Equal-area on the wrong body

This site’s opening result is that the most widely used projection in the world fails the property named in its own title, because it applies a spherical formula to an ellipsoidal latitude. The failure is 0.3848° of angular deformation against a noise floor of $1.6\times10^{-6}$, and the machinery found it without being told to look.

The same mistake is available in the other direction and nobody appears to have measured it. An equal-area projection fed the latitude a coordinate actually carries, and measured against the body that coordinate refers to, is not equal-area.

Why the failure is invisible #

The conformality failure announces itself. Angular deformation is zero for a conformal projection and 0.3848° for Web Mercator, and zero is a number a test can be written against — measuring instead of naming does exactly that and catches it in twenty lines.

This one does not announce itself, because it averages out. The areal factor is 1.00674 at the equator and 0.99332 at 88°, and the area-weighted mean of those is 1.0022 — two parts in a thousand. A check that computed the total area of the world on this projection and compared it with the published figure would find it right to a fifth of a per cent and pass.

So the failure is a redistribution, not a loss. The map has almost the right amount of area in it and puts it in the wrong places, which is precisely the defect an equal-area projection exists to prevent.

The size, in the units the map is used in #

A one-degree cell on the equator is 12,308 square kilometres on WGS84. The naive projection reports it as 12,391 — 83 square kilometres too many, which is a district.

A one-degree cell between 59° and 60° is 6,310 square kilometres, and the same projection reports it 21 too few.

Both errors are around two thirds of a per cent, and they have opposite signs. A thematic map shaded by density on this projection reports equatorial regions as less dense than they are and high-latitude regions as more dense, by 1.3 per cent of the ratio — which is smaller than most of the effects such a map is drawn to show, and is not smaller than all of them.

Beside the errors an equal-area projection is chosen to avoid #

Two thirds of a per cent sounds small until it is compared with the number the projection was chosen for, and then it sounds small in a different way — because the comparison is what decides whether the choice was worth making at all.

So the naive ellipsoidal version is still enormously better than any non-equal-area projection, by three orders of magnitude, and nobody choosing between Gall–Peters and Mercator would be affected by it.

The reason it matters anyway is that an equal-area projection is not usually chosen over Mercator. It is chosen because a number is going to be computed off it — a density, a share, a per-hectare rate — and those numbers are compared with each other at the per-cent level. An error of 1.3 per cent between the equator and the pole is not competing with Mercator’s 1,540; it is competing with whatever the map was drawn to show.

The mechanism is one line #

An equal-area cylindrical projection maps latitude $\varphi$ to a northing $y(\varphi)$, and its areal scale factor is

$$\frac{|\det J|}{M(\varphi)N(\varphi)\cos\varphi} = \frac{y'(\varphi)}{M(\varphi)N(\varphi)\cos\varphi}$$

where $M$ and $N$ are the ellipsoid’s two radii of curvature. On a sphere both are 1 and the condition $y' = \cos\varphi$ gives $y = \sin\varphi$, which is the textbook formula.

On an ellipsoid $MN\cos\varphi$ is not $\cos\varphi$, so $y = \sin\varphi$ is wrong, and the size of the error is the size of $MN$: it runs from $1-e^2 = 0.99331$ at the equator to $1/(1-e^2) = 1.00674$ at the pole. Those are the two numbers in the measurement, reciprocal to each other and 1.34 per cent apart, and they are the same $M$ and $N$ that geodetic against geocentric latitude is about.

The fix, and it has been known since 1772 #

The condition $y' = MN\cos\varphi$ integrates exactly. The result is half the authalic quantity

$$q(\varphi) = (1-e^2)\left[\frac{\sin\varphi}{1-e^2\sin^2\varphi} + \frac{1}{2e}\ln\frac{1+e\sin\varphi}{1-e\sin\varphi}\right]$$

whose derivative is exactly $2MN\cos\varphi$ — which can be checked by differentiating it, and is checked here by measuring the projection’s areal factor and finding it 1 to $1.1\times10^{-10}$.

Normalising $q$ by its polar value and taking an arcsine gives the authalic latitude, the auxiliary latitude of a sphere with the same total area as the ellipsoid, and it is one of the six the site’s ellipsoid machinery already computes. The fix for the whole class of equal-area projections is the same one word: substitute the authalic latitude for the geodetic one.

The fix is one of the six auxiliary latitudes and has been known since 1772. The authalic latitude is the one this argument needs, and it departs from the geodetic by up to 7.70 arcminutes at 45° — 14.3 kilometres of ground, and exactly the quantity a spherical equal-area formula throws away without saying so. Two of the other five auxiliary latitudes are what Web Mercator is not conformal and the meridian arc are about, which is the same confusion arriving in two other places.

