The best flat picture is not a map

The two rungs below this one measure a flat picture of a set of places and never once mention a map. That is deliberate and it leaves an obvious question standing: if the best arrangement of four dots is wrong by six-tenths of a per cent, what is a projection of the same four places wrong by?

The comparison is direct. Take a set of places, put them through every projection in this collection’s library, and score the result with exactly the rule the free picture was scored with — the worst of the ratios between page separation and ground distance, after the picture has been given its one free scale.

Thirteen times better. That is a large enough ratio to be worth understanding, and understanding it turns out to require running the same comparison on a set the reader would not have thought to vary.

The comparison had to be rigged in the projections’ favour #

A local search reports the best arrangement it can find, not the best that exists, and a search that starts somewhere unhelpful can finish above a competitor it should have beaten. That is not a hypothetical: the first version of the machinery here started every search at the classical multidimensional scaling solution and reported 72.6 per cent on sixteen cities against the azimuthal equidistant’s 57.6 — a free picture losing to a map, which is impossible, because a map’s own layout is one of the arrangements the free picture is allowed to be.

So the search is now started from every projection’s layout as well as from classical scaling, and the best of those runs is reported. That makes the claim “the free picture wins” true by construction rather than by measurement, which is exactly what it should be: the interesting quantity is not whether the freedom helps but by how much, and rigging the start removes an artefact of the optimiser from the answer.

The site’s own gate on this file requires it. The refusal is that no projection may score below the free picture, and it is checked on the sixteen-city set every time the machinery runs.

What the scoring rule does and does not reward #

One more thing about the comparison has to be said before the numbers mean anything, because the ordering in the hero figure is not the ordering this collection usually reports.

Gall–Peters comes fourth on four cities, ahead of Mollweide, Winkel tripel and Robinson. On every regional criterion this site computes it is a poor performer, and which projection is best shows that the ordering of projections is a function of the criterion and the region rather than a property of the projections. This figure is one more criterion — the worst relative error over the distances of one specific finite set — and it is a criterion no published ranking uses.

What it rewards is a map whose scale is roughly uniform over the particular quadrilateral those four cities sit in, which is a different thing from being uniform over a continent or over the world. Two of the four are at high northern latitudes and one is well into the south, so a cylindrical equal-area projection with its standard parallels in the right place is not the absurd choice here that a world-wide angular-deformation score would make it. Move one city and the ordering moves.

That instability is not a defect in the measurement. It is the reason a bound is worth more than a ranking, which is what the last section of this essay is about.

Why the azimuthal equidistant is the one that comes closest #

It is not a coincidence that the same projection wins on four of the five sets below, and the reason is a counting argument that the next rung makes properly.

An azimuthal equidistant map centred on a place is exactly right about every distance from that place. Not nearly right: exactly, to the last bit of the arithmetic, because the radial coordinate of the projection is the angular distance and nothing else. So on a set of n places whose centroid happens to be one of them, n − 1 of the n(n − 1)/2 distances are perfect and the map’s error is the worst of the rest.

No other projection in the library has any exact distances at all. The places where a map is exactly right finds the curves on which a projection’s scale factor is one, and a scale factor of one at a point says nothing about the distance between two places on opposite sides of the map. An exact distance is a much stronger property and only the azimuthal equidistant family has it.

That is one exact distance per place and no more, which is why the map wins and does not win by much. Fifteen exact distances out of a hundred and twenty is twelve and a half per cent of them, and the other hundred and five are as wrong as any other map’s.

The advantage collapses, and it is not about maps #

The four-city result invites a conclusion — that cartography has been leaving a factor of thirteen on the table for four centuries — and the conclusion is wrong. Run the same comparison on more places.

The two columns of that figure move together, and the second explains the first.

A flat picture of n places has 2n coordinates. Sliding it costs two and turning it costs one, so 2n − 3 numbers in it are genuinely free. It has n(n − 1)/2 distances to get right. At four places that is five free numbers against six constraints, so the arrangement can very nearly do as it likes and the optimiser has real room. At sixteen it is twenty-nine free numbers against a hundred and twenty constraints, and the freedom per constraint has fallen from 83 per cent to 24.

By the time the set is large, almost everything about the picture is forced by the distances themselves, and a projection — which spends none of its freedom on the particular set — is nearly as good as an arrangement that spends all of it. The next rung takes that counting seriously and finds it is exact rather than heuristic: 2n − 3 distances can be held perfectly and no more, whatever anybody does.

And at the other end of the size axis nothing is at stake either #

The advantage is small on a large set because there is no freedom left. It is small on a small set for the opposite reason: there is nothing to fix.

So the freedom is worth most in the middle — a handful of places spread over a large part of the world — and worth almost nothing at either end. That is an unusual shape for a result in this collection, where most quantities are monotone in the span, and it comes from two mechanisms running in opposite directions: the difficulty rises with span, as how wrong a flat picture has to be measures, and the freedom to do anything about it falls with count.

The middle of the size axis, stated as a rule #

Putting the two mechanisms together gives a rule of thumb with numbers in it, and it is the practical content of this rung.

The freedom is worth having when a picture has enough spare parameters to move the worst edge — which needs 2n − 3 to be a decent fraction of n(n − 1)/2, so n below about eight — and when the sphere is causing enough trouble to be worth moving, which needs a span above a few tens of degrees. Both conditions together describe a small number of widely separated places: an airline’s route map, a set of observatories, the stations of a global network, a table of distances in the endpapers of an atlas.

That last one is where the free picture has in fact been used, without anybody calling it that. A distance table is not a picture at all — it prints the numbers — and printing the numbers is the limiting case of getting every distance exactly right at the cost of having no geometry whatsoever. The free arrangement sits between the table and the map, closer to the table than most readers would guess: it keeps the shape of the relationships and gives up everything else.

