Cartographic Projection
every map is wrong, and by exactly how much
What is taught wrongly

Web Mercator is not conformal

It carries almost every map on the internet, it is named after the projection whose entire purpose is preserving angles, and it does not preserve angles. The machinery here found that without being told to look.
17 min read 9 figures Measured, not namedComputed, not quotedPurpose before property

Almost every map on the internet is drawn in Web Mercator. Google Maps, OpenStreetMap, Bing, Mapbox, every slippy map with tiles — all of them, on a projection registered as EPSG:3857.

Mercator’s whole reason for existing is that it preserves angles exactly, everywhere. Web Mercator does not.

Every projection in the library, measured against both propertiesMaximum angular deformation across the bottom, maximum areal error up the side, both on logarithmic scales and both measured over several hundred sample points rather than taken from the projection's description. Conformal projections lie on the left edge, equal-area ones along the bottom, and the corner where both vanish is empty because a projection there would be an isometry of the sphere onto the plane.10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured
Fig. 1 Every projection in this site’s library, put through both tests its family implies. Angular deformation across the bottom, areal error up the side, both measured over several hundred sample points rather than read off the projection’s description. One point is marked in red: it is the only projection here that fails the property named in its own title.

What it does

Mercator’s formula, on a sphere, is

x=λ,y=ln⁡tan⁡ ⁣(π4+φ2)x = \lambda, \qquad y = \ln\tan\!\left(\frac{\pi}{4} + \frac{\varphi}{2}\right)

That is exact and it is conformal, and it has to be, for the projection to do its job.

The Earth is not a sphere. It is an oblate ellipsoid, flattened by about one part in three hundred, and the correct Mercator projection for an ellipsoid has an extra factor:

y=ln⁡[tan⁡ ⁣(π4+φ2)(1−esin⁡φ1+esin⁡φ)e/2]y = \ln\left[\tan\!\left(\frac{\pi}{4}+\frac{\varphi}{2}\right)\left(\frac{1-e\sin\varphi}{1+e\sin\varphi}\right)^{e/2}\right]

Web Mercator omits it. It takes latitudes measured on the WGS84 ellipsoid — the ones every GPS receiver reports — and pushes them through the spherical formula.

Why that breaks conformality

Conformality is a local condition: the projection must scale by the same factor in every direction at every point. Measuring it means comparing the scale along the meridian with the scale along the parallel, and requiring them to be equal.

On an ellipsoid those two directions have different natural units. A step of one radian in latitude covers a distance M dφM\,\mathrm{d}\varphi, where MM is the meridional radius of curvature; a step in longitude covers Ncos⁡φ dλN\cos\varphi\,\mathrm{d}\lambda, where NN is the radius of curvature in the prime vertical. And M≠NM \ne N anywhere except at the poles.

Working the two scale factors out for Web Mercator, almost everything cancels and what is left is

hk=NM=1−e2sin⁡2φ1−e2\frac{h}{k} = \frac{N}{M} = \frac{1 - e^2\sin^2\varphi}{1 - e^2}

At the equator that is 1/(1−e2)=1.006731/(1-e^2) = 1.00673. The projection stretches the meridian direction two-thirds of a per cent more than the parallel direction, and a projection that stretches unequally is not conformal.

The measured maximum angular deformation is 0.3848°, at the equator, falling to zero at the poles.

Why the machinery caught it

Nothing in this site was written to look for this. The library holds a set of projections, each declaring what it claims to be, and a function that measures angular deformation from the projection’s own derivatives. Running the claims against the measurements is one loop.

Web Mercator declares itself conformal because that is what it is called, and the loop reported that it is not. The same loop confirms the genuine cases: the spherical Mercator, the correct ellipsoidal Mercator, the stereographic and the Lambert conformal conic all measure their angular deformation at around 1.5×10−61.5\times10^{-6} degrees, which is the noise floor of a numerically differentiated Jacobian rather than a real quantity.

The gap between that noise and Web Mercator’s 0.3848° is a factor of about 240,000. This is not a marginal call.

How Web Mercator distorts, by latitudeAngular deformation and areal scale plotted against latitude along the meridian at 0°. On Web Mercator the angular deformation reaches 0.4° and the areal factor reaches 90.9.-60°-30°0°30°60°angular deformation, to 2°areal factor, to 90.9×latitudetwo independent distortionsalong the 0° meridian
Fig. 2 Web Mercator’s two distortions against latitude. The areal factor runs away toward the poles, as it must for any Mercator. The angular deformation should be flat on zero and is not — it peaks at the equator, which is the opposite of where a Mercator’s problems usually live.