The repair that does not work, and it is the obvious one #

Faced with a projection whose total area is 0.22 per cent too large, the natural fix is a constant: multiply the northing by $q_p/2$ so the map’s area matches the body’s.

It works exactly, for the total. The map’s area then equals the ellipsoid’s to the last digit a double carries.

And it makes every individual cell worse. The worst pointwise error goes from 0.674 per cent to 0.890, because rescaling shifts the whole curve down: the equator improves from +0.674 to +0.449 and 88° degrades from −0.668 to −0.890.

That is a genuinely instructive failure. The constant that makes the total right is the one that sets the area-weighted mean of the areal factor to one, and most of a sphere’s area is near the equator, so the constant is dragged towards the equatorial value and the poles pay for it. A correction fitted to an aggregate makes the aggregate exact and the extremes worse, which is a general property of fitting a constant to a varying quantity and is worth having a case of.

The failure is sharper than worse, and the sharpening is an exact symmetry rather than an accident of these numbers.

The naive construction’s two extreme areal factors are 1.00674 and 0.99332, and those are reciprocals — because MN runs from 1 − e² to 1/(1 − e²) and nothing else enters. So their product is one, their geometric mean is one, and the constant that minimises the worst-case departure of a positive quantity from unity is exactly the geometric mean of its extremes.

The naive map is therefore already the minimax-optimal member of its own family of rescalings. No constant multiplier can bring its worst pointwise error below 0.674 per cent, and the area-matching constant — which is the arithmetic, area-weighted mean rather than the geometric one — is a move away from an optimum rather than a step towards one.

That is why the repair costs what it does, and the numbers say which mean is which. The area-weighted mean of the areal factor is 1.0022, dragged towards the equatorial value because most of a sphere’s area is near the equator; dividing by it improves the equator to +0.449 and degrades 88° to −0.890, which is the 1.0022 being subtracted from a symmetric pair and making it asymmetric.

So the choice between the two constants is a choice between two statistics, and it is the ordinary one: the geometric mean is the right centre for a ratio and the arithmetic mean is the right centre for a total. A map whose job is to have the right total area should take the second and accept the worse cells; a map whose job is to have every cell right within a bound should take the first — and, having taken it, discover it was already there.

The real instance, and it has a date #

This is not a hypothetical. NSIDC’s EASE-Grid, which carries three decades of polar satellite products, was defined on a sphere: the cylindrical and azimuthal equal-area projections in its definition take a spherical radius and spherical formulae, applied to geodetic latitudes from the satellites.

EASE-Grid 2.0, published in 2012, redefines the same projections on the WGS84 ellipsoid. The documentation gives the reason plainly — the spherical definition misplaced areas — and the transition required reprocessing a long archive.

That is the sequence this essay’s arithmetic describes: a formula that is right for a sphere, applied to coordinates that are not spherical, in a product whose whole purpose is that a cell’s area is a known constant.

The polar case is the one where it bit hardest, and the reason is in the profile above: the departure is largest and most one-sided at high latitude, where the areal factor sits 0.67 per cent below one across a wide band rather than crossing through it. A product covering only the polar regions therefore carries a nearly constant bias rather than a redistribution, and a constant bias in a cell area is a constant bias in every concentration, flux and extent computed from it — which is datum shifts dwarf projection errors inverted, a case where the projection detail is the one that survives into the answer.

The cells the error is measured in #

The quantity being got wrong is the area of a piece of the world, and this site computes those from a closed form rather than from a dataset, which is what makes the comparison exact rather than a difference of two approximations.

The circularity there is only apparent and is worth naming, because it is the kind of thing that can hide a mistake. The cell area comes from integrating the ellipsoid’s own area element; the projection’s areal factor comes from differentiating the forward map against the ellipsoid’s metric. The two do use the same $q$, but one integrates it and the other differentiates a map that happens to contain it, and it is the sphere control — the same formula on a sphere, measured against a sphere, at $7\times10^{-12}$ — that establishes neither is quietly comparing something with itself.

The same cell measured five ways settles it. A shoelace in degrees, a shoelace in three projected planes, the spherical closed form and the ellipsoidal one give five answers; the equal-area plane is the only projected one that gets it right, and it is right against the ellipsoidal figure only if it was built with the authalic latitude. The difference between those last two columns is this essay’s 0.67 per cent.

What the audit machinery does with it #

The site’s founding rule is that no property is printed until it has been computed from the projection’s own derivatives, and the four constructions above are exactly the case that rule exists for. All four claim to be equal-area. One is, on a sphere; one is, on the ellipsoid; two are not.

