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

Every projection in the library, measured against both properties

Drawn here at the parameters it defaults to, with every essay that calls it.
Every projection in the library, measured against both properties. Maximum 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. All 26 are plotted; the six named here are Mercator, Web Mercator, Gall–Peters, Mollweide, Equirectangular, Stereographic.

Maximum 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. All 26 are plotted; the six named here are Mercator, Web Mercator, Gall–Peters, Mollweide, Equirectangular, Stereographic.

It is one of 8 figures drawn by the same construction, and the drawing above is this one with nothing changed.

16 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

Construction against property. Every projection in the library, sorted by the shape it was notionally rolled from and by what it actually preserves. The families do not line up with the properties: cylindrical and pseudocylindrical and pseudoazimuthal and azimuthal and conic each contain projections of more than one kind, which is why the cylindrical–conic–azimuthal taxonomy answers a question nobody has. Three entries are picked out: Mercator, Albers equal-area conic, Orthographic. 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.

 Six surfaces and their Gaussian curvature. Curvature computed at the centre of each surface, by both available routes — from the way the surface sits in space, and from distances measured inside it alone. The two agree, which is Gauss's theorem. A surface with K = 0 can be unrolled flat without stretching: here the plane, the cylinder and the cone do, and the sphere, the torus and the pseudosphere do not. 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.

 Every projection in the library, measured against both properties. Maximum 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. All 26 are plotted; the four named here are Mercator, Mercator (ellipsoidal), Web Mercator, Gall–Peters. 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 test that found it was not looking for it.

 Every projection in the library, measured against both properties. Maximum 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. All 26 are plotted; the five named here are Mercator, Stereographic, Gall–Peters, Mollweide, Winkel tripel. 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.

 6 projections of the same sphere. The same graticule under Robinson, Winkel tripel, Mollweide, Mercator, Gall–Peters, Eckert IV. Every one of them is a faithful drawing of the same object and no two agree, because each has chosen a different thing to preserve. 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.

 Angular deformation against latitude, four projections. The same quantity for Mercator, Gall–Peters, Mollweide, Winkel tripel, 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. 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.

 Every projection in the library, measured against both properties. Maximum 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. All 26 are plotted; the six named here are Mercator, Web Mercator, Gall–Peters, Mollweide, Equirectangular, Stereographic. 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.

 The same six projections, ranked over Europe and Chile. Each column orders the projections by Kavrayskiy's regional criterion — the root-mean-square departure of the two principal scales from unity, integrated over the region with the area element. The lines cross, which is the point: Lambert conformal conic leads over Europe and comes fifth over Chile. A table of projections ordered by distortion is a table about somebody's region. Measuring distortion

Distortion over a region

An indicatrix describes a point and every practical question is about a country. Going from one to the other means integrating, integrating means choosing a weighting, and the weighting is the step that turns a measurement into somebody's opinion.

 The scale factor across Europe, from the middle outwards. Each band is the range the larger principal scale factor takes on the ring at that distance from the centre of Europe: 0 is the middle, 1 the frontier. two of the three projections drawn are conformal, and every one of those reaches its maximum at the right-hand edge, because the logarithm of a conformal map's scale factor is subharmonic. None of them reaches a maximum inside the region. Measuring distortion

Where the worst point is

The largest scale error on a conformal map of a country is always on the frontier, never inside it, whatever the country's shape and whichever conformal projection was chosen. It is a theorem rather than a tendency, and it is the reason Chebyshev's criterion works.

 Every mixture of Equirectangular at 50.46° and Aitoff. Airy's criterion over the whole sphere, for every weighted average of the Winkel tripel's own pair. Each mixture is normalised to its own best constant scale first, so the comparison is about shape rather than size. The curve has its minimum at w = 0.45, where the criterion is 0.3037 against 0.4114 and 0.4290 at the two ends. That is a projection nobody constructed beating both projections somebody did, by 26 per cent. What each projection optimises

The average of two projections

The Winkel tripel is literally the arithmetic mean of two other projections, and this site's implementation of it agrees with that mean to the last bit. Averaging beats both ingredients by 26 per cent — and it preserves conformality exactly, destroys equal-area completely, and can turn eighteen per cent of the world inside out without either distortion measure saying so.

