The plate carrée, the projection nobody chooses

Take a table of latitudes and longitudes. Put longitude on the horizontal axis and latitude on the vertical. Plot the points.

That is a map projection. It has a name, it is about nineteen hundred years old, it preserves nothing, and it is very likely the most frequently produced map in the world — because it is what happens when nobody makes a decision.

What it is #

The equirectangular projection, or plate carrée in its simplest form, is

$$x = \lambda, \qquad y = \varphi$$

Longitude across, latitude up, at the same scale, with no function applied to either.

There is a one-parameter family of them — a standard parallel $\varphi_1$ can be chosen, giving $x = \lambda\cos\varphi_1$ — and the plate carrée proper is the member with $\varphi_1 = 0$. The general form is sometimes called the equidistant cylindrical projection.

What it preserves #

Two things, both narrow, and naming them precisely is the fairest thing this essay can do for it.

Distances along the meridians are true. The vertical scale is uniform, so a north–south distance measured on the map is correct anywhere. That is the “equidistant” in the name, and it is a real guarantee.

Distances along the standard parallel are true. For the plate carrée that is the equator.

That is the whole list. It is not conformal, it is not equal-area, it does not straighten great circles or rhumb lines, and it does not preserve distances in any direction other than due north–south.

What it costs #

The site measures the equirectangular projection at 108° of maximum angular deformation and an areal error reaching eightfold.

Both numbers deserve a moment. 108° of angular deformation is better than Mollweide’s 130° and Gall–Peters’ 146° and worse than Winkel tripel’s 86°; eightfold areal inflation is a fraction of Mercator’s ninety and is still enormous. What is unusual is not the size of either number but the fact that there are two of them: Mollweide pays 130° and buys exact areas, Mercator pays ninetyfold and buys exact angles, and the plate carrée pays a substantial amount of both and buys nothing.

That is the state most projections avoid by choosing one, and it is what the audit plot shows at a glance.

In the audit it sits well out along the angular axis and well up the areal one — the region of that plot occupied by projections which have not chosen anything, and a small and unfashionable neighbourhood.

That figure makes the point cleanly. The projections on the left edge bought conformality with area; the ones along the bottom bought area with shape; the compromises sit in the middle having deliberately balanced. The plate carrée is out in the open, and it got there by not deciding.

The mechanism #

Why the distortion is what it is follows in one line, and it is a good illustration of how much a projection’s behaviour is determined by very little.

On the sphere, a degree of longitude covers $\cos\varphi$ times the ground distance that a degree of latitude does. The plate carrée draws them the same width. So the horizontal scale is stretched by $\sec\varphi$ relative to the vertical one, and

$$h = 1, \qquad k = \sec\varphi$$

At 60° that is a factor of two: every shape is drawn twice as wide as it should be relative to its height. At 80° it is 5.76. The areal factor is $\sec\varphi$ too, since $h = 1$, and the angular deformation follows from the ratio.

The whole behaviour of the projection is that one $\cos\varphi$, left out.

The mechanism is visible in its indicatrices, which are all ellipses of the same height and increasing width, because the meridian scale is exactly one everywhere and the parallel scale is $\sec\varphi$. An undistorted map would draw every one of them as a circle of the same size.

Where it is genuinely right #

Being fair to it, because the essay is largely a complaint and there are real cases.

A raster grid. A global dataset stored as an array with rows of constant latitude and columns of constant longitude is a plate carrée, and storing it that way is the right decision: the indexing is trivial, the interpolation is separable, and every value’s position is recoverable by arithmetic rather than by a projection library. Almost every global elevation, climate and satellite dataset is stored this way.

Equatorial regions. The distortion is zero on the equator and small near it. Within about ten degrees the scale error is under 1.5 per cent, and for a map of the tropics the projection is entirely reasonable.

As a coordinate display. A plot whose purpose is to show where numbers are, rather than what a place looks like, is not making a cartographic claim, and the plate carrée’s transparency — a point’s map position is its coordinates — is a genuine advantage.

Thumbnails and locator maps. At a size where nothing can be measured, the choice does not matter and the cheapest one is correct.

The site’s own region ranking bears the second case out: over a band of the tropics the equirectangular projection does perfectly respectably, and over the whole world it ranks third of seven on the Kavrayskiy criterion — better than Mollweide and better than Mercator — which is a genuine surprise and worth reporting rather than suppressing.

Where it is wrong and used anyway #

The dominant case, and it is the reason the essay exists.

Every plotting library that draws geographic points on Cartesian axes produces a plate carrée. Every scatter of coordinates, every heat map of locations, every choropleth assembled from a shapefile without a projection step. Nobody selected the projection; it is what treating the numbers as numbers produces.

