The two ways a map is wrong

“How distorted is this map” sounds like a question with a number for an answer. It is not, and the reason is that there are at least two independent things going wrong and no principled way to add them.

Two measurements, two rankings, opposite conclusions. That is not a paradox; it is what independence means.

What the two quantities are #

From the indicatrix come the principal scale factors $a$ and $b$ at each point — the largest and smallest amounts by which the map stretches.

Angular deformation $\omega$ depends on their ratio:

$$\sin\frac{\omega}{2} = \frac{a-b}{a+b}$$

It is zero when $a = b$, whatever their common value. A projection can double every distance at a point and still have zero angular deformation there, because doubling everything preserves every angle.

Areal error depends on their product, $ab$, and is zero when the product is one. A projection can have $a = 4$ and $b = 1/4$ and be perfectly equal-area at that point while destroying every shape.

Ratio and product are independent — knowing one says nothing about the other — and that is the whole of the independence. There is no hidden relationship, no conservation law linking them, and no way to trade a known amount of one for a known amount of the other.

What is genuinely forced #

Only one thing: both cannot be zero at once over a region, because that would make the map an isometry.

Everything else is free. A projection can be:

• zero angular error and unbounded areal error (Mercator)
• zero areal error and large angular error (Gall–Peters)
• moderate amounts of both (Winkel tripel, Robinson)
• large amounts of both (equirectangular, which measures 108° and eightfold)

That last case is worth dwelling on because it disposes of an intuition. The trade-off does not mean that being bad at one buys being good at the other — a projection can be bad at both, and the plate carrée is. Being good at something requires choosing it.

That scatter is the essay’s argument in one picture. If the quantities were related, the points would lie on a curve. They do not.

Why a single number fails #

Attempts to summarise a projection’s distortion in one figure exist and are used, and their limitations are structural rather than fixable.

The obvious approach is to combine $\omega$ and the areal error into a weighted sum, integrate it over the map, and compare. Airy’s criterion and its descendants do this. Two problems follow immediately:

The weighting is a choice. How many degrees of angular deformation equal a doubling of area? There is no principled exchange rate, because they measure different things. Any weighting encodes a purpose, and once a purpose has been chosen the comparison is nearly settled anyway.

The region and the measure are choices too. Integrating over the whole sphere weights the empty ocean equally with everything else. Weighting by land area, or by population, gives different rankings — and each is defensible for a different map.

So a single number is a purpose in disguise. That is not a reason never to use one; it is a reason to state the weighting, which almost nobody does.

Where each failure shows #

Because the two are independent, they are visible in different ways and matter to different users.

Angular deformation shows locally. A shape looks wrong — a country looks stretched, a circular lake looks elliptical, streets meet at wrong angles. It is noticeable at any scale and is what makes a map look distorted.

Areal error shows globally. No individual shape looks wrong; the relative sizes are wrong. This is much harder to see and much easier to be misled by, because there is nothing locally odd to notice. The Greenland effect is invisible in any single region and enormous across the map.

That asymmetry has a consequence worth naming: areal distortion is the more dangerous of the two, precisely because it does not look like anything. A reader can see that Gall–Peters has stretched Africa. A reader cannot see that Mercator has enlarged Greenland fifteenfold, because Greenland’s own shape is fine.

A third quantity, occasionally #

Some treatments add a third failure: distance distortion, whether lengths are preserved.

It is not independent of the other two — it is bounded by them, since $a$ and $b$ are the extreme scale factors — but it is a different question, because a projection can preserve distances from a particular point or along particular lines without preserving them generally.

Azimuthal equidistant is the standard example: distances measured from its centre are exact, in every direction, and distances between any other pair of points are not. That is a genuine and useful property and it does not appear in either of this essay’s two measurements, which is a reminder that the two-quantity picture is a simplification chosen because those two are the ones that can hold everywhere at once.

Where the two failures are worst #

The two distortions do not merely differ in kind; they differ in where on a map they concentrate, and the pattern is worth knowing.

Angular deformation for a cylindrical equal-area projection is worst at the equator and at the poles, and zero at the standard parallel. For a pseudocylindrical projection it is worst at the outer edges, where the meridians are most oblique.

Areal error for a conformal projection is worst wherever the scale factor is largest — the poles for Mercator, the antipode for stereographic — and grows without bound.

So the two failures are usually in different places, which is one more reason a single summary number is misleading: it averages a defect at the edges with a defect in the middle.

