Which projection a weighting can make best

The projections that are beaten on both counts computes the Pareto front of a twenty-member library and reports what it excludes: two projections dominated outright, and Robinson on no regional front in any population. What it says about the survivors is the standard thing to say — nothing beats them on both criteria, so any of them might be the right answer for a reader who weighs the two differently.

Four of the seven are not the right answer for anybody.

The front, as the previous rung leaves it #

The exclusion is not this essay’s work and is worth having in view before the weighting arrives, because everything below is about the gap between what the front rules out and what a reader with a preference actually gets.

That is one region. Asking the same question of a population of regions turns a yes-or-no into a count, and the count is what gets quoted as though it were a ranking.

Both pictures answer what is not ruled out. Neither answers the question a reader with a preference actually has, which is what does my preference select, and the two answers are not the same set.

What a weighting is, geometrically #

A reader who cares about both criteria and has to choose is minimising some combination of them. The usual combination is a weighted sum,

$$s(w) = w\,\hat\omega + (1-w)\,\hat a,$$

with the two errors normalised to the library’s own range — because a weighting between a quantity in degrees and a quantity in log-area is otherwise a statement about the units.

In the plane of the two normalised errors, the set of projections with a given score is a straight line, and minimising the score means sliding that line up from the origin until it first touches a projection. A line sliding in from below touches the convex hull of the point set, never a point sitting in a dent between two others.

So a projection can be undominated — nothing is better on both counts — and still be beaten, for every weighting, by a mixture of two others’ positions. That is a strictly stronger exclusion than domination and it is the whole of this rung.

The four the weighting cannot reach #

Over the world, with twenty projections scored on angular and areal error:

Web Mercator, Miller, the equirectangular and the Winkel tripel are on the front and are first for no weighting at all.

The Winkel tripel is the interesting name in that list. It is the projection National Geographic adopted for its world maps in 1998, it is a deliberate compromise between the two failures, and it is exactly the kind of projection this analysis was invented to recommend. On this library, over the whole world, on these two criteria, a reader who states any weighting of angle against area is told to use something else.

That is not a verdict against the Winkel tripel, and the next two sections are about why: the exclusion turns out to be a statement about the arithmetic of the trade-off rather than about the map.

What the three winners own #

Robinson’s window is twenty-eight thousandths wide. A reader whose weighting falls inside it has Robinson as their answer; a reader a thirtieth of the way to either side has Eckert IV or Mercator, and neither of those looks anything like Robinson.

That is a much stronger version of the instability which projection is best opens the collection with. There the point is that the answer depends on the purpose; here the answer depends on the purpose to three decimal places over one narrow band, and is stable over the rest.

Nobody knows their own weighting to three decimal places, so the honest reading of that table is that Robinson is not selected by this procedure at all — it is selected by an accident of where a number nobody can state happens to fall.

How many answers there are #

The first place is one reading of the table. The whole ordering is another, and there are more of them than there are projections.

Over the world a weighting produces 121 distinct orderings of the twenty projections, out of a possible 191. Over Europe 101, over Chile 112, over Japan 91.

That is the number to hold beside any published ranking of projections. There are 2.4 × 10¹⁸ orderings of twenty things; a weighting of two criteria reaches 121 of them; and which of the 121 a reader gets is decided by a number they were never asked for.

The same arithmetic on five regions #

The world is the region with the most structure in it, and the pattern is not peculiar to it.

Two things in that table are worth separating. The gap between the front and the hull opens where the front is large, which is where a reader most needs help choosing — over the world and over Chile, and not over Japan, where the front has three members and all three are selectable.

And the number of first places is three almost everywhere, whatever the front does. A two-criterion trade-off over a library of twenty offers a reader a choice between three maps, and the choice between them is a number the reader has to supply.

Why the front is still the right first cut #

The exclusion above is real and it is narrow, and it is worth being precise about what it does not say.

A dominated projection is beaten by a single other one on both counts — the orthographic and the gnomonic, everywhere. That is a verdict a reader cannot argue with, whatever they care about, and it is what rung 8 established.

A projection off the hull is beaten by a combination of two others’ scores, which is not the same thing at all, because a combination of two scores is not a map. Nothing exists at that point on the line. What the exclusion says is that a reader minimising a weighted sum will always prefer one of the two neighbours — and that is a statement about the reader’s arithmetic rather than about the projection.

The distinction matters because the arithmetic is a choice.

A reader who says “minimise whichever of the two is worse, weighted” — the Chebyshev scalarisation, and a perfectly ordinary way to state a preference — is minimising a corner rather than a line. A corner reaches into a dent. Under it all four of the excluded projections become somebody’s answer, and the Winkel tripel is one of them.

So the honest statement is: which projections a stated preference can select depends on the form of the trade-off and not only on its weights, and the weighted sum that everybody uses is the form that can reach the fewest.

The corner cannot rescue a dominated projection #

The refusal that makes the last section a finding rather than a loophole: no scalarisation that is monotone in both criteria can put a dominated projection first, and the measurement confirms it. Every projection the corner selects is on the front; the two that rung 8 found dominated outright over every region stay out under every weighting of either form.

That is what separates the two exclusions. Domination is a property of the projections. Hull membership is a property of the projections and of the arithmetic somebody chose to combine their scores with.

A correction the measurement forced #

Computing the front on the raw scores turns out to be the wrong instrument for this question, and the reason is worth recording.

Every equal-area projection scores an areal error of zero, so what separates two of them is the last bits of a quadrature — numbers of order 10⁻¹⁷. A domination test with a relative tolerance treats those as real, and over Europe it put Mollweide on the front behind the sinusoidal, which beats it on angle and ties it on area.