The consequences are the ordinary ones and they are large. A density map on this projection has high-latitude regions inflated by up to eightfold, so the visual weight contradicts the data — which is exactly the error an equal-area projection exists to prevent. Countries above about 45° look stretched sideways. Distances measured with a ruler are wrong except due north.

None of that is the projection’s fault. It is a specification met exactly, and it was applied to a job whose specification is different.

The family it belongs to #

The plate carrée is one member of a one-parameter family, and the others are worth knowing because the parameter is free and is almost never exercised.

The equidistant cylindrical projection with standard parallel $\varphi_1$ is $x = \lambda\cos\varphi_1$, $y = \varphi$. The meridian scale is 1 everywhere; the parallel scale is $\cos\varphi_1/\cos\varphi$, which equals one on the standard parallel. So the projection is true to scale along the meridians and along one chosen parallel.

The whole map is $2\pi\cos\varphi_1$ wide and $\pi$ tall, giving an aspect ratio of $2\cos\varphi_1$: exactly 2:1 for the plate carrée, $\sqrt2$:1 at 45°, and square at 60°.

The member at $\varphi_1 = 45°$ has a name — Gall’s isographic projection — and is a straightforwardly better default than the plate carrée for a world map, because it halves the worst-case parallel stretch by putting the true-scale line in the middle of the inhabited latitudes rather than at the equator. It is almost never used, for the same reason the plate carrée is: nobody is choosing.

That is the standard-parallel argument in its cheapest possible form. One number, freely available, never set.

Ranked over the whole sphere and then over the tropics, it is mid-table on the first and near the top of the second — because its distortion is concentrated exactly where an area-weighted integral discounts it.

The surprise in the ranking #

Something worth reporting because it cuts against the essay.

The site’s regional criterion integrates both principal scales’ departure from unity over a stated region, and over the whole sphere Kavrayskiy’s version ranks the plate carrée third of seven — behind Winkel tripel and Robinson, and ahead of Eckert IV, Mollweide, Mercator and Gall–Peters.

Airy’s version, on the same seven projections, puts it sixth. The two criteria differ by one function — a squared difference against a squared logarithm — and they disagree about this projection more than about any other in the set, because its failures are large inflations at high latitude and the two measures price an inflation quite differently.

That is not what the worst-case figures suggest, and the reason is instructive. The plate carrée’s failures are concentrated at high latitude where a world map has very little land and where the area element $\cos\varphi$ is small, so an area-weighted integral discounts them heavily. Its behaviour through the middle latitudes, where most of the world is, is mediocre rather than bad.

So a projection can be a poor choice and score respectably, and the gap between those two statements is a purpose. The criterion is measuring average distortion over the sphere; nobody’s map has that as its objective; and a scalar summary is a purpose in disguise. The essay’s case against the plate carrée is not that it scores badly. It is that it was not chosen.

The Euclidean distance problem #

One specific failure deserves its own note because it happens in code rather than in pictures.

Computing $\sqrt{(\Delta\lambda)^2 + (\Delta\varphi)^2}$ between two coordinate pairs — treating latitude and longitude as plane coordinates — is measuring distance on the plate carrée. The answer is wrong by a factor that depends on latitude and on the direction of the separation, and it is wrong in a way that produces plausible small numbers rather than errors.

At the equator, for an east–west separation, it is right. At 60° it overstates an east–west distance by a factor of two. For a north–south separation it is right everywhere. A nearest-neighbour search using it will return different neighbours at different latitudes and will never fail visibly.

The fix is the haversine formula, or a projection chosen for the region, and both are a line of code. The reason the wrong version persists is that it is the version that requires no decision.

The projection Ptolemy preferred instead #

The oldest recorded objection to this projection came with an alternative attached, and the alternative is a good one.

Ptolemy’s first projection, described in the Geography around AD 150, keeps the parallels as circular arcs rather than straight lines, spaced at true distance along the meridians, with the meridians converging toward a point above the pole. It is a simple conic, and it is true to scale along the meridians and along two chosen parallels rather than one.

That is a strictly better construction for the region Ptolemy was mapping — the inhabited world, from about 16° south to 63° north — and the argument he makes for it is the argument this site keeps making: the map should be shaped to the region it shows, and the convenience of the draughtsman is not a cartographic criterion.

His second projection goes further, curving the meridians as well, and he says plainly that it is harder to draw and better, and that a reader in a hurry may use the first. Nineteen hundred years later the field is still choosing the version that is easier to draw, and the version that is easier to draw is now the one that requires no drawing at all.

The name #

Plate carrée is French for “flat square”, and the projection’s oldest attribution is to Marinus of Tyre, around AD 100, who used it for the sailing charts that Ptolemy criticised half a century later.

