The projections that are beaten on both counts

The rule of thumb was scored over thirty regions and scored again out of sample over forty-five, and both times the question was the same: given a region, does the advice name the family that wins?

There is a harder question the same machinery can answer and the ladder has never asked it. Not which projection wins here, but which projections never win anywhere — which library members are beaten on every criterion by something else, over every region in a population, so that no reader holding any weighting of the criteria should ever choose them.

That is an audit of a recommendation rather than of a claim, and it is the natural place for this ladder to end up: the earlier rungs audited what a projection is called, and this one audits what it is for.

The two axes, and why they are the right two #

The criteria are the collection’s own: the mean angular deformation over the region and the mean absolute logarithm of the areal factor. Neither is a weighted sum of the other, which is what makes dominance — being worse on both — a statement with content rather than a restatement of a single score.

Every projection is normalised before it is scored, exactly as ranking does: a projection is free to choose its overall scale, and an unnormalised comparison mostly measures that choice.

A projection on the Pareto front is one nothing else beats on both axes. It might be a terrible choice for a given purpose; what it cannot be is strictly worse than an alternative in every respect. A projection off the front is exactly that: there exists another library member with less angular deformation and less areal error over the same region, so no weighting of the two criteria can rescue it.

Over Europe the front holds four members: Mercator at zero angular deformation, the sinusoidal at zero areal error, and the conformal conic and the polyconic between them. Robinson sits at 15.85° and 0.088, and the polyconic — at 1.51° and 0.025 — is better than it on both axes at once, by a factor of ten on one and three and a half on the other. That is what being off the front means, and it means it for every weighting a reader could hold between the two.

Which projections are dominated outright #

Over the thirty regions of the training population, two library members are beaten on both counts in every single region by the same projection:

• the orthographic, beaten everywhere by the sinusoidal, the Hammer, the Lambert azimuthal, the azimuthal equidistant and the polyconic;
• the gnomonic, beaten everywhere by those and by Mercator, Miller and the stereographic as well.

Neither is a surprise and both are worth stating, because they are the cases where the audit’s verdict is unambiguous. The orthographic is a picture of a globe rather than a map of a region; the gnomonic exists because it makes great circles straight, which is a property neither of these axes can see. Both are chosen for reasons outside the criteria, and the audit correctly reports that inside the criteria they are never competitive.

The margin is worth quoting: at the region where each does best, the beating projection is still better by a factor of three on one axis or the other. This is not a close call anywhere.

The projection that is on no front and is not dominated #

The more interesting column is the one that is empty for a different reason.

Five library members are on no region’s front over the training population — Web Mercator, Robinson, the Winkel tripel, the orthographic and the gnomonic — but only the last two are dominated by a single rival everywhere. For the other three, the projection that beats them changes from region to region.

That is a weaker claim and a more realistic one. It says: for every region in this population there is something better on both counts, and there is no single alternative to recommend.

Run the same audit over the other two populations and the list moves:

One name is in all three. Robinson is on no regional Pareto front in any population tried — thirty synthetic extents, forty-five different synthetic extents, and the seven real regions this collection’s library carries.

And it is on the world’s #

Score the same library over the whole sphere and the front holds nine members, and Robinson and the Winkel tripel are both on it.

That is not a contradiction and it is not a rescue. It is the answer, and it is the same answer the rule of thumb got two rungs down this ladder: a recommendation without a population behind it is not a measurable claim.

Robinson was constructed in 1963 for Rand McNally’s world maps, by eye, adjusting a table of parallel lengths until the whole sphere looked right. It is on the world’s front. Asked to serve a region of thirty degrees’ extent it is beaten on both counts by projections built for that job, and it would be strange if it were not, because it was never offered for it.

So the audit’s verdict has to be stated with its scope attached, and the scope is the entire content:

• on a world map, Robinson and the Winkel tripel are undominated and the recommendation to use them is defensible on these axes;
• on a regional map, they are beaten on both counts by something in every case tried, and the recommendation is not.

Why the front changes shape from region to region #

The two fronts drawn here look different and the difference is structural rather than incidental.

Over a broad east–west region the front is long: there are many ways to trade angle against area and several projections occupy the middle of the trade. Over a compact region the front collapses towards the corner, because a well-fitted conic can get both errors small and there is little left to trade.

That has a consequence for how the audit should be read. A projection off the front over a compact region is off it by very little, because everything is near the corner; a projection off the front over a broad region can be off it by a great deal. So the count of regions a projection is undominated on is a coarse summary, and the honest version — which the chart above draws — is the front itself, region by region.

It also explains why the never-on-a-front list is longer for the synthetic populations than for the named one. Seven named regions give seven chances to be on a front; thirty give thirty, and a projection that is marginal everywhere fails more of them.

What was computed, and how #

Each population is generated by a stated construction rather than chosen. The training set is a grid of five centre latitudes, two heights and three ground aspect ratios; the held-out set is a different grid — latitudes offset by half a step, heights and shapes from different values, running to more extreme aspects than the training set contains; the named set is the library’s own seven regions.

Every projection is scored over every region with its scale normalised, and a projection that cannot cover a region is recorded as absent rather than scored badly. That distinction matters: the Aitoff projection is absent from the whole audit because it fails to cover at least one region in the population, and reporting it as dominated would be reporting a failure to appear as a failure to compete.

The dominance test is deliberately strict. A projection is dominated only if the same rival beats it on both axes in every region of the population — not on average, not in most. That is why the list of dominated projections is two names long while the list of never-on-a-front projections is five.

The refusal is built in: every dominated projection must appear on no region’s Pareto front, which is a consequence of the definition and would fail if the front computation and the dominance computation had drifted apart.

