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

Fields

A field is the kind of question an essay answers. The split is by question rather than by object, which is why the families of construction and what a projection optimises are separate — one is how a map is built, the other is what it was built for.
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

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.

22 essays · 2 ladders 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

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 · 7 ladders 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

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

29 essays · 3 ladders Angular deformation against latitude, four projections. The same quantity for mercator, gallPeters, winkelTripel, 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. Naming the objective is more honest than naming the shape of the paper it was notionally rolled from.

32 essays · 2 ladders 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

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?

20 essays · 2 ladders 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

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 · 5 ladders 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 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.

28 essays · 2 ladders 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 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 · 4 ladders 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

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.

28 essays · 3 ladders

Not written yet

the two divisions the schedule still owes

Practice — grids, zones and what a working surveyor actually holds in their hands — and applied cartography are the remaining two divisions, and neither has an essay here yet. Each needs its own generator families before it can be written to the standard the rest of the collection is held to, because an essay here exists when a figure explains something better than prose does, and not before.

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