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

Axis order — where it appears

Which of a coordinate's two numbers comes first, latitude or longitude — a convention two standards disagree about and most files do not carry. A file that carries coordinates without stating it is a file whose positions can be read two ways, and the two readings differ by a reflection about the diagonal.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

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.

practice · Grid 53° north, 2° west: four places one pair of numbers could be. The same two numbers, read as three datums and as the pair in the other order. The centre is the WGS84 reading; the marks are where the ground is if the numbers were measured against OSGB36, ED50, NAD27 instead, at 103 m, 138 m, 195 m. Every one of those is a plausible position — near enough to look like a survey difference and far enough to be a different field. Reversing the two numbers instead moves the point 7942 kilometres, and is the mistake nobody worries about because it is usually obvious.

A coordinate without its system is not a location

The same two numbers name ground 103 metres apart on one datum, 195 on another, and 7,942 kilometres away if the pair is read in the other order. The datum errors are the dangerous ones because they are plausible — and the axis swap, which everyone calls obvious, does nothing at all along a line that crosses three continents.

applied · Dataset Two latitudes for one point on Mars. A cross-section of Mars with its flattening exaggerated 6× so the two angles can be told apart. The line to the centre defines the planetocentric latitude and the outward normal defines the planetographic one; at 45° they differ by 0.338° on the real body, which is 20 kilometres along the surface. Both numbers are published for Mars, and a coordinate that does not say which it is is ambiguous by that much.

A coordinate on another body

Mars publishes two latitudes for every point and they differ by up to 0.338°, which is twenty kilometres of ground — almost exactly the same distance as the Earth's own 0.192°, because Mars is smaller by nearly the same factor. The libraries that compute either have known both numbers since this collection's early essays and had never been pointed anywhere but here.

datums · Bodies

Named alongside it

The objects these essays reach for when they reach for this one.

ConventionCoordinate reference systemDatumEPSGRealisationUnit of measureCentral meridianClosed formEllipsoidFalse originFlatteningGeodetic latitude

All concepts