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

The most-used map in the world fails the property in its own name.

Web Mercator carries almost every map on the internet, and it is not conformal. It applies spherical formulae to ellipsoidal latitudes, so the one property Mercator exists to provide — that angles are preserved, everywhere, exactly — does not hold. The machinery here found that without being told to look: a projection is called conformal on this site only after the angular deformation has been measured at several hundred points and found to be zero, and this one is not. These are essays about map projections with the distortion computed rather than described.

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 the library put through the two tests its name implies. Maximum angular deformation on the left, areal scale error on the right, both measured over several hundred sample points rather than taken from the projection’s description. Mercator passes the angle test and fails the area test; Mollweide does the reverse; Web Mercator fails both. Nothing passes both, and nothing ever can.

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19 essays

10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured 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.

9 figures planeK = 0 — unrolls flatcylinderK = 0 — unrolls flatconeK = 0 — unrolls flatsphereK = 1.00torusK = 1.79pseudosphereK = -1.00K computed at each centretwo routes, one answer The impossibility

No map is faithful

Not "no map yet", and not "no map at page size". Gaussian curvature can be computed from inside a surface, a sphere has some and a plane has none, and that closes the question permanently.

7 figures 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 Measuring distortion

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.

8 figures LondonTokyosolid: shortest · dashed: constant bearing+18.2% for the rhumbdrawn in Mercator Paths and directions

The shortest route is not straight

The shortest path between two points on a sphere is an arc of a great circle, and on almost every map it is a curve. The straight line on a Mercator chart is a different route entirely, and on some journeys it is twenty-eight per cent longer.

7 figures conformalequal-areacompromiseclaim failscylindricalMercatorMercatorLambertGall–PetersBehrmannEquirectangularMillerWebpseudocylindricalSinusoidalMollweideEckertRobinsonpseudoazimuthalHammerWinkelazimuthalStereographicLambertOrthographicGnomonicAzimuthalconicLambertAlbersrows: how it is builtcolumns: what it preserves The families

Cylinders, cones and planes

The standard taxonomy sorts projections by the shape of paper they were notionally rolled from. It is memorable, it is how everyone is taught, and it says almost nothing about the properties anyone actually chooses on.

6 figures 0°20°40°60°80°MercatorGall–PetersWinkelRobinsonlatitudeangular deformationalong a meridian What each projection optimises

Every projection minimises something

A projection is the solution to an optimisation problem, and naming the objective explains more than naming the family. Some objectives are exact constraints, some are least-squares fits, and one is a table of numbers a man adjusted until it looked right.

6 figures New YorkMadridsolid: shortest · dashed: constant bearing+3.0% for the rhumbdrawn in Mercator Paths and directions

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.

8 figures equator23°45°60°70°1×MercatorEquirectangularMillerGall–Peterslatitude of the cell1× means treated fairlyrelative to the equator What is taught wrongly

Mercator against Peters

The most-argued question in cartography, conducted almost entirely without anyone measuring anything. Both projections are exactly what they claim, each destroys what the other keeps, and the numbers are computable in either direction.

8 figures EquirectangularMercatorMollweideSinusoidalRobinsonWinkel tripelsame sphere, same graticuleno two agree What each projection optimises

Which projection is best

An incomplete question, and the incompleteness is the answer. Every projection preserves something and destroys something else, so the comparison worth making is between a projection and a purpose, not between two projections.

8 figures wrappedunrolled — every distance unchanged→K = 0 on bothcircumference 2π = width 2π The impossibility

What can be unrolled

A cylinder is obviously curved and is intrinsically flat, so it lays out on a table with nothing stretched. A sphere is not. The distinction is exactly zero Gaussian curvature, and it is why paper tubes exist and paper globes do not.

8 figures a = 1.08b = 0.92an infinitesimal circle, projectedmeasured at this pointh1.0834scale along the meridiank0.9231scale along the parallela1.0834larger principal scaleb0.9231smaller principal scalea·b1.0000areal scale factorω9.16°maximum angular deformationdashed: undistorteddrawn in Gall–Peters Measuring distortion

What survives a change of coordinates

The scale along the meridian is a property of the map and the grid together. The principal scale factors are a property of the map alone. Only the second kind describes the projection, and the two are routinely quoted as though they were the same thing.

