distortion is a field: a number at every point of the region. A criterion is a number. Something has to turn one into the other, and the thing that does it is a choice.

One quantity, one region, ten different answers #

Fix everything a comparison usually argues about. Take one pointwise quantity — the departure of the principal scale factors from one, which is what measuring instead of naming computes at every point. Take one region, the whole world. Take ten projections, normalised so that none of them is being scored on its arbitrary overall size.

There is one decision left: how to summarise several hundred numbers as one. The obvious summaries form a family with a single parameter,

$$\lVert q \rVert_p = \left( \frac{\int_R q^p \, \mathrm{d}A}{\int_R \mathrm{d}A} \right)^{1/p}$$

where $p = 1$ is the area-weighted mean, $p = 2$ is the root-mean-square that both classical indices use, and $p \to \infty$ is the maximum. Everything between them is a legitimate index of distortion, and every one of them is computed from exactly the same field.

The ordering at $p = 1$ is Eckert IV, Robinson, Winkel tripel, Miller, Mercator, equirectangular, Mollweide, Gall–Peters, Hammer, sinusoidal.

The ordering at $p = \infty$ is Winkel tripel, Hammer, sinusoidal, Eckert IV, Robinson, Mollweide, Gall–Peters, Mercator, Miller, equirectangular.

All ten projections change position. Spearman’s rank correlation between the two orderings is −0.042, which is the number one gets from shuffling. The sinusoidal is last on one and third on the other. The equirectangular is sixth on one and last on the other. Both orderings are correct, both are computed from the same measured field, and neither of the two published summaries that produced them says which it is.

The family the four published indices belong to #

The reason this is worth an essay rather than a footnote is that the family is not a hypothetical. This collection’s own tournament treats seven criteria as seven independent voters, and six of them are three pointwise quantities read at three exponents.

That placement is a reconstruction rather than a resemblance. Each of the six is recomputed here as a p-norm of a stated pointwise quantity and required to agree with the criterion’s own independently written implementation to twelve significant figures, which it does for every projection tested. Airy’s index is the $p = 2$ norm of $\sqrt{\tfrac12((a-1)^2 + (b-1)^2)}$; Kavrayskiy’s is the $p = 2$ norm of the same expression in logarithms; the mean angular deformation is $p = 1$ on $\omega$; the worst areal error is $p = \infty$ on $|s - 1|$.

So the seven voters whose disagreements produce the majority cycles are not seven opinions about maps. They are two or three opinions about maps and a spread of opinions about aggregation, which is a different claim about what the cycling means — and a weaker one, because a cycle among genuinely different criteria is a fact about the maps while a cycle among three exponents of one quantity is a fact about a distribution’s shape.

What the exponent is actually saying #

An exponent looks like a technical detail because it has no units and appears in no caption. It is making a substantive statement, and the statement can be measured directly: how much of the region does the score depend on?

Take the region’s area, ordered from the worst-distorted point down, and ask what share of it carries half the mass of the integral.

At $p = 1$, 21.1 per cent of the sphere carries half the answer. At $p = 8$ it is 0.97 per cent. At $p = 64$ it is 0.21 per cent — about the area of Greenland, and specifically the worst 0.21 per cent.

That is the honest reading of the exponent, and it is a purpose statement in disguise. An atlas designer choosing a world projection cares about the whole sheet and wants a low $p$. A navigator who will not use the chart outside one band cares about nothing but the worst point in that band and wants $p = \infty$; that is what Chebyshev’s criterion is, stated as a norm rather than as a theorem. Both are defensible. Neither is stated, and a reader handed the number cannot tell them apart, because both are printed as distortion.

The ordering, watched as the exponent moves #

The two end points understate what happens between them, because the ordering does not slide smoothly from one to the other.

Nine of the eleven exponents produce a distinct ordering. Twenty-five of the forty-five pairs change places somewhere in the range, and two of them do it twice — Robinson passes Mollweide at $p = 4.35$ and Mollweide takes it back at $p = 22.4$, so there is a window of exponents in which the answer is one thing and windows on both sides where it is the other. A crossing is not a threshold beyond which one map is simply better.

The first place moves too, which is the part a reader actually sees. Eckert IV is the best world projection on this quantity at $p = 1$. The Winkel tripel is the best at every exponent from two upwards. A published table would show one of those and no indication that the other exists.

Why a pair can swap at all #

The mechanism is entirely in the distributions, and once it is seen the whole family becomes predictable.