The two-sided form is what makes the test a measurement rather than a preference, and both sides are asserted. The authalic construction must pass on the ellipsoid and fail on a sphere; the spherical construction must pass on a sphere and fail on the ellipsoid. A test that only ever rejected the second would be consistent with the machinery simply disliking a particular formula.

Where the same mistake is not being made #

Three places, and saying which is as useful as saying where the error is.

The library’s own equal-area projections — Gall–Peters, Behrmann, Mollweide, the sinusoidal, Lambert’s cylindrical and azimuthal, Hammer, Eckert IV, Albers — are all defined on a sphere and are all measured against a sphere. They are exactly equal-area, to $10^{-11}$, and nothing here touches them. The error is not in a formula; it is in a pairing of a formula with a body.

Albers on an ellipsoid is normally implemented with the authalic latitude already, in every serious library, because a conic equal-area projection is used for national statistical mapping where the areas are the product.

A projection used on a sphere on purpose. Plenty of products are defined on a sphere and mean it — a global climate grid, a spherical harmonic model, a graphic. There the spherical formula is the right formula and the coordinates are spherical too, and no error arises. The mistake needs a mismatch, which is why the Earth is a sphere, and when it is not is the essay this one depends on: the question is never whether a sphere is accurate, it is which body the numbers refer to.

The cell areas this site computes come from a closed form on the ellipsoid — $A = a^2\Delta\lambda\,[q(\varphi_2)-q(\varphi_1)]/2$ with the same $q$ as above — which is why computing an area needs a surface can state a cell’s area exactly rather than as a projected approximation.

What it would take to notice, from inside #

A reader handed one of these maps has no way to tell which construction produced it, and the practical question is what test would separate them.

Not the total area: that is right to 0.22 per cent on the naive one and exactly right on the rescaled one, so a check against the published surface area of the Earth passes on both.

Not a single cell: 0.67 per cent is well below the accuracy of anything a reader would compare a cell against, and any real region’s published area carries a larger uncertainty than that from its boundary definition.

What does separate them is a ratio of two cells far apart in latitude. The equatorial cell is 0.67 per cent too large and the polar one 0.67 too small, so their ratio is 1.34 per cent out, and that is a quantity nothing else in the map’s construction can produce. A test written as “the areal factor at the equator equals the areal factor at 80°” catches all three failures and needs no external data at all.

That is the same shape as the check measuring instead of naming uses for conformality — compare the projection with itself at two places rather than with anything outside it — and it is the only kind of test available to somebody holding a map and no ground truth.

What the reader should take from the pattern #

Two projections, two properties, one mistake. Web Mercator is conformal on a sphere and is used on an ellipsoid. The cylindrical equal-area projection is equal-area on a sphere and is sometimes used on an ellipsoid. In each case the formula is right, the coordinates are right, and the pairing is wrong.

The difference between them is only in how loudly the failure speaks. Angular deformation is zero or it is not, so the conformality failure is a qualitative change that any test detects. Areal factor is one or it is not, but its departure integrates to nearly nothing, so the equal-area failure hides inside every aggregate anybody would check it with.

The general rule the pair suggests: a property that is exactly zero when satisfied is easy to audit, and a property that is exactly one is not — because a quantity that should be one can be wrong in both directions and average to one, and a quantity that should be zero cannot.

That is a statement about how to design a check rather than about cartography, and it explains why this site’s assertions are written against zero wherever the choice is available: the two ways a map is wrong measures the angular failure directly and the areal one as $|\text{areal}-1|$, which is the same trick applied by hand.

The equal-area projections on this site, checked #

Since the pairing is the mistake rather than the formula, the useful thing to publish is which pairings this site’s own library holds and what each measures.

The convention this suggests, and which the site now follows, is that a projection’s name is incomplete without its body. “Lambert cylindrical equal-area” names a formula; “Lambert cylindrical equal-area on WGS84 via the authalic latitude” names a map. EPSG codes carry the distinction — 6933 is the ellipsoidal version and 3410 the spherical one — and almost no prose does.

Where this ladder goes #

The audit ladder has taken Web Mercator’s failure, the Mercator–Peters argument, the phrase “true size”, the plate carrée that nobody chooses, and the conformal projection whose drawn angles are wrong. This adds the twin of the first, and closes a symmetry the field has been missing since foundation: both exact properties can be broken by the same mistake, and only one of them complains.

What is left in this field is the class of claims that are not about a projection at all but about a product — the statistics computed off a map, the areas quoted from a shapefile, the densities shaded from a grid. Every one of them inherits whatever the projection did, and none of them says which projection it was.