Compromise projections

Every projection so far in this collection preserves something exactly and pays for it elsewhere. There is a third option, and it is what the two most widely used world maps of the last forty years both are: preserve nothing, and distort everything a little.

The case #

The trade-off is forced, so a projection preserving angles exactly must accept unbounded areal error, and one preserving area exactly must accept large angular error.

But nothing requires a projection to preserve anything exactly. Give up both, and the freedom bought can be spent reducing both — so a compromise projection can have less angular distortion than any equal-area projection and less areal distortion than any conformal one, simultaneously.

That is not a paradox. It is what happens when a constraint is relaxed: the feasible set gets larger, and the best available point on any single axis gets worse while the best available combination gets better.

For a map whose purpose is “look at the world”, there is no property that must hold, so the constraint was never earning anything.

Measured on angular deformation against latitude, Mercator is flat on zero — exact, at a price paid on the other axis — and Gall-Peters rises steeply. Measured on areal inflation the two swap: Gall-Peters is flat and Mercator runs to fifteenfold. The compromises sit between on both plots at once, and being between on both at once is the whole of the case for them.

Read those two figures together. A curve flat in the first is steep in the second; the compromise curves are moderate in both; and there is no curve flat in both because there cannot be.

Robinson #

Arthur Robinson was commissioned by Rand McNally in 1963 to produce a world map that looked right. He worked backwards from appearance rather than forwards from a criterion — adjusting the spacing and length of the parallels, printing, looking, adjusting again.

The published definition is a table of nineteen numbers. There is no formula, and the table is not an approximation to one. Robinson described the process as beginning with the visual and ending with the mathematical, which is the reverse of how the subject usually works and is the reason the projection is sometimes treated as slightly disreputable.

It is also the reason it works. The criterion was “a competent cartographer looking at it thinks the world looks like that”, which is a real objective, is the actual objective of a general-purpose world map, and is not expressible as an integral.

National Geographic adopted it in 1988, replacing the Van der Grinten they had used since 1922.

Winkel tripel #

Oswald Winkel’s projection of 1921 is the arithmetic mean of two others: the equirectangular projection with a standard parallel at $\arccos(2/\pi)$, and the Aitoff. Average their coordinates and the result is the Winkel tripel.

The name means “triple” and refers to the three distortions Winkel wanted to control at once — area, direction and distance. It is a compromise in the most literal available sense: two projections, averaged.

Averaging two projections is a slightly startling operation. It has no geometric interpretation, it does not preserve any property either parent had, and it produces something better than either. That it works is an argument for treating projections as functions to be combined rather than as constructions to be respected.

National Geographic replaced Robinson with Winkel tripel in 1998, after studies comparing distortion measures ranked it better on most of them. It remains their standard.

How to evaluate something with no exact property #

This is the genuine difficulty with compromises, and it should be stated rather than glossed.

A conformal projection can be verified: measure the angular deformation and require it to be zero. The claim is falsifiable and this site falsifies it for one projection.

A compromise projection claims only to be reasonable. There is no measurement that confirms or refutes it, because every scalar summary of distortion encodes a weighting somebody chose, and different weightings rank the compromises differently.

So compromise projections are the one part of this subject where the site’s central discipline has nothing to bite on. Their distortions can be measured exactly — and they are, in the figures above — but whether those numbers constitute a good compromise is a judgement.

The honest position is that this is a judgement, made by people who make maps for a living, and that the measurements inform it without settling it.

The pole problem #

One structural choice divides the compromises, and it is worth understanding because it is the most visible difference between them.

The poles are points. A projection can render them as points, which is topologically correct and squeezes the high latitudes into an increasingly narrow wedge; or as lines, which keeps the high latitudes readable and asserts something false.

Mollweide, Hammer and the azimuthals render them as points. Robinson, Winkel tripel and Eckert IV render them as lines.

Neither is right. A pole line makes Antarctica and the Arctic legible at the cost of stating that the pole is a place with extent; a pole point is correct and makes the polar regions unreadable. General-purpose maps mostly choose the line, because their readers mostly want to see Greenland.

What the compromises actually measure #

Since the case for them is quantitative, the numbers are worth putting side by side.

Over the sample this site takes, Robinson reaches 107° of maximum angular deformation and 1.7 of areal error. Winkel tripel reaches 86° and 2.8. Mercator reaches zero and ninety. Gall–Peters reaches 146° and zero.