That last point is worth dwelling on. Every other defect of Mercator is worst at high latitudes. This one is worst at the equator, because the ellipsoid’s two radii of curvature differ most there and agree exactly at the poles. It is a different kind of error with a different signature, which is part of why it went unnoticed for so long.

How big is 0.38 degrees

Small. Genuinely small, and the essay would be dishonest if it implied otherwise.

For displaying a map on a screen, 0.38° of angular deformation is invisible. Nobody has ever looked at Google Maps and noticed that the streets meet at slightly wrong angles, and nobody ever will. In terms of position, the discrepancy between Web Mercator and true ellipsoidal Mercator reaches about 20 km at the equator when both are scaled to agree at the poles — which sounds enormous and is, for surveying, and is nothing for finding a restaurant.

The reason it matters is not the magnitude. It is that conformality is a yes-or-no property, and a great deal of software assumes it holds.

Anything that computes a bearing from a map, rotates a symbol to match a heading, or draws a circle of fixed radius on the ground is relying on angles being preserved. Anything that reprojects between EPSG:3857 and a real coordinate system and expects the transformation to be conformal is relying on it too. The error is small, but it is a systematic error in a property that was supposed to be exact, and systematic errors accumulate in a way random ones do not.

The registry refused it for years

The institutional history is the most revealing part.

EPSG is the authority that assigns codes to coordinate reference systems, and for a long time it declined to give this one a code at all. Its position was straightforward: a coordinate reference system is defined by a datum and a projection, this thing uses a datum from one and a projection from another, and the combination is not internally consistent. Assigning it a code would legitimise it.

Practice won, as it does. The system was in use everywhere, files needed to declare what they contained, and refusing a code meant everyone invented their own. It was eventually assigned EPSG:3785 and then, after that was deprecated, EPSG:3857 — and the official name carries the qualification in it: WGS 84 / Pseudo-Mercator.

The word pseudo is the standard admitting exactly the point of this essay.

The same point at 30°, 5° under four projectionsOne circle on the sphere, four projections, four ellipses. A conformal projection keeps it circular and lets the area run; an equal-area projection keeps the area and lets the shape go. Nothing keeps both, and the dashed circle shows what keeping both would look like.Mercatorω 0°area 1.01×Webω 0°area 1.01×Mercatorω 0°area 1.01×Gall–Petersω 39°area 1.00×dashed: undistortedscaled to fit
Fig. 3 The indicatrix near the equator under four projections. The spherical Mercator and the correct ellipsoidal Mercator produce circles, as a conformal projection must. Web Mercator produces an ellipse — very slightly, and measurably.

Why it was done

Not carelessness. The spherical formula has one property the ellipsoidal one lacks: it is closed-form in both directions.

The ellipsoidal inverse requires iteration. That is nothing on a modern processor and it was not nothing on a web server in 2005 serving map tiles to thousands of clients, where the projection has to run for every tile boundary, every marker, every pan and zoom. Dropping the correction made the arithmetic cheap and made the inverse exact rather than iterative, at the cost of a property nobody displaying a street map needed.

Given the purpose, it is a defensible engineering trade. Naming the purpose before arguing about the property is the whole method this site recommends, and by that standard Web Mercator is a reasonable choice for slippy maps and a bad choice for anything measuring the ground.

What is not defensible is the silence. A projection called Mercator, used by everyone, quietly lacking Mercator’s defining property, is a trap for exactly the people most likely to walk into it — the ones who know what conformal means and assume the name is telling the truth.

What it inherits from Mercator, which is worse

The conformality failure is the interesting defect. It is not the one that affects most readers, because Web Mercator also inherits every property of ordinary Mercator, including the one everybody complains about.

Five identical cells on Web MercatorFive patches, each 20° of longitude by 10° of latitude. On the sphere the higher ones are genuinely smaller, because the meridians converge. On Web Mercator the cell at 70° comes out 15.4 times larger than the equatorial one relative to its true size.1.0×1.3×2.4×5.6×15.4×each cell is 20° × 10°drawn in Web Mercator
Fig. 4 Five patches of the sphere, each 20° of longitude by 10° of latitude. On the sphere the higher ones are genuinely smaller, because the meridians converge as they approach the poles. On Web Mercator the patch at 70° comes out fifteen times larger than the equatorial one relative to its true size.

Fifteen-fold. That figure is computed exactly rather than estimated, because a latitude–longitude cell has a closed-form area and no coastline dataset is involved. It is the mechanism behind every argument about the apparent size of Greenland, and it is worth a separate essay because the argument is usually conducted without anybody measuring anything.

The two defects are independent, which is the point of measuring them separately. Web Mercator is wrong about angles by a third of a degree and wrong about areas by a factor of fifteen, and neither number says anything about the other.

