Distortion over a region

Tissot’s construction answers “what does this map do here”. Nobody has that question. The question people have is which projection should carry this country, this ocean, this continent — and going from a point to a region means integrating.

Integrating means choosing a weighting, and that choice is the moment a measurement becomes a judgement. This essay makes the step explicit and then shows what it costs.

The criteria #

Two classical ones, differing in a single choice, and worth writing out because the difference between them is the whole subject.

Airy’s, from 1861, integrates the squared departure of each principal scale from unity:

$$E_A = \sqrt{\frac{1}{S}\int_\Omega \tfrac12\left[(a-1)^2 + (b-1)^2\right]\mathrm{d}A}$$

Kavrayskiy’s integrates the squared logarithm instead:

$$E_K = \sqrt{\frac{1}{S}\int_\Omega \tfrac12\left[(\ln a)^2 + (\ln b)^2\right]\mathrm{d}A}$$

The difference is that Airy treats a doubling and a halving as different sizes of error — $(2-1)^2 = 1$ against $(0.5-1)^2 = 0.25$ — while Kavrayskiy treats them as equal, since $|\ln 2| = |\ln 0.5|$.

Both are defensible. Neither is more correct. And they disagree about the answer.

The weighting nobody states #

Three separate choices hide inside those formulae, and published rankings state none of them.

Which function of the scale factors. Airy’s or Kavrayskiy’s or something else. The site computes both.

Which region. The integral is over $\Omega$, and $\Omega$ has to be named. A ranking over the whole sphere, over the land, over a continent and over one country are four different quantities.

Which measure on the region. The site integrates with the area element $\cos\varphi\,\mathrm{d}\varphi\,\mathrm{d}\lambda$, so equal areas of sphere count equally. Integrating over the coordinate rectangle instead — which is easy to do accidentally — weights the poles as heavily as the equator, and is one of the ways two published tables come to disagree without either being wrong.

There is a fourth, which is whether to weight by land, or by population, or by anything else the map is about. Each is defensible for a different map and each gives a different order.

The ranking is a statement about a region #

The clearest demonstration is to run the same six projections over two differently shaped regions.

Over Europe — wide in longitude, shallow in latitude, mid-latitude — the Kavrayskiy scores are:

Over Chile — long in latitude, narrow in longitude — the same six come out:

The order has inverted almost completely. The conic that wins Europe by a factor of three is fifth over Chile; the projection that loses Europe outright wins Chile.

Nothing about either projection changed. What changed is the shape of the region relative to the projection’s own axis: a conic’s zero-distortion lines run east–west, which is right for Europe and wrong for a country running 39° north–south. That is the aspect argument with a number attached.

The site asserts that the two orderings differ. It is an unusual assertion — a check that fails when a result is too clean — and it is the site’s evidence that a regional criterion is measuring the region.

And a statement about the criterion #

The second disagreement is smaller and is the one that matters for reading published tables.

Over the whole sphere, Airy’s criterion and Kavrayskiy’s rank six of the world projections in almost the same order, and swap two of them: Mercator and Gall–Peters change places. Airy puts Gall–Peters ahead; Kavrayskiy puts Mercator ahead. That pair is what the site’s assertion is written on, because it is the smallest disagreement the two measures produce and therefore the one most likely to disappear if either were implemented wrongly.

The reason is the logarithm. Mercator’s failures are large inflations — areal factors of 90 at the top of the map — and Gall–Peters’ are large compressions of shape. A squared-difference measure penalises the inflation far more heavily than the compression; a squared-logarithm measure treats them symmetrically.

So which of the two projections is “less distorted” over the world depends on whether stretching by a factor and squeezing by the same factor are counted as equally bad. That is not a fact about the projections. It is a preference, it is the entire content of the disagreement, and no published ranking states it.

The site asserts that the two criteria do not always agree, for the same reason it asserts the regions disagree: if they always agreed, the choice between them would not be a choice, and the essay’s argument would be false.

What the numbers still buy #

The essay is sceptical about rankings and the measurements are worth having. Three things they do settle.

Bounds. A criterion puts a number on how much worse one option is than another over a stated region, and an order-of-magnitude difference survives any reasonable reweighting. Albers over Chile scores 0.596 against Mercator’s 0.147 — a factor of four — and no choice of criterion reverses that.

Elimination. A projection that scores badly on every criterion over the region in question is a bad choice, whatever the weighting. The disagreements are between adjacent entries, not across the table.