So on angular deformation the compromises are better than either equal-area projection and worse than Mercator; on areal error they are better than Mercator by more than an order of magnitude and worse than the equal-area projections. That is the shape of a compromise, stated in numbers.

The worst-case figures overstate the case against them, because the extremes occur at the map’s corners where little land is. A land-weighted average would flatter the compromises further, and would be a weighting somebody chose.

The interrupted alternative #

There is a different way to reduce both distortions at once, and it is worth mentioning because it is the only one that genuinely works.

Cut the map. Goode’s homolosine is an equal-area projection assembled from sinusoidal and Mollweide sections, interrupted through the oceans so that each landmass sits near a central meridian where the shape distortion is small.

The result preserves area exactly and keeps shape distortion far lower than any uninterrupted equal-area projection — because each lobe covers less of the sphere and carries less curvature.

The cost is that the map is torn. The oceans are cut, distances across the cuts are meaningless, and the whole is no longer a continuous picture of a continuous surface. For a thematic map of land that is often an acceptable price, and for anything about oceans it is not. Interruption is a design variable in its own right, and the number of lobes trades shape fidelity against the number of places the map has been cut.

The criterion the compromises lack, and where one exists #

The un-checkability above is worth localising, because it is a property of world maps rather than of compromise as an idea.

Restrict the question to a bounded region and an exact optimum reappears. Chebyshev’s criterion says that among all conformal projections of a region, the one minimising the variation of scale across it is the one whose scale is constant on the region’s boundary — a statement with a proof, a unique answer, and a computable test.

That is the shape a defensible “best” takes: a stated family of candidates, a stated objective, and a theorem picking one out. Robinson and Winkel tripel have none of the three. Their candidate family is “any smooth map of the whole sphere”, their objective is appearance, and there is no theorem.

The difference is not rigour against sloppiness. It is that a bounded region has a boundary for the criterion to live on, and the sphere does not.

Why averaging works #

Winkel tripel is the arithmetic mean of two projections, and that construction deserves more attention than it usually gets.

Averaging two maps is not a geometric operation. There is no surface being wrapped, no light source, no construction — just two sets of coordinates added and halved. It has no interpretation beyond the arithmetic.

And it works, for a reason worth naming: the two parent projections fail in different directions. The equirectangular stretches the high latitudes horizontally; the Aitoff compresses them. Averaging cancels part of both errors while keeping neither parent’s exact property.

That is a general technique and it is under-used. A projection is a function, functions can be combined, and the result is a projection whose properties are whatever they turn out to be. Nothing requires a map to be constructible.

The evaluation problem, restated #

Since this is the one place the site’s method does not reach, it is worth being precise about why.

An exact property is a universally quantified statement: the angular deformation is zero at every point. That is falsifiable by a single counterexample, and this site looks for counterexamples.

“Reasonable overall distortion” is not universally quantified and not falsifiable. It is a judgement about an aggregate, and every aggregate encodes a weighting.

So the compromises can be measured precisely and cannot be checked. The measurements are still worth having — they bound the judgement even where they cannot settle it — and the honest caption reports the numbers and stops.

Robinson’s indicatrices show moderate shape distortion, moderate areal variation, no exact property anywhere and no clean pattern. That is what the measurements can say about it, and it is less than a verdict.

The name is doing them a disservice #

“Compromise” sounds like a failure to commit, and it is worth noticing that the word is doing rhetorical work the measurements do not support.

A compromise projection is not a projection that failed to achieve a property. It is one designed against an objective that no exact property serves, and it meets that objective better than any exact projection does. Calling it a compromise is like calling a general-purpose tool a compromised specialist tool — accurate about the mechanism and misleading about the intent.

The alternative term sometimes used is minimum-error, which overclaims in the other direction since the error being minimised is a weighting somebody chose. Neither word is quite right, and the honest description is the long one: a projection with no exact property, whose distortions are moderate everywhere, chosen because the map has no property requirement.

That is the situation of most world maps ever printed, which makes the compromises the normal case rather than the exception the vocabulary implies.

A last observation about their reception. Compromise projections are treated with faint suspicion in technical writing — they have no theorem attached, no exact property to state, and their designers appealed to judgement. That suspicion is misplaced. Judgement exercised against a real objective is a better guide than rigour applied to a proxy for it, and the two projections most people have actually looked at the world through were both chosen that way.

The measurements in this essay are also worth reading as a boundary on the site’s own method rather than as a verdict. Everything else here ends in an assertion that either holds or stops the build. This field ends in a table of numbers and a judgement, and marking where the machinery stops is part of using it honestly — a site that measured everything and claimed to have settled everything would be making exactly the kind of unchecked claim it was built to avoid.