The four things the winner cannot do #

Everything above scores one quantity, and a projection is not chosen on one quantity. The free picture wins on distance error by definition and loses on everything else, and the list is worth stating in full because it is the whole reason nobody uses one.

It has no inverse. A projection is a rule that takes a coordinate to a page position, and it can be run backwards: a point on the page is somewhere. A free arrangement is a table of dots and the space between them means nothing. A map that cannot be read backwards prices this for a projection whose inverse is merely hard; here there is no inverse to be hard.

It says nothing about a place that is not in the set. There is nowhere to draw a coastline, a river, a boundary or a grid, because none of those things is in the distance table. A map of sixteen cities in this sense is sixteen dots on white paper, and the ink that makes a map recognisable is entirely absent.

It is not stable. Add a seventeenth city and every one of the sixteen dots moves, because the optimisation is over the whole configuration and the new distances re-balance the worst case. A projection’s answer for London does not depend on whether anybody asked about Auckland. This is the property that makes an atlas possible: sheets drawn at different times by different people join up.

It has no properties. A projection can be conformal, or equal-area, or true along a stated line, and those are the terms in which every requirement in this subject is written. The free picture is optimal for one scoring rule on one finite set and has no angular or areal behaviour at all, because it has no derivatives — there is nothing between the dots to differentiate.

And it cannot be checked. A projection is a formula, so a second person with the same formula and the same coordinates draws the same map, which is what makes a published map an object anybody can audit. Report the map, not the parameters is about the weaker version of that failure — a map whose parameters are stated in a form that does not reproduce it — and a free arrangement fails it outright: reproducing one means re-running a stochastic search from the same starts to the same tolerance, and two implementations of the same pattern search will not agree in the last digits.

That list is what every projection minimises something is really about. A projection is an answer to a question about the whole sphere, and the free picture answers a question about sixteen points; the second is an easier question and the answer to it is not a map.

What the free picture is actually good for #

It is not useless. It is the bound — the thing that says how much of a map’s error is the map’s fault and how much is the sphere’s.

Handed the statement that the azimuthal equidistant is 8.28 per cent wrong about four cities, a reader has no way to know whether that is a bad projection, a bad centring or an impossible request. The free picture answers it: 0.63 per cent is available, so 8.28 is mostly the projection’s doing. Handed the statement that it is 57.6 per cent wrong about sixteen, the same comparison says the opposite — 53.3 is available, so nothing at all is the projection’s doing and the request was impossible.

That is exactly the role the Chebyshev optimum plays for conformal maps in Chebyshev’s criterion, where the best possible conformal projection of a region is known in closed form and every other conformal map is measured against it. This is the same manoeuvre for distances rather than for scale, with a pattern search in place of a closed form, and it extends to a case the Chebyshev result cannot reach: a finite set of places rather than a region.

There is a second use, and it is the one that made the fourth rung of this ladder necessary. A bound that is nearly attained by an existing map is evidence that the map is doing the right thing; a bound that is far from attained is an invitation to ask what the gap is made of. The azimuthal equidistant sits at 8.28 against a bound of 0.63 on four cities, and the gap is not mysterious — the map holds three distances exactly and the picture holds five — but nothing in this rung establishes that five is the ceiling, or that a cleverer map could not hold four. That is a counting question and it has an exact answer.

It is also the honest way to state a compromise. A world map that is 140 per cent wrong about some pair of distances sounds indefensible until the bound says 53 is the floor, at which point the interesting question stops being why is this map so bad and becomes which pairs did it choose to be bad about — which is the question compromise projections is written to answer and the one a single error figure hides.

The one place the comparison is unfair to the projections #

Everything above scores each projection at its own defaults, with one exception made deliberately: the azimuthal members are re-centred on the set’s centroid. Nothing else is fitted, and a reader entitled to be sceptical will notice that this is not how a cartographer would work.

A cartographer choosing a map for four cities would fit it. The aspect is free — the aspect is a free choice prices what rotating a projection’s axis buys and finds it is the cheapest available improvement and the one most often left unmade — and the standard parallels of a conic or a cylindrical are free too. Fitting those on the four-city set would move the numbers, and it would move them in one direction only.

How far is a question this rung does not answer, and the omission is recorded rather than hidden. What can be said is a bound on the bound: no amount of fitting takes a projection below the free picture, because a fitted projection’s layout is one of the arrangements the free search is allowed to reach. So the ratio of 13.17 is an upper bound on what the freedom is worth, and the true figure for a well-fitted projection is somewhere between 1 and 13.17.

That is a weaker claim than the figure suggests and it is the correct one. The two claims that do not weaken are the ones at the ends of the size axis: on sixteen cities the ratio is 1.08 before any fitting, so fitting can win at most eight per cent and the case is closed; and on five towns in Britain the absolute numbers are metres, so the ratio of 1.80 is a ratio of two quantities nobody cares about.

The counting that has now been made twice without being stated #

Two rungs of this ladder have leaned on the same arithmetic without doing it properly. The first counted five free numbers against six distances for four places and used it to explain why the sixth constraint fails. This one counted 2n − 3 against n(n − 1)/2 to explain why the free picture’s advantage collapses.

Neither established that the count is tight — that 2n − 3 distances can actually be held exactly, all at once, by a construction rather than by an optimiser that gets close. Neither said which 2n − 3, which turns out not to be a free choice. And neither noticed what the azimuthal equidistant’s exact star means in the same units: n − 1 distances held, against 2n − 3 available, so a map centred on a place is leaving n − 2 exact distances unspent that a picture could have had.