Every projection minimises something

Projections are usually introduced by construction — this one wraps a cylinder, that one fits a cone. A more useful question is what each was trying to achieve, because that is what a map user is choosing between.

Read that way, every projection is the answer to an optimisation problem, and the problems fall into three kinds.

Kind one: an exact constraint #

The strongest form. A property is demanded to hold everywhere, and the projection is whatever satisfies it.

Mercator demands conformality with straight parallel meridians. That determines the parallel spacing uniquely — it must be $\ln\tan(\pi/4 + \varphi/2)$, and there is no freedom left.

Equal-area cylindricals demand $ab = 1$ with straight parallel meridians. That also determines the spacing, up to the choice of standard parallel.

Gnomonic demands that great circles map to straight lines, which forces the light source to the centre of the sphere.

Azimuthal equidistant demands that distances from the centre be true.

These projections have no tuning parameters worth speaking of. The constraint is the design, and the distortion elsewhere is a consequence rather than a choice — which is why arguing that Mercator should have less areal distortion is arguing that it should not be conformal.

Kind two: a minimised functional #

Weaker and more interesting. No property is demanded exactly; instead a measure of overall badness is written down and minimised.

The general form is to integrate some combination of the distortion measures over a region:

$$E = \int_{\text{region}} w_1\,f(\omega) + w_2\,g(ab)\;\mathrm{d}A$$

and choose the projection’s free parameters to make $E$ smallest.

Airy proposed the first such criterion in 1861. Kavrayskiy, Jordan, Klingach and others proposed variants. They differ in the weighting, in the region, and in what function of the distortion is integrated — and they do not agree on the rankings.

That disagreement is not a failure of any of them. The two distortions have no natural exchange rate, so every criterion has to invent one, and inventing one is choosing a purpose.

Which means a “minimum-distortion projection” is always minimum with respect to something, and the something is the interesting part. It is rarely quoted alongside the claim.

Kind three: fitted by eye #

The most honest of the three about what it is doing, and the one that makes mathematicians uncomfortable.

Robinson was commissioned by Rand McNally in 1963 to produce a world map that looked good. He worked backwards from appearance: adjusting the length and spacing of the parallels, printing the result, looking at it, and adjusting again.

The published definition is a table of nineteen values. Interpolating it is not approximating a closed form, because there is no closed form. Robinson described his approach as starting from the visual and working toward the mathematical rather than the other way round.

This site’s implementation quotes the table, and that is consistent with its own rule: a table that IS the definition is a measurement of sorts, and measurements may be quoted. What could not be quoted is a formula somebody derived, and there is none.

Winkel tripel is a hybrid of the two kinds — the arithmetic mean of an equirectangular and an Aitoff projection, chosen because that combination happened to score well on the three distortions Winkel wanted to control. The name means “triple”, for those three.

The fourth kind: an objective met approximately, on purpose #

The three-way classification is missing a category, and it is the one most working coordinate systems belong to.

Transverse Mercator on an ellipsoid has an exact objective — conformality — and no closed form. The projection that satisfies the constraint exists as an elliptic integral, and what is actually computed is a truncated series in the flattening, carried to four, six or eight terms depending on the accuracy required.

So the projection as specified is conformal and the projection as implemented is not, by an amount that is known, bounded and chosen. The Gauss–Krüger series to fourth order is accurate to a few millimetres within a three-degree zone and degrades rapidly outside it; Karney’s eighth-order version holds to nanometres over the same range. Both are called transverse Mercator.

That is a genuinely different kind of design. Kinds one to three fix an objective and accept whatever distortion follows; this one fixes the objective, accepts that it cannot be met exactly, and then makes the shortfall an explicit engineering parameter with a stated error budget.

It is also the kind almost every legally defined national grid belongs to, which means the projections most survey data actually lives in are approximations to projections nobody computes. What the series is and how far it can be trusted is the piece usually left out.

What naming the objective explains #

Three things that the construction does not.

Why the distortion is where it is. Mercator’s areal distortion is at the poles because conformality plus straight meridians forces it there. That is a consequence of the objective, and knowing the objective predicts it.

Which parameters are free. A projection defined by an exact constraint has few or none; one defined by a minimised functional has whatever the functional left open. Knowing which is which tells a user what can be tuned.

