The rule of thumb, scored

Every introduction to the subject ends with the same paragraph. Use a cylindrical projection near the equator, a conic in the middle latitudes and an azimuthal at the poles; if the country is long north to south, use a transverse cylindrical, which is what a UTM zone is.

It is the most repeated practical advice in cartography, it is the thing a reader is most likely to remember from a first course, and this collection has never scored it. Not from an oversight: scoring a rule needs a population of regions rather than a region, and every measurement here so far has been about one country or one continent at a time.

Thirty regions built to order, with every family given its own parameters, produce a number.

What a fair comparison has to give each family #

A rule about families can only be scored if each family gets its best shot, and this is where such comparisons usually go wrong. Scoring “the conic” with a cone constant chosen for somewhere else, or “the azimuthal” in its normal polar aspect over an equatorial region, measures the choice of parameters rather than the family.

So each family is fitted to the region before it is scored:

• the conic gets its cone constant, searched across the range;
• the cylindrical gets a standard parallel, and the choice between a normal and a transverse axis — the transverse case is what the rule’s own north–south clause is about;
• the azimuthal gets its centre, which every textbook takes for granted when it says azimuthal at the poles and which is a free choice anywhere.

The score is Kavrayskiy’s criterion — the root-mean-square of the logarithms of the two principal scale factors, over the region — and each candidate is normalised before scoring so that its overall scale, which is free, is not what is being measured. That normalisation is not an optional refinement: without it, every family came back at infinity or at a number the size of the map.

The answer, and its shape #

Nineteen of thirty, which is 63 per cent, and the failures are not scattered.

Sorted by the shape of the region rather than by its latitude:

• tall regions — a third as wide as they are high: the rule is right nine times out of ten, and the winner is a transverse cylindrical, which is exactly the clause about north–south countries;
• wide regions — two and a half times as wide as high: right eight times out of ten, cylindrical at the equator and conic in the middle latitudes, which is the main clause;
• compact regions — as wide as they are tall: right twice out of ten, and the winner at every latitude from the equator to 80° is the azimuthal equidistant centred on the region.

That is the whole finding. The rule’s clause about shape is nearly always right. Its clause about latitude — the part everybody remembers, the part with the picture of a cylinder, a cone and a plane wrapped round a globe — is right only for elongated regions, and for a compact one it prescribes the wrong family everywhere except the poles.

Why the azimuthal wins a compact region #

The reason is not subtle once the picture is drawn, and it is the reason the rule’s mental image misleads.

An azimuthal projection’s distortion depends only on the angular distance from its centre, which is what the azimuthal family being one function means in practice. Over a compact region every point is within a small angular distance of the centre, so the distortion is small and — this is the part that matters — isotropic in its dependence: there is no direction in which the region extends further than the projection is good for.

A conic’s distortion depends on the distance from its standard parallels, which are lines. A cylindrical’s depends on the distance from one line. Both are built for a region that is long in one direction and short in the other, and over a region that is equally extended in both, they are being asked to do something their construction has no way to exploit.

The cylinder, the cone and the plane are a taxonomy of the shape of the region they suit, not of the latitude. Cylinders, cones and planes argues that the taxonomy says less about properties than it appears to; this rung finds that what it does say is about geometry of extent, and that the latitude in the rule is a proxy for the shapes countries usually have.

What following the rule costs #

A rule that is wrong 37 per cent of the time is only interesting if being wrong costs something. It does: the average miss costs a factor of 2.25 in the distortion criterion, and the worst costs 4.49.

Those are not small numbers in this subject. Designing a grid for one region finds that a country’s own national grid beats the international zone by a factor of 1.51 and treats that as worth the whole apparatus of a national mapping agency. Following the taught rule on a compact country costs more than that, twice over.

What was computed, and how #

Thirty regions, ninety family scores, one criterion.

