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

EPSG — where it appears

The registry of coordinate reference systems, datums, projections and transformations that most software identifies them by. A code names a definition and does not carry its uncertainty, which is the practical reason a transformation's covariance is discarded at the first hop.

Named by 6 essays across 5 fields — each of them below, with the objects they name alongside 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 four named here are Mercator, Mercator (ellipsoidal), Web Mercator, Gall–Peters.

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 machinery here found that without being told to look.

wrong · 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.

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 Two feet, and the one that is not a rounding. The US survey foot is 1200/3937 m and the international foot is exactly 0.3048 m, so the first is longer by 2.0000 parts per million. Read a coordinate in the wrong one and the error is proportional to the coordinate, not to any distance in the job: at 2.5 million feet from the false origin it reaches 1.52 m, and it crosses a survey's own closing tolerance of twenty millimetres at 33 thousand feet. Across a 300-foot site the same mistake changes every dimension by 0.18 mm — below every tolerance there is, which is why a plan in the wrong foot passes every check and is in the wrong place.

The units are part of the coordinate

Two feet were in use in the United States until 2022, differing by exactly two parts per million. A plan drawn in the wrong one has every dimension right to a fifth of a millimetre and sits three-quarters of a metre from where it belongs, which is why it passes every check anybody runs.

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 What a solved map is, as a list of numbers. A map with no formula is a list of coefficients, and this is the list. The Chebyshev map of an elongated region, 30° by 10° has its coefficients falling by a factor of 1.5e+13 from the first to the fourteenth, so a table of a dozen numbers carries the whole projection; the conformal cube face's coefficients, marked separately, fall far more slowly because the map has a singularity at each corner. How fast this line falls is exactly how portable the map is — and neither map has a name, an inverse in closed form, or a formula anybody could quote.

A map with no formula

The solved projection has no name, no formula and no closed-form inverse. It is fourteen numbers — and the rate at which those numbers fall away decides whether a map can be shipped at all: geometrically for a smooth region, and like a power for one with corners.

choosing · Condition What reading the seven parameters under the wrong convention costs. The distance between the two published conventions' answers for the same parameter set, at the worst point of the world. The difference is 2s(r × X) exactly — twice the rotation, crossed into the position — so a parameter set with no rotations is immune and one with a large rotation is not: DHDN reaches 152.3 metres and NAD27, whose published transformation is three translations and nothing else, reaches zero. The note beside each bar is the size of that datum's rotation.

The rotation has two sign conventions

Eleven essays price a datum transformation's parameters, their fit, their residuals and their uncertainty. None asks what the numbers mean: three of the seven are published under two conventions whose rotations differ in sign, and reading one set with the other formula costs exactly twice the rotation — 55 metres on OSGB36 and 152 on DHDN.

datums · Datum

Named alongside it

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

ConventionCoordinate reference systemDatumRealisationUnit of measureAxis orderFalse originHelmert transformationNational GridOSGB36Scale factorTolerance

All concepts