A p-norm is monotone in $p$ for each projection separately, so no curve in the hero figure ever falls. Two curves can only cross if one projection is better than another on the small values and worse on the large ones — which is to say, if their distributions cross. Where they do not, no exponent can produce a swap, and the ordering is a fact about the maps rather than about the summary.

The equirectangular is the cleanest case. Its scale departure is 0.147 at the median, the best of the ten, and 1.655 at its worst, also the worst of the ten. It is the projection that is excellent nearly everywhere and catastrophic in a small place, which is exactly the shape a low exponent rewards and a high one punishes. The sinusoidal is its mirror: mediocre over most of the sphere, and comparatively restrained at its worst.

Neither of those is a defect in a measurement. They are two different maps, and the question of which is better has no answer until somebody says how much of the sheet has to work.

The crossing exponent is where that decision bites, and it is not evenly spread. Of the twenty-five pairs that cross, fourteen do so below $p = 3$ — which is to say inside the range every published index already occupies, between the mean and the root-mean-square. Those are the swaps that matter in practice, because nobody has to be persuaded to consider $p = 3$; two authors using Airy and a mean respectively have already chosen exponents on opposite sides of the crossing without either of them mentioning an exponent. The three pairs whose first crossing is above $p = 8$ are a different kind of fact: they separate maps only for a reader who cares exclusively about the worst place, and such a reader is usually not choosing between world projections at all.

The quantity decides whether it matters #

The exponent is not always free in practice, and the third measurement of this rung is the one that says when to worry.

Aggregated over the areal departure, the exponent decides almost nothing: two distinct orderings across the whole range, one crossing pair of forty-five, and a rank correlation of 0.988 from the mean to the worst case. Two of the ten projections move, they are adjacent, and they swap once. The reason is that the areal fields of these ten projections are close to nested — an equal-area projection has none, a compromise has some everywhere, and Mercator has more than Miller at essentially every point — so there is nothing for an exponent to reorder.

Aggregated over the angular deformation it decides a little: eleven crossing pairs, correlation 0.782. Aggregated over the scale departure it decides everything.

So the exponent’s freedom is a property of the field being summarised, and it is cheap to check. Count the pairs whose distributions cross. If the answer is zero, the aggregation is not carrying the argument and the ranking can be published without qualification. If it is twenty-five of forty-five, the ranking is a statement about an exponent that has not been stated.

There is a reason the areal field is the stable one, and it is worth naming because it is the opposite of what a reader would guess. Area is the quantity these ten maps differ over most dramatically — Mercator inflates a cell at seventy degrees fifteenfold, and the equal-area projections inflate it not at all — and that enormous spread is exactly what makes the ordering robust. Large, consistent differences are nested differences: a map that is worse on area at the median is worse at the ninety-ninth percentile too, because the mechanism producing the error is the same one everywhere. The scale departure is stable in size across the ten and unstable in shape, which is the combination that lets an exponent decide. The criterion whose numbers are closest together is the one whose ordering is least secure, and it is also the one a comparison is most likely to be reaching for, since a criterion that separates the library by a factor of fifteen was never going to be the interesting question.

What was computed, and how #

Everything above is one integral evaluated at several exponents, and the care is in the three places it could go wrong quietly.

The projections are normalised before comparison, by the same scaling rank uses: each is scaled so the geometric mean of its areal factor over the region is one. Without that step the comparison is mostly a comparison of arbitrary overall sizes, and the effect grows with the exponent, so an unnormalised version of this figure would show a much larger reordering that meant nothing.

The samples carry the sphere’s area element, not the coordinate rectangle’s. A p-norm over $\mathrm{d}\varphi\,\mathrm{d}\lambda$ weights the poles as heavily as the equator, and since the poles are where most of these projections do their worst, that convention alone shifts every high-exponent score upward by a different amount for each map.

The crossings are found by bisection on the difference of the logarithms, scanned over twenty-five exponents first so that a pair which crosses twice is found twice rather than averaged into one. The gate requires that the bisected exponent be a genuine crossing — the two scores must agree there to a part in a million — and that no p-norm anywhere fall as $p$ rises, which is the arithmetic identity the whole family rests on.

The control is a projection with nothing to reorder. Mercator’s angular deformation is zero at every point, so its $\omega$ score is zero at every exponent, and no choice of aggregation can put a non-conformal projection above it on that quantity. The gate checks that too, because an aggregation that could move a result which is exactly zero would be an aggregation with an arithmetic error in it.

