What a grid is made of

Here is a coordinate: 429157.31, 623471.08.

It is a real position on the Earth’s surface, stated to a centimetre, and there is nothing whatever that can be done with it. Not because the numbers are wrong — they are exact — but because a projected coordinate is the output of five separate decisions and the pair carries none of them.

The five declarations #

The datum. Which model of the Earth’s shape and placement the latitude and longitude were measured against before anything was projected. Change it and the ground under the coordinate moves — by 106 metres between OSGB36 and WGS84 in the English Midlands, and by nearly 200 metres between NAD27 and WGS84 in North America. This is the declaration what a coordinate refers to is about, and it is the largest term in most real confusions.

The projection. Which surface the ellipsoid was flattened onto. Almost every national grid is a transverse Mercator, which makes this the declaration least likely to be wrong and the one most likely to be assumed.

The projection’s parameters. The central meridian, the scale factor, the latitude of the true origin, and the false easting and northing. Four numbers and a fifth that does nothing, which is the subject of a grid has an origin that is not there.

The unit. Metres, international feet, or US survey feet — and the last two differ by two parts per million, which is the subject of the units are part of the coordinate.

The axis order. Whether the first number is the easting or the northing, and, upstream of the projection, whether a geographic pair is latitude-then-longitude or longitude-then-latitude.

Only one of the five is in the number #

The projection is sometimes inferrable. A transverse Mercator’s coordinates have a characteristic shape — eastings clustered around a false origin, northings running monotonically with latitude — and a practitioner who knows the country can usually name the grid from a coordinate in it.

The other four cannot be recovered at all. A datum shift moves a coordinate by a hundred metres, which is smaller than the grid squares most people work in and larger than any tolerance any job has. A wrong scale factor changes distances by a few hundred parts per million, which is invisible in a coordinate and is metres on a line. A wrong unit displaces the point and leaves every dimension intact. And a transposed axis order produces a coordinate that is somewhere else entirely, unless the two happen to be similar, in which case it produces a coordinate that is nearly right — which is worse.

The pattern across all four is the same and it is the reason this essay exists: the failures are large enough to matter and small enough to look plausible. A coordinate that arrives at the wrong end of the country announces itself. A coordinate a hundred metres out does not.

The registry, and what it does and does not do #

The European Petroleum Survey Group began, in the 1980s, to number coordinate reference systems. The registry it produced is now maintained by the successor body and is universal: EPSG:27700 is the British National Grid, EPSG:4326 is WGS84 in latitude and longitude, EPSG:32631 is UTM zone 31 north on WGS84.

What the code does is bind all five declarations to a single integer. That is a real and enormous contribution, and it converts a class of error from unrecoverable to checkable: a file that carries 27700 beside its coordinates can be read correctly by anybody, forever, with no local knowledge.

What the code does not do is travel with the coordinate unless somebody puts it there. A pair of numbers in a spreadsheet cell, a text message, a site diary or a hand-drawn sketch carries no code, and most coordinates in the world live in exactly those places.

Nor does the code settle the fifth declaration. EPSG:4326 specifies latitude first and longitude second, and the overwhelming majority of software that claims to implement it puts longitude first, because that is the order that matches an x-then-y convention. The registry is right and the practice is not, and a reader who trusts the registry against a file produced by that software will read every coordinate in it transposed.

Measuring each declaration, in metres #

The point of a figure-first treatment is that these can all be put on one scale.

The datum term, computed from the published seven-parameter transformation, is 106 metres at 52.5°N, 1.9°W. The difference the two grids’ scale factors make at that point is half a metre. Everything else — 5,544 kilometres — is where the two grids put zero.

And that is the finding, once the terms are separated properly: only one of a grid’s five parameters changes the map. The central meridian decides where easting zero falls, the true origin’s parallel decides where northing zero falls, and the two false constants move both again; all four move every coordinate and no distance. The scale factor is the whole of the geometry. Four parameters of bookkeeping and one of shape, and the bookkeeping is four orders of magnitude larger.