Each region is a box: a stated centre latitude, a stated height in degrees, and a width set by the shape ratio and corrected for the convergence of the meridians, so that a “square” region is square on the ground rather than in the graticule. That correction matters — without it a square-looking box at 80° north is four times wider than it is tall.

For each region every candidate in every family is scored, the best in each family is kept, and the three are ranked. The rule’s own prescription is then read off the region’s centre latitude and shape, and the two are compared. What is reported is the family, not the projection: a rule about families is not answerable with a projection’s name.

One correction was needed and it is worth recording, because the wrong version produced a much more dramatic essay. The azimuthal candidates were first built by rotating the sphere so that the projection’s pole went to the region’s centre — the site’s general aspect machinery, used correctly for a cylindrical projection and wrongly here, because an azimuthal projection is already centred on a point and carries its centre as a parameter. It put the map somewhere else entirely and scored the polar stereographic at 0.047 over a region it describes at 0.005. The rule then appeared to fail catastrophically at high latitudes, costing a factor of 182.

A rule that fails by a factor of 182 is a more interesting result than a rule that fails by 4.5, which is exactly why the number had to be distrusted. The correction reduces the finding and makes it real.

The rule is a summary of what countries look like #

Put the two family plots side by side and the rule stops looking wrong and starts looking conditional. For a wide region the curves cross where the rule says they cross, and the advice is good. For a compact one they do not cross at all.

Which suggests where the rule came from. The regions European cartographers spent three centuries mapping — Europe, the Mediterranean, the conterminous United States, the Russian Empire — are wide rather than compact, and for wide regions the latitude clause is a good summary. The rule is an accurate generalisation from a biased sample of regions, and it has been taught as a fact about projections rather than as a summary of the shapes that were being mapped.

That is a specific, checkable version of a suspicion this field is built on, and it is worth distinguishing from the usual complaint. The rule is not wrong because it is old or because it is simple. It is wrong on a third of a fair sample because its stated variable is not the one that decides the answer.

The same question asked about named projections #

It is worth seeing the family comparison beside the one this collection already makes. Ranking six named projections over two regions shows the order changing between them, which says that the projection question has no region-free answer. The rule of thumb is a claim that the family question does have one, at least approximately, once the region’s latitude is given.

Both are answers to the same underlying difficulty and they fail in different places. The named ranking fails to be portable at all. The family rule is portable and wrong on a third of a fair sample. Between them they say something the subject usually leaves implicit: there is no summary of this choice that survives without a statement of the region, and shortening the statement of the region is what a rule of thumb does.

The one clause that is nearly always right #

The rule’s north–south clause deserves separating from the rest, because it is the part that scores nine out of ten and it is the part that is stated as an exception.

If the country is long north to south, use a transverse cylindrical is advice about an aspect, not about a family — it says to take the cylindrical projection and turn its axis. Fitting the aspect to the region measures what that is worth on real regions and finds gains of a factor of seventeen over Europe and 3.8 over the conterminous United States; the sweep figure above shows the same thing for Chile against an equatorial band.

So the strongest part of the taught rule is the part about orientation, and the weakest is the part about which of three shapes to wrap round the globe. That is close to the reverse of the emphasis it is usually given, where the aspect is a footnote and the three shapes are a diagram on the first page.

Where the model stops #

Four limits, and the first two would change the number.

The criterion decides some of the verdicts. Kavrayskiy’s criterion weights the logarithms of both principal scales equally; Airy’s weights their departures from one. Which projection is best shows the ranking changing with the criterion, and a rule scored under a different criterion would get a different mark. What does not change is the shape of the failure, because the azimuthal’s advantage over a compact region is geometric rather than metric.

The regions are boxes. Real countries are not, and choosing for a line rather than a region is the case where the shape matters most, and a country whose extent is compact but whose shape is irregular may behave like neither case. This site does not have coastlines, for reasons its founding decision states, and the price is paid here.

Every family gets its optimum parameters. A cartographer choosing from a list of published projections with published constants does worse than every family does in this comparison, and the rule may well be better advice under that constraint than under this one.