 Four ways to build the same equal-area projection. The cylindrical equal-area projection's own areal scale factor, measured from its Jacobian against the metric of the body it is drawn for. On a sphere with the spherical formula it is one everywhere, which is the control. Feed the same formula the geodetic latitude a coordinate actually carries and measure against the ellipsoid it refers to, and it is 1.00674 on the equator and 0.99332 at 88° — a spread of 1.34 per cent, on a projection whose entire purpose is that there is no spread. Rescaling to make the totals agree does not repair it. The authalic northing q/2 does, exactly. What is taught wrongly

Equal-area on the wrong body

The site's headline is that Web Mercator puts geodetic latitudes into a spherical conformal formula and stops being conformal. The same sentence is true with "equal-area" in it and nobody says it: the areal factor is 1.00674 at the equator, 0.99332 at 88°, and averages to almost exactly one — so every check that adds up areas passes while every cell is wrong.

 The azimuthal family, as one function of one variable. Every azimuthal projection is a rule for how far from the centre to draw a point at angular distance ρ, and nothing else. five named projections are plotted as their radial functions — tan ρ, 2 tan(ρ/2), ρ, 2 sin(ρ/2), sin ρ — and the two dashed curves are the solutions of the two conditions, integrated from the centre with no more than "true scale at the centre" to start from. They land on the stereographic and the Lambert azimuthal to 8e-10. The names are the answers to the equations, not descriptions of viewpoints. The families

The azimuthal family is one function

Five azimuthal projections are taught as five viewpoints — from the centre, from the far pole, from infinity. They are one projection with five choices of how far out to draw a point at angular distance ρ, and each named property is a differential equation in that one function: f′ = f/sin ρ integrates to the stereographic projection and f f′ = sin ρ to Lambert's, from nothing but true scale at the centre.

 Every library projection over Europe, on the two axes it can be wrong on. Each dot is one projection, scored over Europe on the two independent failures: how much it turns angles and how much it changes areas. The lower-left corner is the isometry that does not exist. The line joins the five projections nothing beats on both counts — the rest are inside it, and a reader who prefers either failure to the other should still not choose one of them, whatever weighting they hold. What is taught wrongly

The projections that are beaten on both counts

A rule of thumb has been scored over thirty regions and then forty-five. The same populations answer a harder question nobody has put: which library members are never the right answer at all. Two are beaten outright on both criteria everywhere, one is on no regional front in any population — and it is on the world's.

 Three implementations of one projection, and three different distortions. The spread in angular deformation between linear, Catmull–Rom and Aitken interpolations of Robinson's own table, at 60° of longitude. It is exactly zero at every fifth degree, because all three pass through the tabulated rows, and it is not zero anywhere else: 37 of 87 latitudes differ by more than half a degree and the worst is 8.42°. The graticules the three draw differ by 4.2 parts per thousand of the map's own span, which is under the width of a printed line. The families

A projection defined by a table has an interpolation in it

One member of this library has no formula: Robinson set nineteen pairs of numbers by eye and the table is the definition. Three published interpolations of it draw graticules that differ by four parts in a thousand of the map's span and report angular deformations that differ by 8.4 degrees.

 Every projection's angular deformation, and the tolerance that judges it. The whole library on one logarithmic axis, with the tolerance drawn as a line. The population is bimodal: the projections that satisfy the condition sit at 2.09e-6 and below, the ones that do not at 3.85e-1 and above, and there is nothing between. The tolerance could be moved anywhere in that gap — a factor of 1.84e+5 — without changing one verdict. Larger marks are projections that claim the property. What is taught wrongly

The tolerance that decides the verdict

Every verdict handed out about a projection's claimed property rests on a tolerance chosen once, long ago, at sixty times a measured noise floor. Swept, it decides nothing: the population is bimodal, the tolerance sits in a gap 238,000 times wide for conformality and 646 million times wide for equal area, and the verdict with the least room is a passing one whose margin is the arithmetic's rather than the map's.

 The same patch, pinned six ways. Two vertices have to be held or the conformal energy has a similarity's worth of null space. Which two turns out to decide two of the three numbers reported. The median angular deformation is the same to 8 per cent across all six — that is the map. The areal spread runs from 2.68 to 7.44, so the 3.07 reported for this body was a statement about its corners. And pinning two adjacent vertices, which fixes the similarity through a very short lever, ruins the worst point without touching the median: a badly conditioned constraint pays for its scale in one corner. What the numbers refer to

The map depends on where it was cut

Solving the discrete conformal equations on a triangulated body gave an areal spread of 3.07, with the number possibly belonging to the patch, the boundary and the two pinned vertices rather than to the surface. It belongs to the pins: hold a different pair and it runs from 2.68 to 7.44, while the typical angular deformation does not move at all.

The whole library