The unit, which is the quietest of the five #

Of the five declarations this is the one that most reliably survives every check a practitioner runs, because every check a practitioner runs is on dimensions and this error is not in the dimensions. A site plan in the wrong foot closes perfectly, scales perfectly, and is in the wrong place — the same signature as the scale error that a traverse must close shows a closed figure is blind to.

The axis order, and the transposition that nearly works #

Of the five, the axis order is the only one with no argument behind it, and it produces the most instructive failures.

A transposed grid coordinate is usually absurd. Swapping 429157 and 623471 on the British grid puts a point in Britain into the North Sea, which somebody notices. But swapping a latitude and longitude pair is absurd only when the two are far apart: at 52.5°N, 1.9°W the transposition gives 1.9°N, 52.5°E, which is in the Indian Ocean and is obvious. At 45°N, 45°E it gives 45°N, 45°E and is undetectable, and anywhere the two happen to be close the result lands a plausible distance away in a plausible direction.

Worse, a transposition is a reflection, so a set of points that has been through one is not merely displaced but mirrored. Distances within the set are preserved exactly, which means every internal consistency check passes, and every bearing is reflected about the 45-degree line, which means no bearing check passes for a reason anybody can interpret. It is the one failure on this list that changes the shape of the data rather than its position.

Where the declarations came from #

Each of the five is a decision somebody took, and reconstructing the reasoning is possible for all of them.

The datum was chosen by fitting an ellipsoid to a country’s own triangulation, which is a datum is fitted to a region, and it was chosen before satellites made a global fit possible. The projection was chosen because a transverse Mercator is the conformal projection whose distortion can be bounded over a north–south strip, and most countries are strips or can be cut into them. The parameters were chosen by an optimisation this collection can run again — see the scale factor was chosen.

The unit was chosen by whatever the country’s surveyors were already using, which is why the United States has grids in feet and the reason the feet are the awkward kind. The axis order was chosen by whether the specification was written by a geodesist, who says latitude first because that is how a position has been stated since antiquity, or by a programmer, who says x first because that is how a plot works.

Only the last of the five is arbitrary in the strong sense — there is no argument for either order beyond precedent — and it is the one that produces the most spectacular failures.

What was computed, and how #

The three terms of the difference, each isolated by holding the other two fixed. The datum term comes from the published seven-parameter Helmert transformation applied to the geographic coordinates; the projection term comes from projecting the same geographic coordinates through the second grid’s projection on the first grid’s ellipsoid, so that only the central meridian, scale factor and true origin differ; the false-origin term is the difference of the two constant pairs.

That decomposition is a choice and the choice matters. The three terms do not add to the total, because a datum shift and a projection change do not commute — the same non-commutation that costs 1.3 centimetres in the seven parameters when a Helmert transformation is undone in the wrong order. What is being measured is the size of each declaration’s contribution when the others are held, which is the question a practitioner asks, and it is stated rather than presented as a partition.

The assertion. The false-origin term must exceed the datum term by more than a factor of ten, and the datum term must exceed fifty metres. The pair is the check. Without the second half, a comparison between two grids that happened to share a datum would pass while proving nothing; without the first, a grid pair with similar origins would fail a claim that is true in general.

Reading the coordinate at the top of this page #

429157.31, 623471.08, with all five declarations supplied, is a point in the Peak District.

Supplied means: OSGB36; transverse Mercator; central meridian 2°W, true origin 49°N, scale factor 0.9996012717, false easting 400,000 m, false northing −100,000 m; metres; easting first. That is EPSG:27700, and the whole of it collapses to five digits.

Now change one declaration at a time and see where the point goes. Read against WGS84 rather than OSGB36 and it moves about a hundred metres north-east. Read as UTM zone 30 it lands in the sea west of Ireland. Read in US survey feet it moves 1.2 metres — the only change on this list that a reader would mistake for a rounding. Read northing-first it lands in the North Sea.

Four wrong answers, and they span five orders of magnitude. The one that is hardest to catch is the smallest.