Whether a complaint is legitimate. Complaining that Mercator distorts area is complaining that it is conformal, which is what it is for. Complaining that Robinson distorts everything a little is complaining about its objective, which is a matter of taste and was always going to be.

Sorting the library by construction and by property makes the gap between the two columns plain: the property column is close to the objective and the construction column predicts almost nothing about it. The two projections fitted to a brief rather than derived from a property have no column of their own at all, which is the omission this essay is about.

The objective nobody optimises #

There is a natural question the criteria do not answer: what is the minimum possible distortion for a given region?

It is a well-posed problem. For a fixed region and a fixed measure, there is some projection minimising it, and one could ask what that projection is rather than which of the named ones scores best.

Chebyshev proposed a partial answer for conformal projections in 1856: the best conformal projection of a region is the one whose scale factor is constant along the region’s boundary. That is a genuine optimality theorem and it is constructive, and it is used — the Laborde projection of Madagascar and several national grids descend from it.

For the general case, with both distortions and an arbitrary region, the problem is open in any practical sense. Which is a reasonable place for a subject to be: the exact constraints are solved, the fitted compromises work well, and the optimisation in between is hard.

The objective as a design brief #

Reading a projection as an optimisation makes its history legible, because the objectives were usually written down as requirements before anyone had a formula.

Mercator’s brief was a sailing chart. Lambert’s briefs, stated explicitly in 1772, were conformality and equal-area as properties worth having in themselves. Robinson’s brief was from a publisher and was about appearance. Snyder’s brief for the Space Oblique Mercator came from NASA and specified a satellite’s ground track.

In every case the projection is downstream of a requirement, and the requirement is the thing worth knowing about it. A reader who knows Mercator was built for compass navigation can predict its areal distortion; a reader who knows it is cylindrical can predict nothing.

Objectives that conflict within one map #

A complication the neat three-way classification hides.

A national mapping agency’s brief is usually several requirements at once: conformal for survey work, low distortion across the country, a grid that is legally defined and cannot change, compatibility with neighbouring countries’ grids, and continuity with whatever was published before.

Those pull in different directions, and the resulting systems are compromises between engineering and administration rather than between geometric properties. The oddly specific scale factors and central meridians in national grid definitions are the residue of exactly this — each one is a negotiated position rather than an optimum.

Which is a reminder that “every projection minimises something” is true of the classical ones and understates what a working coordinate system is doing.

Reading a projection backwards #

A useful exercise: given an unfamiliar projection, work out what it was for.

Look at where the distortion is zero. That is where the designer cared. A standard parallel in the mid-latitudes suggests a map of a temperate country; one at the equator suggests a world map or a tropical region.

Look at which quantity is flat. Zero angular deformation everywhere means the brief demanded conformality, and the areal behaviour is a consequence. Zero areal error means the reverse.

Look at the boundary. A projection truncated at 84° was built for something that does not go to the poles. One that shows a hemisphere was built for a hemisphere.

Look at the pole treatment. A pole line means the designer wanted the high latitudes legible; a pole point means correctness was preferred.

Four observations, and they usually recover the brief.

Albers is the worked example of reading a brief off a profile. Two zeros in the angular deformation and an areal error flat on one throughout says, without any documentary evidence: equal-area was required, and the map is for a region between those two latitudes.

The objective that cannot be written down #

Robinson’s case deserves defending, since “adjusted until it looked right” invites dismissal.

The objective of a general-purpose world map is that a competent reader forms an accurate impression of the world from it. That is a real objective. It is not expressible as an integral, because “accurate impression” is not a functional of the distortion field — it involves recognition, familiarity, and what the reader already believes.

So a projection fitted by an experienced cartographer’s judgement is optimising something genuine that no formula captures. The method is unrigorous and the target is real, and the alternative — optimising a formula that stands in for the target — is rigorous about the wrong thing.

That is worth stating on a site otherwise devoted to measuring everything. Some objectives are not measurable, and pretending otherwise by measuring a proxy is a worse error than admitting it.

The brief is the useful thing to record #

A closing observation about documentation, since this is where the practical value of the framing sits.

Projection libraries record parameters: the projection’s name, its standard parallels, its central meridian, its datum. They do not record the brief — what the projection was chosen to achieve — and that is the piece a later reader most needs.