What a dominated projection is still good for #

The verdict on the orthographic and the gnomonic needs a paragraph of defence, because “dominated on both counts everywhere” sounds like a recommendation to delete them and it is not.

The gnomonic is the only projection on which every great circle is a straight line. That is a third property, orthogonal to both axes here, and it is the entire reason the projection exists: a navigator laying off a great-circle route with a ruler is using a property no amount of angular or areal fidelity substitutes for. The gnomonic companion is about exactly that use.

The orthographic is a picture of a globe from far away. Its purpose is recognition rather than measurement — it shows a reader what part of the sphere is being discussed — and it is dominated on both axes in the same sense that a photograph is dominated by a plan.

So the audit’s finding about them is best read as a boundary marker: these two projections are outside the criteria, and any comparison that scores them on angular and areal error alone is scoring the wrong thing. The finding that matters is the one about Robinson and the Winkel tripel, because those two are offered as general-purpose maps, which is a claim these axes can test.

Where the model stops #

Two criteria are not all the criteria. A projection can be chosen because it makes great circles straight, because it tiles, because it is fast to compute, because a standard mandates it, or because a reader recognises it. None of those is on either axis. The gnomonic is dominated on both and is the only sensible choice for a great-circle chart, and the audit does not contradict that — it measures what it measures.

The mean is one summary. These are means of angular deformation and log areal factor over a region. A projection with a low mean and a terrible worst point would look good here, which is why the worst point is its own essay and why a serious selection uses both.

A population of synthetic boxes is not a population of countries. The training and held-out sets are rectangles and caps generated by a rule. Real regions have coastlines, holes and awkward diagonals, and the aspect ratio of a bounding box is a poor summary of them. The named population is the honest one and it has seven members.

Nothing here scores the world map’s other purposes. A world map is chosen for how it reads as much as for what it measures, and this collection has no way to score that. The claim is only that on these two axes the compromise projections earn their place on a world map and not on a regional one.

The generalisation #

The finding is a pair and the pair is what generalises.

Dominance is measurable and it is rare. Over thirty regions, exactly two of twenty projections are beaten on both counts everywhere by the same rival, and both are projections nobody recommends for general use. The library is not full of dead weight.

Being off the front is common, and it is population-dependent. Five projections are off every front over one population, four over another and three over a third, with one name common to all. What a projection is for decides whether the population it is being scored over is the one it was offered for.

Which turns the ladder’s own question round. The earlier rungs asked whether a piece of advice was right. This one says that a piece of advice cannot be right or wrong until somebody names the population, and that the strongest possible verdict — beaten on both counts, everywhere, by the same thing — applies to almost nothing.

That is a more useful result than a list of bad projections would have been. There is no list. There is a scope, and most published advice omits it.

Who found it, and when #

Pareto dominance comes from Vilfredo Pareto’s economics of the 1890s and reached engineering design in the 1960s; the phrase Pareto front is standard in any multi-objective optimisation.

Applying it to projections is not new in principle — every discussion of compromise projections is implicitly about a trade-off surface — but published comparisons almost always reduce the two criteria to a single composite score first. Airy’s 1861 criterion, Kavrayskiy’s, Airy–Kavrayskiy and the Goldberg–Gott flexion measures are all single numbers, and a single number cannot express dominated, because it has already chosen the weighting.

The population dependence is the older lesson. Tobler’s work on projection selection in the 1960s and Snyder’s in the 1980s both frame the choice as a function of the region, and the failure mode this essay measures — advice given without a region — is the one those authors spent their careers arguing against.

What dominance survives a change of criterion #

The open question at the end of this rung has a partial answer available from the structure of the argument, and it is worth stating because it says how much of the verdict is safe.

Dominance is a stronger claim than ranking, and it degrades more gracefully. A ranking by a composite score is entirely a function of the weighting: change the weights and the order changes. A dominance verdict says one projection is worse on both axes, which survives any weighting of those two axes whatsoever — the whole family of composite scores built from them agrees.

So the dominated set is robust across every criterion that is a function of these two invariants, which is a large class: Airy’s, Kavrayskiy’s, Airy–Kavrayskiy and every weighted mean of angular and areal error. None of them can rescue a dominated projection, because there is nothing to rescue it with.

What is not robust is dominance against a criterion measuring something else. A worst-point score is not a function of the two means; neither is a distance criterion, nor a second-derivative one. A projection dominated on the means may be the best available on the worst point, and nothing in this rung’s arithmetic forbids it.

Which sharpens the question the ladder is leaving open. It is not do the fronts agree, which is too coarse; it is whether any projection dominated on the two mean invariants is undominated on a criterion of a different order. That is a specific thing to look for and it has a specific expected answer — the second-derivative measures are the likeliest place, since the best compromise for angle is not the best for bending already found the two orders’ optima far apart on one blend path.

That is worth separating from the stronger thing it might be mistaken for, which is a claim that these projections should not be used.

Meanwhile the verdict as stated is narrow and sound. These projections are beaten on the two quantities that describe a map’s first derivative, by projections that are also in common use, and no weighting of those two quantities changes it.

Where the ladder goes next #

Eight rungs have audited a projection’s stated property, its behaviour on the wrong body, its rule of thumb, that rule out of sample, and now the recommendation itself.

What none of them has audited is the criterion. Every verdict above depends on scoring by mean angular deformation and mean log areal factor, and those two were chosen by this collection because they are the two independent invariants. A reader who cares about the worst point, or about distance, or about the second derivative, would get a different front — and whether the fronts agree with each other is a question the machinery could answer and has not been asked.