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 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 three named here are Mercator, Mollweide, Web Mercator.
Fig. 1 Every projection in the library put through the two tests its name implies. Maximum angular deformation across the bottom, areal scale error up the side, 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.

Nine fields

what the collection is divided into

The impossibility

Gaussian curvature is intrinsic, a sphere has some and a plane has none, so no map can be faithful. A theorem rather than an engineering limit.

37 essays

Measuring distortion

Scale along the meridian, scale along the parallel, and the two invariants that survive a change of coordinates. Tissot's indicatrix drawn from the derivatives and labelled with its numbers.

61 essays

The families

Cylindrical, conic, azimuthal — a taxonomy of construction that says surprisingly little about the properties anyone actually cares about.

37 essays

What each projection optimises

Every projection minimises something. Naming the objective is more honest than naming the shape of the paper it was notionally rolled from.

37 essays

Paths and directions

Great circles, rhumb lines, and the question that produced Mercator: what has to be true of a map for a constant compass bearing to be a straight line?

38 essays

What the numbers refer to

A latitude and a longitude are numbers about a model, and the model has a shape, a placement, a set of marked points and a date. Change any of the four and the ground under the number moves.

49 essays

Grids, and what a survey does

A coordinate is the end of a chain of conventions and the conventions are not in the number. What a grid declares, what a tape has to be corrected by before it becomes one, and which corrections a stated tolerance allows a job to drop.

38 essays

What a machine does with it

A coordinate in a file is drawn on a screen and asked questions by programs that must decide what straight, near and how large mean before they can answer. Those decisions are geometric, and every one of them has a size.

50 essays

What is taught wrongly

The most-argued subject in cartography, and most of the argument is conducted without measuring anything — including the projection that fails the property named in its own title.

37 essays

Start anywhere

where each field begins

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.

8 figures The indicatrix at 45°, 55° on Mercator. A circle on the sphere becomes this ellipse on the map. Its semi-axes are the two principal scale factors, a = 1.743 and b = 1.743; their product is the areal factor 3.040; and the maximum angular deformation is 0.00°. h and k are shown too, and depend on the coordinates rather than on the map. 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.

10 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.

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

5 figures London to Tokyo on Mercator. Two routes. The great circle is 9559 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 11296 km — 1737 km further, or 18.2 per cent. On Mercator the rhumb line departs from straight by 5.0e-9 of its own length. 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.

8 figures The same coordinate on four datums. One pair of numbers — 2.0° west, 54.5° north — read as a coordinate on OSGB36, ED50, NAD27 and on WGS84, and plotted where each reading puts the mark on the ground. The spread runs to 195 metres. The numbers are identical; only what they refer to differs. What the numbers refer to

What a coordinate refers to

A latitude and a longitude are not a place. They are a claim about a model, and the model has a shape, a position, a set of marked stones and a date — change any one of the four and the ground under the numbers moves by hundreds of metres.

8 figures A false origin moves every number and no geometry. British National Grid, drawn twice at the same scale. On the left the coordinates are measured from the projection's own origin, where the central meridian meets the true origin's parallel, and 52% of the country takes a negative easting or northing — the worst reaching -323 km. On the right the authority's published false origin of 400 km east and -100 km north has been applied, and none of them does. Every distance computed from the two sets of numbers agrees to the last bit a double has left after carrying six figures, and every bearing agrees exactly; the only thing that changed is that no coordinate carries a sign. The margins say the origin was fitted to the land rather than placed arbitrarily below it: 77 km spare in the west and 50 m in the south, on a grid 988 km tall. Grids, and what a survey does

A grid has an origin that is not there

Every published national grid measures from a point in the sea. The British one sits 400 kilometres west and 100 kilometres south of where the projection's own zero is, and moving it changes every coordinate in the country and not one distance or bearing.

7 figures The tile pyramid, four levels down to quadkey 120. The world as a square, quartered three times. Each level's tile is exactly half the width of its parent, so a tile is four tiles at the next level and never needs resampling to serve one — the property the whole scheme rests on, and one that holds only because the projected world is square. Level 3 has 64 tiles at 19567.9 metres per pixel, which at 51.5° north is 12181.3 metres of ground per pixel rather than the number the scheme publishes. What a machine does with it

A screen map is a pyramid of tiles

The scheme every slippy map runs on is a coordinate system with three integers and one projection, and almost all of it is forced. A square world is what makes the quadtree work, the levels are exact powers of two, and the published resolution — 156,543 metres per pixel at zoom zero — is a distance on the ground at exactly one latitude.

7 figures 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.

7 figures

How deep one idea goes

the eight longest of 31 ladders

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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.

323 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.

91 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.

110 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.

27 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.

112 essays

Computed, not quoted

Scale factors, areal ratios, angular deformations, geodesic distances and the areas of countries are all computed from the projection's own formulae. None is a number recalled from a table.

333 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.

147 essays