Where the model stops #

Three of the five declarations are here as single numbers and are really families.

A datum is not one transformation. The seven parameters published for a datum pair are a global best fit, and a national authority usually publishes a correction grid on top of them that is worth metres — which is the argument of when a formula is not enough. Quoting “106 metres” for OSGB36 against WGS84 is quoting the smooth part.

A projection is not always a transverse Mercator. Grids built on a Lambert conformal conic exist, chiefly in countries that are wider than they are tall, and their parameters are a different list — two standard parallels rather than a central scale factor. The five declarations survive; the third one’s contents change.

And the axis order has more than two cases once heights and epochs are in play, at which point a coordinate is four numbers and a date and the ordering question multiplies.

What a code cannot fix #

The registry converts an unrecoverable error into a checkable one, and the residue is the interesting part: it cannot help with data that predates it, and it cannot help with a declaration that was recorded wrongly rather than omitted.

Both classes are large. Cadastral records in most countries run back a century or more, in units and on datums that were never written down beside the numbers because everybody at the time knew them. And a file tagged with the wrong code is worse than an untagged one, because the tag suppresses the suspicion that would otherwise have prompted somebody to ask.

The generalisation #

The structure here is not about maps. A measurement’s meaning lives in a set of declarations that the measurement’s own representation does not carry, and every field that has been through this has ended up in the same place: a registry, a code, and a long tail of data that predates the registry and cannot be repaired.

What distinguishes cartography is the arithmetic. In most fields a missing declaration produces a number of the wrong magnitude, which is obvious. Here every one of the five produces a number of the right magnitude in the wrong place, and the differences run from two parts per million to two hundred metres with nothing in the coordinate to signal which — the property that makes this collection’s insistence on measuring rather than naming useful outside itself.

Who assembled it, and when #

The EPSG registry was begun in 1985 by the European Petroleum Survey Group, for a wholly practical reason: an oil company operating in twenty countries had twenty national grids and no way to be sure a contractor’s coordinates meant what the contract said. It was released publicly in 1993, transferred to the International Association of Oil & Gas Producers in 2005, and is now the closest thing the world has to a definition of what a coordinate means.

The registry’s own history contains the best evidence for this essay’s claim. EPSG:4326’s axis order has been latitude-first since the entry was written, the ambiguity was recognised early, and the practice never converged — so a widely used standard, unambiguously specified and freely available, has been implemented two contradictory ways for thirty years because the declaration is not in the data.

The axis-order history is worth reading as the strongest form of this essay’s argument, because every excuse is unavailable. The specification is unambiguous, it has said the same thing since 1985, it is free, it is universally used, and the ambiguity was recognised almost immediately — and thirty years later two contradictory implementations coexist. Nothing about clarity, availability or awareness would have fixed it, because the problem is not that anybody misunderstood the standard: it is that a file holding two numbers cannot say which convention produced them, so both readings survive in the data and each is confirmed by whatever the reader happens to have. A declaration that is not carried alongside the values it governs is not a declaration; it is a hope about what everybody remembers.

And the same argument decides which of the five declarations are dangerous. A declaration whose value can be inferred from the data is self-correcting: coordinates in the millions are metres and not degrees, and a reader who assumes wrongly gets an answer so absurd that the assumption is corrected immediately. A declaration whose two readings both produce plausible numbers is the dangerous kind, because nothing in the output objects — and axis order is the purest case, since swapping two numbers of similar magnitude gives a position somewhere else on Earth that is a perfectly good position.

Which gives a way to rank the five without any measurement. Ask, for each declaration, what a wrong assumption produces: an absurdity, a small error, or a plausible wrong answer. The first is safe, the second is a tolerance question, and the third is the one that needs the declaration carried in the data rather than agreed by convention.

Where this goes next #

Of the five declarations, the scale factor is the one that repays being taken apart, because it is the only one where the designer’s reasoning can be reconstructed and rerun: the scale factor was chosen, and the choice has a closed form. The unit is the one with the smallest number and the strangest history, at the units are part of the coordinate.