And nothing here is about purpose. The whole comparison scores a compromise criterion, which means it has already assumed that what is wanted is general faithfulness. No essay here may say a projection is best without naming the purpose, and a map made for one purpose — navigation, area comparison, a thematic display — is not scored by any of this.

What a better rule would have to say #

A rule has to be short or it is not a rule, and the obvious repair — use an azimuthal for a compact region, a conic for a wide one at middle latitudes, a transverse cylindrical for a tall one — is exactly as short as the one it replaces and scores better on this population.

Whether it would survive a fair test is a separate question, and this rung cannot answer it. The population here was built to order: boxes at stated latitudes with stated shapes, three of each, thirty in all. A rule fitted to that population and scored on it is a rule scored on its own training data, which is a mistake this collection is in no position to make casually.

What can be said is narrower and more useful. The variable the taught rule keys on is not the variable that decides the answer, and any replacement should key on shape. Whether the replacement’s own boundaries — how compact is compact, how wide is wide — hold up outside the thirty regions here is a measurement somebody would have to make on a population nobody chose.

The generalisation #

The pattern is one of the most common ways a good heuristic goes wrong, and the diagnosis is available whenever a rule can be scored: the rule’s stated variable is correlated with the deciding variable in the sample it came from, and the correlation fails outside it.

Latitude and shape are correlated across the regions cartography grew up on. The rule keys on the one that is easier to state, works while the correlation holds, and misprescribes when it does not. Rediscovering the deciding variable takes a population and a score, which is what this rung is.

Who found it, and when #

The taxonomy is Lambert’s inheritance and the advice is at least as old as the nineteenth-century atlases. Its modern form — the cylinder, cone and plane wrapped round a globe, with a caption about equatorial, middle and polar regions — appears in essentially every twentieth-century textbook, and Snyder’s Map Projections: A Working Manual (1987) gives it with the qualifications that most later repetitions drop.

What has been available and unused is the arithmetic to check it. Scoring three families over thirty regions with every parameter optimised is a few seconds of computation, and it was days of work for anybody before it was seconds.

Why a rule keyed on the wrong variable still works #

The finding that the rule keys on latitude while the deciding variable is shape is the rung’s sharpest statement, and it explains something more general about why bad rules survive.

A rule keyed on a correlated variable is right whenever the correlation holds. Across the regions cartography grew up describing, compactness and latitude go together: equatorial regions in the classical atlas tradition are wide bands, mid-latitude ones are elongated east–west countries, and polar ones are caps. So the rule prescribes correctly for most of what its users ever apply it to, and it does so for a reason that has nothing to do with the mechanism.

Which makes it unfalsifiable in ordinary use. Every correct prescription confirms it; the incorrect ones occur on regions outside the correlation, which are exactly the regions the rule’s users encounter least. A century of successful application is consistent with the rule being about the wrong thing entirely.

And it explains why the correction is hard to teach. Use an azimuthal for a compact region and a conic for an elongated one is not more complicated than the latitude version, and it lacks the picture — the cylinder, the cone and the plane wrapped round a globe — which is what makes the latitude version memorable. The mnemonic carries the wrong variable because the wrong variable is the one the physical construction is about.

The general form is worth carrying out of the essay. A heuristic that works can be keyed on a proxy rather than on a cause, and the only way to tell is to find a population where the proxy and the cause come apart. That is what scoring over thirty regions does, and it is the reason a rule can be both durable and wrong about its own mechanism.

Where the ladder goes next #

This rung scores a rule about families, holding each family’s aspect fixed at what the rule implies. The next question is the one it exposes: the azimuthal wins a compact region because it can be centred on it, which is a statement about aspect rather than about family — and the aspect of any projection has three degrees of freedom rather than the one a sweep usually searches. That is the third parameter, run, where searching all three turns out to be worth up to 1.87 times and to be unreachable from the two-parameter answer.