A dataset in Albers with standard parallels at 29.5° and 45.5° is a dataset someone chose an equal-area projection for, fitted to the conterminous United States. That is recoverable by inference and is not written down anywhere, and if the dataset is later reused for something the equal-area property does not suit, nothing warns anybody.

Recording the objective alongside the parameters would cost a line and would answer the question that actually recurs: not what projection is this but why.

The framing also predicts where new projections come from. They appear when a new purpose appears — satellite imagery produced the Space Oblique Mercator, web tiles produced the dominance of one particular Mercator variant, and the growth of thematic mapping is what made equal-area projections ordinary rather than specialist. A projection is a specification met, so new specifications produce new projections and nothing else does.

There is also a category of projection with no objective at all, invented to demonstrate that a construction is possible or to fill a gap in a classification. Most of them are never used. That is worth noticing, because it confirms the essay’s framing from the other direction: a projection without a purpose has nothing to recommend it, and the ones that survive are the ones somebody needed.

The objective also determines what a projection cannot be fixed for. Mercator’s areal distortion is not a bug that a variant might address — it follows from the constraint, and any projection meeting the same constraint has it. Asking for a Mercator with better areas is asking for a non-conformal projection with straight parallel meridians, which exists, is called Miller, and does not straighten rhumb lines.

The framing has one further use: it identifies which complaints about a projection are coherent. A complaint about a consequence of the objective is really a complaint about the objective, and should be argued there. A complaint about a free parameter — a standard parallel, an aspect — is a complaint about a choice that could have gone differently, and is the kind worth making.

The three kinds also differ in how they age. An exact constraint is permanent — Mercator will always be the conformal projection with straight parallel meridians. A minimised functional ages with its weighting, since the weighting encoded somebody’s sense of what mattered. A fitted projection ages with taste. Robinson looked right to a cartographer in 1963, and whether it still does is a question about readers rather than about geometry.

It also explains why the field keeps producing projections despite the classical problems being solved. New purposes arrive continuously, and each one is a new brief. Web tiles, satellite swaths, spherical panoramas and planetary bodies with different flattening have all generated projections in the last fifty years, and every one of them is a specification met rather than a gap in a taxonomy filled.

Naming the objective is half the work. The other half is noticing that the objective is usually optimised over the wrong variable.

What was computed here #

Every distortion curve is measured from the projections’ own derivatives at one-degree intervals, and each projection’s claimed property is asserted before any caption is written.

The Robinson table is the one genuinely quoted object in the library, and it is quoted because it is the definition. The implementation interpolates it linearly rather than with the cubic spline Robinson’s own construction implies, which keeps the derivative finite — a spline through nineteen hand-chosen points has curvature artefacts, and the difference between the two interpolations is well below the width of a drawn line.

That is a compromise and it is stated in the code. Robinson’s projection is the one place on this site where the honest answer to “what was computed” is “an interpolation of somebody’s judgement”.

One consequence of that unfairness is worth keeping, because it generalises past this figure. A projection designed to balance three quantities is being scored on one of them, and the score is therefore an upper bound on how badly it does at its own job rather than a measurement of it. Any single-criterion comparison flatters the specialists and understates the compromises, by an amount equal to whatever the compromise was buying in the criteria the chart does not plot — which is why a compromise projection’s case can only be made on a chart with more than one axis.

What the pictures cannot show #

An objective function. Every figure shows a result, and the thing this essay is about is the criterion that produced it — which lives in the derivation rather than in the map.

The comparison figure is also unfair to the compromises in one respect: it plots angular deformation, which is one of the three quantities Winkel tripel was designed to balance, so it shows a projection being beaten at a game it was not only playing.

Who found it, and when #

Lambert’s 1772 treatise is the first work to state properties as design goals and then construct projections to meet them, and it produced the conformal conic, the cylindrical equal-area, the azimuthal equal-area and the transverse Mercator in one publication.

Airy’s minimum-error criterion is from 1861, and the tradition of integrated distortion measures runs from there through Kavrayskiy in the 1930s to the present.

Chebyshev’s theorem on optimal conformal projections is from 1856, proved by Grave in 1896 — one of the few genuine optimality results in the subject, and a reminder that the general question is harder than it looks.

Where this goes next #

The practical version of the question is which projection is best. The projections that optimise a mixture are compromise projections. And the reason no objective can drive both distortions to zero is the trade-off is two lines.