Ptolemy’s objection was that the projection distorts, that Marinus knew it distorted, and that Marinus used it anyway because it was easy to construct. The Geography is fairly pointed about it.

That criticism is very nearly the argument in this essay, which suggests either that the field has made no progress in nineteen hundred years or, more charitably, that the trade-off between convenience and fidelity has always resolved the same way whenever nobody has a stake in the fidelity.

What a default should be #

The essay’s recommendation is not “use something else”. It is narrower.

A default is a decision made once by whoever wrote the tool, applied thereafter by people who did not know a decision was being made. That makes the choice of default consequential in a way an individual choice is not, and worth arguing about.

For a data store, the plate carrée is the right default and should stay. Its transparency and index arithmetic are worth more than its distortion, and the data is not a picture.

For a displayed map, it is a poor default, and the alternatives are cheap. A Robinson or Winkel tripel for a world map, an equal-area projection for anything per unit area, a local conformal projection for a region: each is one argument to a library that already supports it.

What is not defensible is the current arrangement, where the display default is inherited from the storage default and nobody has noticed that the two questions are different. That is not a cartographic error; it is a default doing a job it was not chosen for, which is the whole pattern this site keeps finding.

The projection nobody chooses has a distortion pattern as definite as any other, and drawing it is the shortest way to say what accepting the default costs.

Two defaults side by side: the one that software falls back to and the one the web fell into.

Two things about that pattern are worth separating. Its graticule stays orthogonal everywhere, so every direction of maximum stretch lies along the graticule and the classical scale along the parallel really is a principal scale — one of the very few things the projection gets right without being asked. And across a region its largest scale factor is on the frontier, as Mercator’s is; the difference is that Mercator’s is pinned there by a theorem and the plate carrée’s is there by accident.

What was computed here #

Both distortions come from the projection’s own derivatives, and the equirectangular projection is one of the four for which analytic scale factors are also implemented — $h = 1$, $k = \sec\varphi$ — so the numerical and analytic routes are compared and agree to better than 3×10⁻¹¹.

The projection is asserted to fail both tests on every build. That is unusual: most assertions here confirm a property. This one confirms the absence of two, and it is the check that keeps the essay’s central claim honest, because a projection that quietly started passing one of them would make the argument false.

The regional comparison is computed by integrating the two principal scales’ departure from unity over a stated region with the area element, normalised so that the geometric mean areal factor over the region is one. The normalisation matters: a projection is free to choose its overall scale, and without it the comparison would largely measure that choice.

The equirectangular projection is also one of the four in the library with analytic scale factors implemented, so its numerical and analytic routes are compared on every build and agree to better than 3×10⁻¹¹. That is worth noting for a projection whose whole defect is one missing cosine: a sign or a placement error in the analytic form would be caught by the comparison rather than by inspection.

What the pictures cannot show #

That anybody chose this. Every figure here shows a projection behaving exactly as its formula says, and the essay’s subject is a decision that was not taken — which has no visual signature at all.

The indicatrix figure comes closest: the ellipses are all the same height and increasingly wide, which is the visible form of one missing cosine, and it is as near as a picture gets to showing an omission.

The projection nobody chooses, tiled #

The argument of this essay has an exact modern counterpart. The plate carrée is the second most widely tiled projection there is, for the reason given above — a coordinate on it is the coordinate in the file, so serving it needs no transformation and no resampling at all.

Its world is 2:1, which a two-tile root covers with squares, so it needs no polar cut: the geographic tiling carries the whole globe while the Mercator one discards 1,901,487 square kilometres. What it gives up is measured in the applied field: 38.9° of angular deformation and a factor of two in area over a mid-latitude region, and 103 of 400 nearest-neighbour queries answered with a different site.

Two closing observations, one about the medium and one about the numbers. A tiling scheme makes a trade between keeping angles and keeping areas, and this projection sits at neither end of it — which is the essay’s whole point arriving in the medium where the projection is most used. And it shares one failure with every other geographic storage: longitude is a circle and the numbers are an interval, so a shape across the cut has a bounding box eighteen times too wide and a midpoint at the antipode — the antimeridian is a cut in the numbers.

Who found it, and when #

Marinus of Tyre, around AD 100. Ptolemy describes and criticises it in the Geography around AD 150, which makes it both the oldest projection in continuous use and the oldest one to have been publicly complained about.

Its modern dominance is entirely a consequence of computing. Storing a global grid as an array indexed by latitude and longitude is the obvious representation, and every raster format, every plotting library and every naive coordinate plot inherits it. The projection did not win an argument; it is what the storage format looks like when it is displayed.

Where this goes next #

The framework for choosing rather than inheriting is which projection is best. The other projection nobody chose, and the one this site’s machinery caught, is Web Mercator is not conformal. And the way to compare projections over a region rather than over the world is distortion over a region.