6 figures 0°20°40°60°80°exact at equatorexact at 30°exact at 45°latitudeangular deformationall three are equal-area The families

What a standard parallel buys

A standard parallel is a line where the projection is exact. Choosing one does not reduce the distortion — it decides where the distortion is zero and lets everything grow away from it.

6 figures 10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured The impossibility

The trade-off is two lines

Conformal means the two principal scales are equal. Equal-area means their product is one. Both at once forces both to one, which is an isometry, which the curvature forbids. That is the entire argument.

9 figures 0°20°40°60°80°MercatorGall–PetersMollweideWinkellatitudeangular deformationalong a meridian Measuring distortion

The two ways a map is wrong

Angles and areas fail independently. A projection can be perfect about one and catastrophic about the other, and a single number summarising "how distorted" a map is has already thrown away the distinction that matters.

8 figures Mercatorgreat circle bows 3.7e-1Gnomonicgreat circle bows not at allEquirectangulargreat circle bows 2.0e-1Orthographicgreat circle bows 9.0e-2solid: shortest · dashed: constant bearingone pair of routes Paths and directions

The gnomonic companion

One projection turns every great circle into a straight line, and it is the only one that does. It shows less than half the sphere, distorts enormously, and was indispensable for three centuries because of that single exact property.

6 figures RobinsonWinkel tripelMollweideMercatorGall–PetersEckert IVsame sphere, same graticuleno two agree What each projection optimises

Compromise projections

A projection that preserves nothing exactly can distort everything less than one that preserves something exactly. For a general-purpose world map that is the right trade, and it is why the two most widely used ones today have no exact property at all.

7 figures Orthographicgreat circle bows 9.0e-2Gnomonicgreat circle bows not at allStereographicgreat circle bows 9.3e-2Azimuthal equidistantgreat circle bows 1.5e-1solid: shortest · dashed: constant bearingone pair of routes The families

The aspect is a free choice

A projection's distortion pattern is fixed relative to its own axis, and where that axis points is entirely up to the cartographer. Rotating it is the cheapest available improvement and it is the one most often left unmade.

6 figures Mercator5.62× areaGall–Peters1.00× areaMollweide1.00× areaEquirectangular2.36× areathe cell at 60°scaled to fit, areas as stated What is taught wrongly

The projection that shows true size

There is no such thing, and the phrase hides a real question. Equal-area projections preserve area and destroy shape; nothing preserves size in the sense the phrase implies; and the tools that make the point best are not maps at all.

6 figures 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 distortion

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.

7 figures

Threads running through

themes, not chapters

Measured, not named

A projection is usually called conformal because that is its name. Here it is called conformal only after the angular deformation has been computed at several hundred points and found to be zero.

11 essays

A theorem, not a limitation

No map is faithful, and this is not a shortcoming awaiting a cleverer cartographer. Gaussian curvature is intrinsic, a sphere has some and a plane has none, and that settles it.

3 essays

What survives a change of coordinates

Scale along the meridian and scale along the parallel depend on how the sphere was parameterised. The principal scales, the areal factor and the angular deformation do not, and only those are properties of the map itself.

4 essays

Every figure here is a projection

The page is flat, so there is no way to show the sphere except by projecting it. Even the globes are projections with their own distortion, and no picture here is the undistorted truth.

2 essays

The trade-off is forced

Conformal means the principal scales are equal; equal-area means their product is one. Both at once forces them both to one, which is an isometry, which cannot exist. The trade-off is two lines of algebra.

9 essays

Computed, not quoted

Scale factors, areal ratios, angular deformations, geodesic distances and the areas of countries are all computed while the figure is drawn. None is a number recalled from a table.

14 essays

Purpose before property

Asking which projection is best is asking an incomplete question. Every one of them minimises something, and the only useful comparison is between a projection and the job it was chosen for.

12 essays