What it does to this collection’s own rankings #

The default ranking used throughout these essays sorts on Kavrayskiy’s index, which is the $p = 2$ norm of a logarithmic scale measure. It is a defensible choice and it is a choice.

At $p = 1$ on the same pointwise quantity, Robinson is first and Mercator sixth. At $p = 2$ — the published index — the Winkel tripel is first and Mercator seventh. At $p = \infty$ the Winkel tripel is still first, Mercator is tenth of ten, the Hammer has climbed from ninth to third, and the sinusoidal from tenth to fifth. Twenty-one of the forty-five pairs cross somewhere in between.

That is a real qualification on a real number, and it is narrower than it looks. The top of the table is stable: the Winkel tripel is first at every exponent from two upwards, which is why the compromise projections keep winning these comparisons whoever runs them. What moves is the middle and the bottom, where the differences between maps are small and the shape of each map’s tail is what decides the order.

Where the model stops #

A p-norm is not the only aggregation. A trimmed mean, a median, a quantile at a stated level and an integral against a population density are all summaries of the same field, and none of them is in this family. The rule scored out of sample uses a different one again. What the family does is contain every summary this collection has published, which is enough to make the point without being a taxonomy of everything possible.

The region is fixed here and it is the other free parameter. Distortion over a region already shows that the ranking depends on the region, and nothing in this rung separates the two freedoms. A comparison could in principle be stable under the exponent and unstable under the region, or the reverse, and measuring which of the two carries more of the variation is a rung this ladder has not taken.

Ten projections is a small population. Rank correlation on ten items has a standard error of about a third under the null, so the −0.042 measured here is properly read as indistinguishable from no relationship rather than as a precise zero. The claim that survives is the count: all ten moved, twenty-five pairs crossed, and nine orderings appeared.

And the sampled maximum is a lower bound, which the worst point is not on the grid measures. The $p = \infty$ column of every table here inherits that shortfall, and so does the high end of every curve. It biases the high-exponent scores downward by between four and thirteen per cent, unevenly across the ten, which adds to the reordering rather than explaining it away.

The generalisation #

The pattern is not about maps. It is that a scalar summary of a field has two arguments and reports one of them.

A criterion names the quantity — angular deformation, areal error, scale departure — because that is the part that sounds like a decision about what matters. It does not name the exponent, because the exponent looks like arithmetic. Yet the exponent is the part that says how much of the region has to work, which is the more directly practical of the two questions and the one a purpose actually answers.

This collection’s founding rule is measuring instead of naming, and its second rule is that no map may be called best without naming the purpose it is best for. The exponent is where the purpose enters the arithmetic. A world atlas has $p$ near one written into what it is; a navigational chart has $p = \infty$ written into what it is; and a table of distortion figures with no exponent stated is a table that has quietly answered a question nobody asked it.

Who found it, and when #

The mathematics is old and uncontroversial. Interpolation between the mean and the maximum by a power is standard, and the fact that $\lVert q \rVert_p$ increases with $p$ is an exercise. Chebyshev’s criterion from 1856 is the $p = \infty$ end of it, stated as a theorem about conformal maps; Airy’s of 1861 and Kavrayskiy’s of the 1930s are both at $p = 2$, in two different pointwise measures, and both were presented as the way to score a projection rather than as one member of a family.

What has not been done is the join. Cartography compares projections constantly and reports the comparison as a table; numerical analysis has known for a century that the $L^1$ and $L^\infty$ optima of the same problem are different objects. A reader of the table is not told which of the two they are looking at, and the two turn out to disagree about ten maps in ten places.

The one place the subject does think about it is in projection design, where somebody minimising a criterion over a parameter has to write the criterion down and therefore has to pick the exponent. Every projection minimises something is the record of what those choices were, and the exponents in it are all over the family — which is the same finding arriving from the design side rather than the reading side.

The cheap habit, for a reader of any distortion figure: ask what fraction of the map the number is about. If nobody can say, the number is a mean or a maximum and the difference between those two is, on this evidence, the difference between the best world projection and the fourth best.

Where the ladder goes next #

Twelve rungs have now measured how much of a projection comparison is the comparer’s own freedom: the criteria, the weights, the aggregation, the region, the tolerance and the sample. What none of them has asked is who the map is for. Every score in this essay weights a square kilometre of empty ocean exactly as heavily as a square kilometre of city, because area is the only weight the geometry supplies — and the next rung replaces it with one that is not about geometry at all.