The shape of the trade. Reporting the two components separately — mean angular deformation and mean areal error — instead of the blend says what the reader needs without deciding for them. This site’s tables carry both alongside the blended score, because the blend is somebody else’s weighting and the components are the measurement.

The whole-sphere table, both ways #

The world case is the one published rankings are usually about, and it is worth setting out in full because the two criteria’s disagreement is small, specific and instructive.

Three things are worth reading off it.

The two compromises win on both criteria. Winkel tripel and Robinson lead under either weighting, which is the strongest available evidence that the twentieth-century move to compromise projections for world maps was right on its own terms rather than merely fashionable.

Mercator and Gall–Peters swap. Under Airy, Gall–Peters at 0.498 beats Mercator at 0.663. Under Kavrayskiy, Mercator at 0.379 beats Gall–Peters at 0.383. Fifty years of argument, and the two criteria that were available throughout disagree about which of them is less distorted over the world.

The plate carrée does better than its reputation, on one criterion. Third of the seven under Kavrayskiy, ahead of Mollweide, Mercator and Gall–Peters, because its failures are concentrated where the area element is small. Under Airy it is sixth. That is the largest disagreement in the table and it is instructive: the projection’s error is almost entirely inflation, and a squared-difference measure prices an inflation far more heavily than a squared-logarithm one. It is still a bad default, and the criterion is not the reason.

Measured pointwise along a meridian, the four projections that lead the European ranking tell the same story the integrated scores do — which is not guaranteed and is worth checking on the occasions it happens, because a pointwise reading and an integrated one answer different questions and agree only when nothing is cancelling.

The region that is not a shape #

A weighting the classical criteria do not offer, and the one most world maps actually want.

Everything above weights by area of sphere. A general-purpose world map is looked at by people, and people look at land — so a land-weighted integral is at least as defensible, and it flatters projections that treat the mid-latitudes well and the empty Southern Ocean badly.

Weighting by population is more defensible still for a thematic map, and it moves the answer further: half the world’s population lives between 20°N and 40°N, a band where almost every projection behaves reasonably, so a population-weighted criterion compresses the differences between projections dramatically.

None of the three is the right one. Each answers a different question, and the honest form of a ranking names which. This site weights by area because it is the neutral choice and because a land or population weighting would need a dataset, and the site does not use datasets for quantities it can compute exactly — a coastline’s area depends on its simplification level, and a population raster is a model.

Which is itself an example of the essay’s point. Declining to weight by land is a choice, it changes the answer, and stating it is the only thing that makes the numbers usable by somebody who would choose differently.

The one region with an exact answer #

Restricting the question far enough makes it well posed, and it is worth knowing exactly how far.

Fix the region and fix the property to conformality, and the answer becomes unique with a proof attached. Chebyshev’s criterion says the optimal conformal projection of a region is the one whose scale factor is constant on the boundary, and for a spherical cap that is the stereographic projection centred on it, achieving a scale ratio of exactly $\sec^2(\rho/2)$.

That is a genuine optimum: a stated family of candidates, a stated objective, and a theorem picking one out. Every other comparison on this page is a ranking under a chosen weighting, and this one is not.

The difference is instructive. Chebyshev’s objective — the ratio of maximum to minimum scale — is a worst case, not an integral, so it needs no measure on the region and no weighting. The moment the objective becomes an average, a weighting appears, and it never leaves.

Why the disagreements are between neighbours #

A pattern in the numbers worth extracting, because it says how much the scepticism in this essay should be worth in practice.

Across every comparison the site runs, the criteria disagree about adjacent entries and never about distant ones. Airy and Kavrayskiy swap Mercator and Gall–Peters, which are fifth and sixth of seven; they agree completely about the top three and the bottom one. Europe and Chile invert the whole order, and the inversion is not a disagreement between criteria — it is the same criterion answering about two different regions, which is a different thing entirely.

That gives a usable rule. A ranking is reliable at the resolution of its gaps: differences of a factor of two or more survive any reasonable reweighting, and differences of a few per cent do not survive any.

So the honest reading of a published table is as a partition into bands rather than as an order. Over Europe, the two conics are in one band, the compromises and Mollweide in a second, and the two cylindricals in a third; within a band the order is a property of the criterion and between bands it is a property of the projections.

That is less than a ranking claims and considerably more than nothing, and it is what the measurement can support.

How to report a regional comparison honestly #

The practical recommendation, since the essay is against the usual form.

State the region, by coordinates. State the weighting — area, land, population, or none. Report the two components separately, and the blended score only alongside them.