Reading the two most-used world maps of the last forty years as compromises also reframes the history. The move away from Mercator for general-purpose use did not go to an equal-area projection, despite that being what the argument was about. It went to projections with no exact property at all — which suggests the field’s working answer to the Peters question was that both sides had over-specified the problem.

Their reception also shifted the vocabulary of the field in a useful way. Once a projection with no exact property became the standard for world maps, “what does it preserve” stopped being the only question worth asking about a projection, and “how much does it distort, where” became askable. That is the question this site is built to answer, and the compromises are the reason it is worth asking.

One consequence for anyone specifying a map: a compromise is the correct default only when no property is required, and “no property is required” is a conclusion rather than an assumption. Working out that a map genuinely has no hard requirement takes about as long as working out what its requirement is, and skipping the step is how conformal projections end up carrying density data.

A compromise projection compromises on more than one axis, and the sharpest of them is a decision with only two answers.

A compromise has no exact property, so the usual descriptions have nothing to fix on. The directions of maximum stretch are still perfectly definite.

Winkel tripel’s field of principal directions fans away from the central meridian and lies almost horizontal near the pole line, which is the pattern every pole-line projection has — the compromise’s distortion made visible without a single number.

What was computed here #

Every distortion figure measures the projections rather than describing them, from their own derivatives, at one-degree intervals along a meridian and at five latitude cells for the areal comparison.

The compromises are asserted to fail both tests, which is what “preserves nothing exactly” means and is the one falsifiable claim available about them. Robinson measures 107° of maximum angular deformation and 1.7 of areal error; Winkel tripel 86° and 2.8. If either ever passed a test, its description here would be wrong.

Robinson’s table is quoted, because it is the definition. That is the single largest quoted object in this site’s library and it is quoted for the same reason the colour-matching functions are quoted on a sibling site: it is data rather than a derivation.

What the pictures cannot show #

Whether a compromise looks right, which is its entire objective. The figures show numbers; Robinson’s criterion was visual, and no measurement on this site can evaluate it against its own goal.

The grid figure also shows the projections at thumbnail size, where the differences that most influence a cartographer’s judgement — the treatment of the high latitudes, the shape of the outline, the behaviour at the edges — are least visible.

A compromise measured as an instrument #

Compromise is usually assessed by eye. There is a way to price one exactly: draw a scale bar on it and ask what the bar reads at each latitude, in each direction.

On Robinson, a bar drawn for a thousand kilometres at the equator reads 1,000 · 976 · 902 · 789 · 626 · 384 kilometres along the parallel — better than a cylindrical everywhere — and 1,000 · 1,000 · 1,000 · 1,018 · 1,093 · 1,293 along the meridian. Flat to a tenth of a per cent as far as 30° and within two per cent at 45°, because the projection’s tabulated parallel spacing was built to hold the meridional scale near one. So a north–south bar on this map is a usable instrument and an east–west one is not.

As an instrument the compromise behaves better than its scores suggest — which matters, because a scale bar is right in one place and no more. A bar drawn on it barely moves as far as 45°, which is a real property of the tabulated curve rather than a happy impression; a projection that made the opposite compromise puts the ratio between the printed scale and the actual one on a curve that is flat almost nowhere.

One consequence of that is worth stating for anybody reading a distortion table of compromise projections. Every figure in such a table is an average with a weighting in it, and the weighting was chosen by whoever built the table rather than by whoever designed the projection — so a compromise map’s number is a summary of somebody else’s preference about a map that was itself designed to somebody’s taste. Two conventions, stacked, and neither of them printed.

Who found it, and when #

Winkel published his triple projection in 1921, having produced two earlier compromises before it.

Robinson’s was commissioned in 1963 and published in 1974, and he explicitly declined to give it a closed form. He is also the author of The Look of Maps (1952), which argued that cartographic design is a discipline in its own right rather than a decoration on top of the geometry — a position his projection embodies.

The general acceptance of compromises for world maps is a twentieth-century development, and it arrived alongside the recognition that a general-purpose map has no property requirement. The 1989 resolution by seven North American geographic organisations against rectangular world maps is the clearest institutional statement of it.

Where this goes next #

The framework for choosing is which projection is best. What each projection is optimising is every projection minimises something. And the argument that pushed general-purpose maps toward compromise is Mercator against Peters.