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

Every essay

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

9 fields, 31 ideas with a ladder of their own, 713 named objects threaded through them, and 50 claims the collection puts a number to. This page is the flat list; the others are the ones that keep working as it grows.

Fields31 laddersThreads50 tested claims713 named objects29 generator familiesSearch

The impossibility 37 Measuring distortion 61 The families 37 What each projection optimises 37 Paths and directions 38 What the numbers refer to 49 Grids, and what a survey does 38 What a machine does with it 50 What is taught wrongly 37

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 · Paths 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 · Screen 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 · Tissot 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 · Families 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 · Curvature 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 · Datum 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 · Choosing 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 · Audit 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 · Grid 6 projections of the same sphere. The same graticule under Equirectangular, Mercator, Mollweide, Sinusoidal, Robinson, Winkel tripel. 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

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.

4 figures · Choosing A cylinder unrolls exactly; a sphere does not. Both pictures show the same grid. On the left it is wrapped round a cylinder of radius 1, on the right it is laid flat, and every distance in the grid is the same in both — the circumference is 2π and so is the width of the rectangle, checked to 10⁻⁹. This is possible because a cylinder has zero Gaussian curvature. No corresponding picture exists for a sphere. 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 · Curvature Angular deformation around three standard parallels. Three equal-area cylindrical projections differing only in where they are exact — standard parallels at equator, 30°, 45°. Each has zero angular deformation at its own standard parallel and grows away from it in both directions. Choosing a standard parallel is choosing which latitudes to treat well, and there is no choice that treats them all well. 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.

4 figures · Families The indicatrix at 30°, 40° on Gall–Peters. A circle on the sphere becomes this ellipse on the map. Its semi-axes are the two principal scale factors, a = 1.083 and b = 0.923; their product is the areal factor 1.000; and the maximum angular deformation is 9.16°. h and k are shown too, and depend on the coordinates rather than on the map. 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.

7 figures · Tissot OSGB36 to WGS84, one parameter at a time. Each bar is how far the mark at 2.0° west, 54.5° north moves under one of the seven parameters with the other six set to zero. seven of the seven are not zero for OSGB36. The units hide the comparison: one arcsecond of rotation moves this point 30.8 metres and one part per million of scale moves it 6.36 metres, so OSGB36's scale term contributes 130 metres — more than one of its three translations. What the numbers refer to

The seven parameters, and what each one does

A datum transformation is published as three metres, three arcseconds and one part per million, in three different units — which hides the fact that all seven are the same size of thing. Converted to ground displacement, the scale term beats two of the translations.

8 figures · Datum What "1:136,495" means at each latitude, at zoom 12. A screen map at zoom 12 prints one scale for the whole world. The curve is how much larger in scale the map really is, measured from the projection's own derivatives rather than from a formula: at 60° it is 1.98 times, so the map labelled 1:136,495 is a 1:68,765 map. The hollow marks are sec φ, the textbook answer. They do not sit on the curve — the worst gap is 9949 parts per million at 85° — because Web Mercator puts a geodetic latitude into a spherical formula, and the same spherical Mercator measured the same way reproduces sec φ exactly. What a machine does with it

The scale of a screen map is not one number

A zoom level prints one scale for the whole world, and the map is at that scale along exactly one line. At 60° north the picture labelled 1:136,495 is a 1:68,765 map — and the factor is not quite sec φ either, because the projection puts a geodetic latitude into a spherical formula.

5 figures · Screen New York to Madrid on Mercator. Two routes. The great circle is 5768 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 5939 km — 171 km further, or 3.0 per cent. On Mercator the rhumb line departs from straight by 9.6e-16 of its own length. 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.

5 figures · Paths How much each projection inflates a cell, by latitude. Five patches of the sphere, each 20° by 10°, and the factor by which each projection enlarges them relative to the equatorial one. An equal-area projection sits flat on 1. Mercator reaches 15.4× at 70°, which is the mechanism behind every complaint about the size of Greenland. 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.

5 figures · Audit Only one of a grid's five parameters changes the map. The same ground point written on the British National Grid and on UTM zone 31N, with the difference between the two coordinate pairs taken apart. The largest term by four orders of magnitude is where the two grids put zero — 5544 km, being a different central meridian, a different true origin and different false constants, all of which move every coordinate and no distance. The datum, which is the term everybody names, moves the ground 106 m. And the whole geometric difference between two transverse Mercators is their scale factors — 0.9996012717 against 0.9996 — which at this point is 50 cm. Four of a grid's five parameters are bookkeeping; the fifth is the map. Grids, and what a survey does

What a grid is made of

A projected coordinate declares five things — a datum, a projection, its parameters, a unit and an axis order — and four of them are invisible in the numbers. Every one has a documented case of being guessed wrong, and the cheapest of those cost a hundred metres.

8 figures · Grid London to Tokyo, seen four ways. The same two routes on four projections. The gnomonic projection renders every great circle as an exactly straight line, which is what it is for; Mercator renders every rhumb line straight instead. Neither path changed — only the map did. 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.

8 figures · Families 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.

5 figures · Curvature 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.

5 figures · Choosing Fitting a sphere to 10° of latitude at 54.5°. The Earth's Gaussian radius of curvature across the band, with two candidate spheres drawn against it. The global mean radius is 6371.0 kilometres and sits 2203 parts per million away from the ground here — 2.2 metres in every kilometre measured. The best local radius is 6385.0 kilometres and has no bias at all by construction, leaving 325 parts per million of spread that no sphere can remove, because the curvature varies across the band and a sphere's does not. The gain is a factor of 6.9. What the numbers refer to

A datum is fitted to a region

Forty countries adopted forty ellipsoids, and the usual explanation is that measurement was poor. It was not — a figure fitted to one country beats the global one over that country by a factor of seven, and what it removes is a systematic bias of two metres in every kilometre.

8 figures · Datum A scale factor is a property of a point, and a distance is not. The grid scale factor sampled along a 362 km line on the British National Grid at a bearing of 90°, with the three rules in use drawn as the constants they replace it by. In a transverse Mercator k depends on the easting almost alone and goes as its square, so along an east–west line the curve is very nearly a parabola. The endpoint mean is the chord across it and lands 268.6 ppm high, which is 97.3 m over this line. Simpson's rule integrates a parabola exactly and lands 0.004 ppm out — 1.3 mm. Neither number is about the line's length. Grids, and what a survey does

The scale factor of a line

A scale factor is a property of a point and a distance is not measured at a point. The three rules in use for averaging one along a line differ by a hundred metres on a 362-kilometre baseline, and which of them is adequate is decided by the line's direction rather than its length.

8 figures · Grid A 1,000 km bar on Mercator, drawn at 0° and read elsewhere. The same length of paper, carried up the map. Each pair of bars is the ground distance that length actually spans at that latitude — the filled bar along the parallel, the outline along the meridian. At 75° the reading along the parallel is 259 km against the 1,000 the bar claims, an error of 74 per cent. The two readings agree everywhere, because Mercator is conformal — so one number per latitude corrects any measurement taken off it. What a machine does with it

A scale bar is right in one place

The bar in the corner of a world map is a picture of a distance, and it is a true picture along one line. On Mercator it reads 500 kilometres for a thousand at 60° north — and on an equal-area map it reads 500 one way and 2,000 the other, so the projection recommended for measuring is the one on which no single correction exists.

5 figures · Screen