Something like: over 35°–70°N, 10°W–40°E, area-weighted: Lambert conformal conic, mean angular deformation 0.0°, mean areal error 1.03; Albers, 4.1° and 1.00; Mollweide, 18.3° and 1.00. Longer, and it lets a reader whose map is about area rather than shape reach a different conclusion from the same measurement.

That is the same discipline the captions on this site use: report the number, state what it is a number of, and leave the weighting visible rather than baked in.

What a criterion cannot see #

One limitation that no amount of care about weighting fixes, and it is the reason the compromise projections keep escaping this machinery.

Every criterion here integrates a local quantity — the scale factors at a point — over a region. A reader looking at a map does not perceive local scale factors. They perceive the shapes of coastlines, the relative sizes of landmasses they recognise, and whether the whole thing looks like the world.

Those are global, gestalt properties of a finite figure, and they are not functionals of the distortion field in any straightforward way. A projection can have moderate distortion everywhere and produce a continent shape nobody recognises; another can have larger measured distortion and look right.

That is the gap Robinson’s method fell into deliberately: he optimised appearance directly, by eye, because appearance was the objective and no integral reaches it. The regional criteria rank his projection second of seven over the world, which is a respectable score for something not optimised against the criterion at all.

So the honest position is that these numbers bound the judgement without settling it, and that the residual — the part a criterion cannot see — is exactly the part a general-purpose world map is chosen on.

An integral over a region says how much and not where. The where has an answer for one family of projections and no answer at all for the rest.

What was computed here #

Every score is computed from the projections’ own derivatives at sample points distributed over the stated region, weighted by the area element.

Each projection is normalised before it is scored. A projection is free to choose its overall scale and an unnormalised comparison would largely measure that choice, so each is scaled so that the geometric mean of its areal factor over the region is one — the only normalisation under which “departure from unit scale” means anything.

Two assertions hold the essay’s argument, and both are of the unusual kind that fail when a result is too tidy. The six projections must rank differently over Europe and over Chile, and Airy’s criterion must rank the world projections differently from Kavrayskiy’s. If either ever came out identical, the essay’s central claim would be false and the build would stop rather than print a table contradicting its own caption.

Projections that cover less than ninety per cent of a region are excluded from its comparison. Transverse Mercator was the first candidate tried against a cap and it “won”, because its declared three-degree window means the sampler sees a sliver, and a sliver has almost no distortion in it. A projection covering four per cent of a region is not a candidate to be the best map of it.

What the pictures cannot show #

The weighting. Every ranking figure shows an order and a set of scores, and the choice that produced them — which region, which function, which measure — is in the caption because it has no visual form.

The figures also cannot show what a reader would notice. A projection can score well and look wrong, or score badly and look fine, because the criteria integrate a local derivative and the eye responds to the shape of a coastline. That gap is where the compromise projections live, and no integral reaches it.

The spread, as a bound on an answer #

A region’s distortion is summarised here as a mean and a worst case. The applied field uses the same quantity for something narrower: the spread of the scale factor over a region bounds how wrong a query answered in that region’s plane can be.

Two candidate sites whose true distances differ by more than the spread cannot swap places, whatever the projection does; a disagreement is therefore always a contest closer than the spread, and the assertion is written that way rather than as a rate. Over Europe the spread is 139 per cent and 5 of 400 queries change answer; over a city it is 24,738 parts per million and none do.

And the three regional summaries an operation actually depends on — worst areal error, worst angular deformation, and the scale spread — have a different winner in each column over one and the same region. A ranking is not one ranking even after the region is fixed.

Who found it, and when #

George Biddell Airy, then Astronomer Royal, proposed the first integrated criterion in 1861 and used it to derive a minimum-error azimuthal projection. His paper is explicit that the criterion is a choice and gives his reasons for it, which is more than most of its successors do.

Kavrayskiy’s logarithmic variant is from the 1930s, and Jordan, Klingatsch and others proposed others in between. No two agree on the ranking of the compromise projections, and the literature has treated that as an unresolved problem for a century and a half.

It is not unresolved. It is what happens when several people choose different weightings for quantities that have no natural exchange rate, and the correct conclusion — that the criterion is part of the question rather than part of the answer — was available from Airy’s own framing.

Where this goes next #

The two quantities being integrated are the two ways a map is wrong. The one regional question with a proved answer is Chebyshev’s criterion. And the practical framework the rankings feed is which projection is best.