Where the ellipsoid can be ignored, and where it cannot

The general rule this case illustrates: the difference between a sphere and the WGS84 ellipsoid is about one part in three hundred, and whether that matters is a question about the application rather than about the geometry.

For a world map at page size, one part in three hundred is a third of a pixel. Every projection drawn on this site outside this essay uses a sphere for exactly that reason, and says so.

For a national survey, one part in three hundred is hundreds of metres, and the ellipsoid is not optional. This is why national grid systems — the UK’s Ordnance Survey grid, the various UTM zones — are defined on ellipsoids with the full series expansions, and why their formulae are so much uglier than anything in this site’s library.

Web Mercator sits in the awkward middle. It is used at scales where the ellipsoid matters, by software that assumes it was handled, and it was not.

Angular deformation against latitude, four projectionsThe same quantity for webMercator, mercator, mollweide, winkelTripel, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both.0°20°40°60°80°WebMercatorMollweideWinkellatitudeangular deformationalong a meridian
Fig. 5 Angular deformation against latitude for four projections. Three of these curves behave as their names promise. The Web Mercator curve should lie on the axis with Mercator’s and does not — it lifts away near the equator, which is where the ellipsoid’s two radii of curvature differ most.

What the site does about it

Every projection here declares what it claims. The claim is then measured, and the caption reports the measurement rather than the claim. That is the whole of the discipline, and this essay is what it is for.

The gate goes further and requires the assertion to fail on Web Mercator on every build. An assertion that had quietly stopped rejecting would leave the site’s opening argument false while every other check continued to pass, and no picture would look wrong — which is the general reason for asserting anything at all.

What a coordinate reference system is supposed to be

The registry’s original objection is worth spelling out, because it identifies the defect more precisely than “not conformal” does.

A coordinate reference system is a datum plus a projection. The datum says what shape the Earth is taken to be and where it sits; the projection says how that shape is flattened. The two have to agree about the body, or the composition means nothing.

Web Mercator’s datum is WGS84, an ellipsoid. Its projection is spherical Mercator, defined for a sphere. The composition applies a sphere’s formula to an ellipsoid’s coordinates, and the mismatch is not a rounding error — it is a category error that happens to produce small numbers.

That framing explains why the objection was raised by coordinate-system specialists rather than by cartographers. It is not primarily that the map looks wrong; it is that the definition does not typecheck.

MercatorThe graticule of the Mercator projection at 30° of longitude and 15° of latitude. the only projection on which a constant compass bearing is a straight line. It is conformal.conformaldrawn in Mercator
Fig. 6 The Mercator graticule. Web Mercator draws exactly this, from exactly these formulae, and differs only in what the latitudes fed into them are taken to mean. Nothing about the picture records the difference.

There is a historical parallel worth noting, because it shows what happens when this kind of mismatch is large.

Before satellite geodesy, every country used its own datum, fitted to its own region. Coordinates in one datum could differ from the same point’s coordinates in another by hundreds of metres. Mixing them silently was a routine and expensive error, and the whole apparatus of datum transformations exists to prevent it.

Web Mercator is the same class of mistake with the magnitude turned down. The difference is that datum confusion produced errors big enough to be noticed, and this one does not — so the machinery built to catch the first kind never fires on the second.

Angular deformation against latitude, four projectionsThe same quantity for webMercator, mercatorEllipsoidal, mercator, miller, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both.0°20°40°60°80°WebMercatorMercatorMillerlatitudeangular deformationalong a meridian
Fig. 7 Web Mercator, the correct ellipsoidal Mercator, the spherical Mercator, and Miller for scale. Two of these curves sit exactly on the axis. Web Mercator’s does not, and the departure is largest where the ellipsoid’s two radii of curvature differ most.

What to do about it

Practical guidance, since the essay is otherwise diagnostic.

For display, nothing. The error is far below anything visible and Web Mercator is a reasonable choice for a slippy map.

For measurement on the ground — distances, areas, bearings — reproject to something appropriate first. A local projected coordinate system, a UTM zone, or an equal-area projection depending on what is being measured. Computing an area in Web Mercator coordinates and treating the result as square metres is a common and serious error, and it is wrong by the areal factor at that latitude rather than by the conformality defect.

For anything assuming conformality — rotating symbols to a heading, drawing a circle of constant ground radius, computing a bearing from screen coordinates — be aware the assumption is slightly false and decide whether a third of a degree matters.

The last case is the one this essay is really about, and the honest answer is usually that it does not. What matters is knowing rather than assuming.