The scores here are normalised to the table’s own range before the front is computed, so an absolute tolerance is available and means something: two projections whose scores differ by a billionth of the library’s own spread are tied. Over Europe that takes the front from six members to four.

This is the same lesson the tolerance that decides the verdict draws about this collection’s conformality test, one anchor along: a tolerance chosen against one question is a different tolerance when the question changes, and a relative test on a quantity whose true value is zero is not a test.

What a reader should be told instead #

The practical form of all this is a change to what a ranking prints, and it is three lines rather than one.

The front, which says what is not ruled out. The hull, which says what a weighted sum can select. And the window each hull member owns, which says how precisely a reader would have to know their own preference for the answer to be theirs.

Printed that way the Winkel tripel’s position is honest and legible: on the front, off the hull, selectable under a corner scalarisation and not under a sum. A reader learns that it is a defensible map and that the usual procedure will not recommend it, which is a fact about the procedure and is exactly what they need in order to overrule it.

Printed as a ranking — which is what every published comparison of projections is — the same information reduces to a position in a list, computed at a weighting the author chose and did not state. The list is one of a hundred and twenty-one, and nothing on the page says so.

The narrower point is that a hull is cheap. It is a convex hull over as many points as there are candidates, in as many dimensions as there are criteria — milliseconds, on any comparison anybody has published. Nothing about the extra two lines is expensive; what is missing is the question.

Where the model stops #

The two criteria are the ladder’s own — mean angular deformation and mean log areal error over the region, both from the rule-of-thumb rungs and both region-dependent by construction. A third criterion would make the front a surface and the hull a polyhedron, and a projection excluded in two dimensions can be a vertex in three; nothing here says how often that happens.

The normalisation is to the library’s own range, which makes the weight axis interpretable and makes the answer depend on the library. Adding a very bad projection stretches the range and moves every window; the set of hull members does not change, because the hull is affine-invariant, but the shares in the table are not.

And the scoring is a mean over a region. Where the worst point is makes the case for a maximum instead, and a maximum-based table would have its own front and its own hull.

The generalisation #

A Pareto front tells a reader what is not ruled out. It does not tell them what any stated preference can select, and the gap between the two is usually most of the front.

Seven undominated, three selectable. That ratio is not special to projections — it is the ordinary shape of a two-criterion trade-off with a slightly concave front — and the practical consequence is a habit rather than a calculation: after computing a front, compute its hull, and report both. The members in between are the ones a reader will be told are candidates and will never be given a reason to choose.

The second consequence is sharper and less comfortable. The projections a procedure can recommend are decided partly by the shape of the objective the procedure uses, and that shape is nearly always a weighted sum because a weighted sum is easy. The Winkel tripel is a good world map that a weighted sum cannot recommend, and the right conclusion is about weighted sums.

Who found it, and when #

That a weighted sum can only reach the convex hull of a Pareto set is the standard limitation of linear scalarisation in multi-objective optimisation, known since the 1960s and stated in every treatment of the subject. The Chebyshev scalarisation exists precisely to reach the rest, and its ability to do so is a theorem rather than a measurement.

What is new here is the arithmetic on a real library. Nobody appears to have asked which projections a stated weighting of the classical criteria can actually select, and the answer over the world — three out of twenty, with one of them owning under three per cent of the range — is sharper than the general theory would lead anyone to guess.

What the arithmetic adds to a known theorem #

The limitation is textbook and the number is not, and the gap between those two is worth stating because it is the case for doing arithmetic on a real library at all.

The theorem says a weighted sum reaches only the convex hull of the Pareto set. That is general, it is proved, and it warns that some non-dominated options are unreachable. What it cannot say is how many, or which, or whether the unreachable ones are the interesting ones.

The measurement says three out of twenty. Seventeen projections in a common library cannot be selected by any weighting of the classical criteria whatever — not because they are bad, but because they sit in the non-convex part of the front — and one of the three that can be selected owns under three per cent of the range of weightings, so it appears only for a narrow band of preferences nobody is likely to hold.

Which turns a caution into a verdict about a practice. Choose a weighting and optimise is the standard way a projection is selected in the multi-criteria literature, and the arithmetic says that procedure is choosing between two projections while appearing to choose between twenty. A reader told the method searched a library has been told something true and misleading.

There is a remedy and it is not to abandon scalarisation. The Chebyshev form reaches the non-convex parts of the front by construction, so a selection procedure using it can in principle nominate any non-dominated member — at the cost of a criterion that is less intuitive to state and harder to defend to somebody who wanted to say area matters twice as much as angle. That is a real trade rather than a free improvement, and it is the reason the weighted sum persists.

The cheaper remedy is to report the front rather than the winner. A procedure that returns the non-dominated set, with each member’s two scores, hands the reader every option the criteria admit and lets the preference be applied by somebody who holds it. The weighting is then a reading of a published table rather than a parameter buried in a search, and the seventeen unreachable projections are at least visible.

And the sharpness is what a general theorem could not supply. Some options may be unreachable invites the reasonable assumption that the unreachable ones are marginal. Eighty-five per cent of a working library is not marginal, and the only way to find that out is to run the arithmetic on the library people actually use.

Which is also the honest reading of what this rung has established. Not that weighted selection is wrong — it answers the question it is asked, correctly — but that the question it is asked is much narrower than the framing suggests, and the narrowness is invisible from the output. A procedure returning one projection from a library of twenty looks like a search over twenty whatever the geometry of the front happens to be.

That is a reporting defect rather than a mathematical one, and it is fixed by printing the set.

Where the ladder goes next #

Every score on this ladder is a mean over a region, and every region is a rectangle in latitude and longitude. What the ladder has never audited is its own sampling: a mean over a graticule is not a mean over the ground, and the same instrument that draws an indicatrix field decides how much of the map a reader is being shown.