Mollweide’s indicatrices are all the same area, with shape distortion mild in the centre and severe at the outer edges — a spatial pattern quite different from Mercator’s, where the shape is perfect everywhere and the areas fail towards the poles.

Two failures, two audiences #

A useful way to decide which matters: ask who is reading the map and what they will do with it.

Someone recognising a shape — finding a country, matching a coastline, identifying a region — is affected by angular deformation and essentially unaffected by areal error. A map with perfect shapes and wildly wrong areas serves them well.

Someone comparing quantities — reading a choropleth, judging relative extent, estimating how much of something there is — is affected by areal error and can tolerate a good deal of shape distortion.

Someone measuring — a surveyor, a navigator, an engineer — needs the property their instrument assumes, which is usually conformality because instruments measure angles.

Those three requirements are genuinely independent, and no map serves all of them. Which is why the choice needs a purpose and why the two-quantity framing is the useful one: it maps onto three real user needs rather than onto one abstract notion of accuracy.

The correlation that does exist #

One qualified exception to the independence, because it is worth being precise.

Over a fixed family of projections with a tuning parameter, the two errors often do trade against each other, because the family has already fixed most of the freedom. Sliding a standard parallel changes both, and in opposite directions over part of the range.

But that is a fact about the family rather than about the quantities. Across the whole space of projections there is no relationship, which is what the scatter in the audit plot shows and what makes a projection that is bad at both possible.

The third failure, named properly #

Distance distortion is the quantity most often added to the list, and it behaves differently enough to be worth separating.

It is not a global property. No projection preserves all distances — that would be the isometry the theorem forbids — so every equidistance claim is qualified: distances from the centre, or along the meridians, or along a standard parallel.

That makes it a family of local guarantees rather than a single property, and it does not appear in the two-quantity framing because the two quantities are the ones that can hold everywhere at once.

It is nonetheless the property a great many maps actually want. A map answering “how far is it from here” needs distances from one point, and the azimuthal equidistant projection provides exactly that.

Why the count is exactly two #

The essay’s title makes a claim about a number, and the claim can be made exact rather than left as a convention.

At a point, the projection’s derivative is a linear map, and up to rotations of the plane it is described by two numbers: the principal scales $a$ and $b$. So there are exactly two degrees of freedom, and any pointwise distortion measure whatsoever is a function of them.

The two chosen here recover both. Writing $s = \sin(\omega/2) = (a-b)/(a+b)$ and $A = ab$ for the areal factor, the pair inverts:

$$a = \sqrt{A\,\frac{1+s}{1-s}}, \qquad b = \sqrt{A\,\frac{1-s}{1+s}}$$

Given $\omega$ and the areal factor, $a$ and $b$ follow, and with them every other measure — the scale distortion, the maximum linear stretch, the flattening, any of them.

Which settles what “independent” means here. It is not that the list could not be extended; it is that a third quantity adds no information, because two coordinates already fix the point. The reason there are two rather than one is that the ratio and the product are separately free, and the reason there are not three is that a $2\times2$ Jacobian has only two singular values.

The azimuthal equidistant projection is the clearest case of a third kind of guarantee: distances from its centre are exact in every direction, and distances between any two other points are not. That is genuine, useful and of a different shape from either quantity this essay measures.

What to report instead of one number #

The practical recommendation, since the essay is against scalar summaries.

Report both quantities, at stated locations or as a range over a stated region. “Angular deformation up to 12° and areal error up to 1.4 over the mapped area” says everything a single index would and does not hide the trade.

That is what this site’s captions do, and it is why they are longer than the usual ones. A caption reading “moderate distortion” is a summary somebody computed with a weighting they did not state.

A closing note on vocabulary. “Distortion” as a mass noun — more of it, less of it — is the habit this essay is arguing against, and it is deeply embedded in how the subject is discussed. There is no such quantity. There are two of them, they are independent, and any sentence using the singular has already chosen a weighting without saying so.

The independence also explains why the subject has so many projections. If distortion were one quantity there would be a best projection and the field would have found it. Because there are two, there is a frontier rather than an optimum, and every point on that frontier is somebody’s right answer.

The two-quantity framing is also what makes the site’s audit possible. Both are computable at every point from the same four derivatives, both have a well-defined zero, and both can be asserted against. A single blended index would have neither property — there is no value of it that means “correct”, so there would be nothing to assert.