Five identical cells on MercatorFive patches, each 20° of longitude by 10° of latitude. On the sphere the higher ones are genuinely smaller, because the meridians converge. On Mercator the cell at 70° comes out 15.4 times larger than the equatorial one relative to its true size.1.0×1.3×2.4×5.6×15.4×each cell is 20° × 10°drawn in Mercator
Fig. 8 The larger error, for scale. The conformality defect is 0.38 degrees; the areal inflation at 70° is fifteenfold. Anyone computing areas from Web Mercator coordinates is making the second mistake, which is far larger and far more common.

The general lesson

A name is a claim, and a claim that everyone has repeated for twenty years is still a claim.

Web Mercator’s defect survived because it is small, because the name is reassuring, and because checking would have required somebody to run a computation nobody thought was necessary. All three conditions are common, and the third is the one that can be fixed cheaply.

Running the definition as a computation takes twenty lines and finds this class of error without being told what to look for. That is the whole method this site is built on, and this is the case that justifies it.

How Mercator (ellipsoidal) distorts, by latitudeAngular deformation and areal scale plotted against latitude along the meridian at 0°. On Mercator (ellipsoidal) the angular deformation reaches 0.0° and the areal factor reaches 90.9.-60°-30°0°30°60°angular deformation, to 2°areal factor, to 90.9×latitudetwo independent distortionsalong the 0° meridian
Fig. 9 The correct ellipsoidal Mercator, measured against the same ellipsoid. Its angular deformation sits on the noise floor — which is what Web Mercator’s would look like if the correction had not been dropped.

What was computed here

The angular deformation is derived from the four partial derivatives of the projection, obtained by Richardson-extrapolated central differences and validated against the analytic scale factors where those exist. For Web Mercator the derivatives are taken against the WGS84 metric, using the two radii of curvature above, because measuring a projection’s distortion against the wrong body would answer a question nobody asked.

Three numbers anchor the result. The genuinely conformal projections measure 1.21.2–1.6×10−61.6\times10^{-6} degrees, which is the noise floor. Web Mercator measures 0.38480.3848 degrees. The tolerance sits at 10−410^{-4} degrees, about sixty times above the noise and nearly four thousand times below the failure — and the gate asserts both margins, so a tolerance loosened enough to certify Web Mercator as conformal would itself fail the build.

The correct ellipsoidal Mercator is implemented alongside, and passes. That is what makes the comparison a measurement of Web Mercator rather than a measurement of this site’s arithmetic.

What the pictures cannot show

The effect is too small to see. The indicatrix figure above exaggerates nothing, and the ellipse it draws for Web Mercator is a circle to any eye — the deviation is two parts in three hundred, and the figure has to state the number because the picture cannot carry it.

That is an uncomfortable position for a figure-first site, and it is the honest one. The whole argument of this essay is that a defect can be real, systematic, and invisible, which is precisely the case where measuring beats looking. A figure that made the error visible would have to distort it, and would then be making the same kind of claim it is criticising.

The essay also cannot show the consequences. Whether 0.38° matters depends entirely on what the map is for, and that is a question about the user rather than about the projection.

Who found it, and when

Mercator published his projection in 1569, without a derivation — the mathematics to justify it did not exist yet. Edward Wright supplied it in 1599, and the ellipsoidal correction came much later, once the shape of the Earth was known well enough for the difference to matter.

Web Mercator arrived with Google Maps in 2005 and spread with the slippy-map tile scheme. The objection was raised almost immediately by people who work with coordinate systems professionally; the National Geospatial-Intelligence Agency issued an advisory notice about it; and EPSG’s eventual code carries Pseudo-Mercator in its official name.

None of that stopped it becoming the most-used projection in the history of cartography.

Where this goes next

The method that caught it is measuring instead of naming. What Mercator’s conformality is actually for is why Mercator exists. And for the tool that makes all of this measurable, Tissot’s indicatrix.

Where this goes next

10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured Measuring instead of naming A projection is called conformal because that is its name. Running the definition as a computation over several hundred points takes about twenty lines, catches a projection the whole internet uses, and is almost never done. New YorkMadridsolid: shortest · dashed: constant bearing+3.0% for the rhumbdrawn in Mercator Why Mercator exists A ship can hold a compass bearing and cannot easily hold a great circle. Mercator is the answer to one question — what must a map do so that a constant bearing is a straight line — and it answers it exactly. a = 1.74b = 1.74an infinitesimal circle, projectedmeasured at this pointh1.7434scale along the meridiank1.7434scale along the parallela1.7434larger principal scaleb1.7434smaller principal scalea·b3.0396areal scale factorω0.00°maximum angular deformationdashed: undistorteddrawn in Mercator Tissot's indicatrix A tiny circle on the sphere becomes an ellipse on the map, and the ellipse's two axes are the whole story. Almost every published indicatrix is drawn without them, which discards the content and keeps the decoration.