A last note on how the two quantities are reported in practice. Software and documentation typically give the scale factor and stop, which is one number describing a conformal projection adequately and describing anything else not at all. Where a projection is not conformal there is no single scale factor, and a tool reporting one has picked a direction on the reader’s behalf.

The independence is finally what makes this subject a design discipline rather than an optimisation. If distortion were scalar there would be a best map and the work would be finding it. Because there are two quantities and no exchange rate between them, there is a frontier of non-dominated choices, and picking a point on it requires knowing what the map is for — which is a judgement, informed by measurement and not replaced by it.

Both failures are stated at a point. A third appears only at finite size, and it is bounded below by something no projection controls.

What was computed here #

Both quantities come from the same four partial derivatives, so a systematic error in the differentiation would move both together — which is a reason to check the derivatives independently rather than to trust their agreement. The analytic scale factors are implemented for the four projections that have simple ones and agree with the numerical route to better than $3\times10^{-11}$.

Areal errors here are computed two ways. The pointwise version comes from the Jacobian determinant. The cell version comes from comparing a projected polygon’s planar area against the closed form $R^2\,\Delta\lambda\,(\sin\varphi_2 - \sin\varphi_1)$ for a latitude–longitude cell, which is exact. The cell formula is verified by summing a full covering of the sphere and requiring $4\pi$, to $7\times10^{-15}$.

Using closed-form cells rather than a coastline dataset is deliberate. A polygon’s area depends on its simplification level, so a Greenland-against-Africa figure taken from a shapefile is partly a measurement of the vendor’s generalisation. The cells have no such ambiguity.

What the pictures cannot show #

The independence, directly. Every figure here shows one quantity or the other, and the claim that they are unrelated is carried by the scatter in the audit plot and by the reversal between the first two figures. A picture of two independent quantities is two pictures.

The cell figure also hides the very thing it measures: each panel is scaled to fit, and scaling is what destroys the areal comparison. The numbers are printed because there is no honest way to draw four differently-sized shapes at true relative size and still see any of them.

Both ways, in one operation #

The two failures are usually shown as two pictures. There is one operation that exhibits both at once and returns a number for each: computing something from a stored polygon.

The areal error decides an area — a shoelace in a conformal plane returns 3.06 times the closed form over a mid-latitude cell. The angular error decides a distance ranking, because a plane that shears re-orders candidates at different bearings, which is why the plate carrée answers 103 of 400 nearest-neighbour queries with a different site and the conformal Mercator only 5.

Both halves show up in one instrument. The areal half is arithmetic: three routes agree on the closed form for a cell’s area and two do not, and the two that do not are the two ways of computing an area without naming a surface. The angular half is the bar itself — on an equal-area map the two readings of one bar are reciprocal, short one way and long the other with their product exactly one, so no single correction repairs a distance taken off it.

Who found it, and when #

Tissot’s 1881 analysis separates the two explicitly, and the distinction has been standard in the technical literature ever since.

It is much less standard outside it. Popular accounts routinely speak of “distortion” as a single quantity and rank projections along one axis, which is where the intuition that Mercator is simply worse than Gall–Peters comes from — a ranking that is correct on area, reversed on angle, and meaningless without saying which.

Attempts at scalar summaries began with Airy in 1861 and continue; Kavrayskiy, Jordan, Klingach and others all proposed measures, and no two agree on the ranking of the compromise projections. The disagreement is not a failure of any of them. It is what happens when several people choose different weightings for quantities that have no natural exchange rate.

Which is also why a new one arriving, however carefully constructed, does not settle anything: it adds one more weighting to the pile rather than removing the need for one.

The proliferation of scalar measures is worth reading as evidence rather than as a muddle. Airy, Kavrayskiy, Jordan and the rest were all competent, all working on the same problem, and all producing rankings that disagree — which is what a search for a total order over a genuinely two-dimensional quantity looks like from the outside. The disagreement is not a sign that the right measure has not yet been found; it is a sign that each of them chose a weighting, and that the weightings are where the disagreement lives rather than in the arithmetic.

A reader meeting a new scalar measure is therefore entitled to ask which weighting it encodes, and to expect an answer.

Where this goes next #

The tool both measurements come from is Tissot’s indicatrix. The one relationship between them that is forced is the trade-off is two lines. And for what a projection does with the freedom the independence leaves, every projection minimises something.