What a coordinate refers to

A latitude and a longitude look like a description of a place. They are not. They are two angles measured against a model, and the model is a piece of engineering with four separate commitments buried in it. Any of the four can be changed without changing the numbers, and every one of them moves the ground the numbers refer to.

The figure above is the whole field in one picture. Two degrees west, fifty-four and a half degrees north is a point in the north of England — or it is a point a hundred metres away, or two hundred, depending on which model the reader is supposed to assume. A file of coordinates that does not say which is not a file of locations. It is a file of numbers.

The four commitments #

A geodetic datum is the answer to the question what do these angles measure against, and it has four parts. They are usually presented as one thing with a name, which is why the name is treated as a formality.

First, a shape. An ellipsoid of revolution, fixed by two numbers: the equatorial radius $a$ and the flattening $f$. The Earth is a sphere, and when it is not sets out why an ellipsoid rather than a sphere, and the size of the difference. Different datums use different ellipsoids, and the differences between them are in the hundreds of metres.

Second, a position and an orientation. The shape has to be placed relative to the actual Earth. Modern datums put the centre of the ellipsoid at the centre of mass of the planet and align its axis with the rotation axis. Older ones did not — they placed the ellipsoid so that it fitted the ground under one particular country as closely as possible, which meant putting its centre several hundred metres away from the Earth’s.

Third, a realisation. This is the part that is left out of every summary. A datum is not usable until somebody has computed and published coordinates for a set of physical markers: brass plates, stone monuments, radio telescopes. Those published coordinates are the datum in practice, and they carry the errors of the survey that produced them. The definition is a paragraph; the realisation is a table with tens of thousands of rows.

Fourth, a date. The markers move, because the ground they are set in moves. A coordinate without an epoch is incomplete in the same way a share price without a date is incomplete.

The rest of this essay puts a number on each of the four.

The shape, in metres #

The ellipsoids that national datums were built on are not small variations on each other.

The vertical spread of that figure is the eighteenth century’s entire experimental problem: a degree of latitude is about 111 kilometres and the difference between the equatorial and the polar degree is a little over a kilometre, so the whole signal separating a flattened Earth from a round one is one part in a hundred. The figure of the Earth was measured works out how badly that one part in a hundred determines the flattening, and the answer explains a fifty-year dispute.

Airy 1830, which the Ordnance Survey still uses, has an equatorial radius 574 metres smaller than WGS84’s. Clarke 1866, the basis of the North American Datum of 1927, is 69 metres smaller. The International Ellipsoid of 1924 is 251 metres larger. These were not competing guesses at one quantity; each was fitted to a different part of the Earth, which is the subject of a datum is fitted to a region.

What matters here is that the shape alone changes where a coordinate lands. Hold the two angles fixed and swap the ellipsoid, and the point moves — because the angles are measured against the surface normal, and a different surface has a different normal at the same nominal place.

The placement, and why it dominates #

The shape’s contribution is real and it is not the largest term. The largest term is where the ellipsoid was put.

A datum fitted to Britain in the 1930s had one job: make the survey of Britain close. There was no way to measure the position of the Earth’s centre of mass in 1936, and no reason to want to. So the ellipsoid was placed by fixing one station’s coordinates and one azimuth, and letting everything else follow. The result fits Britain to a few metres and sits about 550 metres from the geocentre.

That figure is the Helmert transformation, which is the standard way one datum is related to another: three translations, three rotations and a change of scale. Its parameters for OSGB36 are a 446-metre shift along one axis, 125 along another and 542 along the third — which is the misplacement of the ellipsoid, stated in the frame where it is simplest to state.

Those three numbers are the second commitment. They are what makes a coordinate on a national datum unusable as a global one without conversion, and they are why a phone that reports WGS84 coordinates and a map printed on a national grid disagree by a distance a walker can see.

The realisation, which is a table and not a formula #

The third commitment is the one that resists being written down.

A datum’s definition can be stated in three lines. Its realisation — the published coordinates of the markers that make it usable — cannot, because those coordinates came out of a triangulation network that took decades to observe and carries the accumulated error of every chain in it. The published position of a triangulation station in Cornwall is not the definition applied to that station; it is the definition applied to it and then adjusted, along with thousands of others, to make the whole network as consistent as it could be made.

The consequence is that the relation between two datums is not a formula. It is a formula plus a residual field, and the residual field is what where a fit leaves residuals is about. Seven parameters can carry the shape and the placement exactly. They cannot carry the distortion of a nineteenth-century triangulation, because that distortion has more degrees of freedom than seven.

This is why national mapping agencies publish grid-shift files rather than parameter sets. A file is a table of displacements at grid nodes, interpolated between them, and it exists precisely because the thing being described is not the kind of thing a formula describes. When a formula is not enough works out what spacing such a table needs, and finds that the answer is set by the roughness of the difference rather than by its size.

The date #

The fourth commitment is the newest, in the sense that nobody needed to state it until measurement got good enough to see it.

The ground moves at between ten and seventy millimetres a year depending on which plate it is on. That was invisible to a theodolite and it is not invisible to a satellite fix. Australia moves about 57 millimetres a year to the north-north-east, which is 1.4 metres in twenty-five years — comfortably larger than any tolerance a cadastral survey works to, and larger than the accuracy of the positioning system that measures it.

So a modern datum has an epoch: a date at which its coordinates are stated to be correct. GDA94 and GDA2020 are the same country’s datum at two different dates, and the 1.8 metres between them is not a correction of an error. It is the continent having gone somewhere. The epoch is part of the coordinate takes this apart.

What was computed, and how #

Every number in this essay comes from the same three-step calculation, and it is worth writing out because it is the only honest way to ask “how far apart are these two readings”.

A geographic coordinate on a given datum is converted to Cartesian coordinates — three distances from the centre of that datum’s ellipsoid, along its own axes:

$$X = (N + h)\cos\varphi\cos\lambda, \quad Y = (N + h)\cos\varphi\sin\lambda, \quad Z = \left(N(1 - e^2) + h\right)\sin\varphi$$

where $N$ is the prime vertical radius of curvature at that latitude. The Helmert transformation is then applied to the Cartesian triple, which is where the seven parameters act. Finally the result is converted back to geographic coordinates on the target ellipsoid, which requires an iteration because the inverse has no closed form.

The distance reported in the hero figure is the ground distance between where the naive reading put the point and where the proper conversion puts it. It is not an estimate. The only inputs are the published ellipsoid constants and the published transformation parameters, and everything else is arithmetic.

Two things fall out of doing it this way rather than approximately.

The first is that the round trip has to close, and checking that it does caught a real error. Negating all seven parameters and re-applying the forward transformation is the obvious inverse and it is wrong, because the scale and the translation do not commute: the forward map scales the rotated point and then translates, so the inverse must translate back before dividing by the scale. Doing it in the wrong order costs about 1.3 centimetres — small enough to look like rounding, and large enough to matter in a land registry.

The second is that the height changes too, by more than most people expect. The transformation acts on a three-dimensional Cartesian point, so it has a vertical component whether or not anybody asked for one — and for the three datums in the hero figure it runs to between 43 and 51 metres, which is a third to a half of the horizontal shift and is almost never quoted.

That is the third coordinate moves too, and the reason it is a separate essay rather than a paragraph is that the vertical has a second reference surface of its own — one defined by gravity rather than by geometry, which height above what? takes apart.

The angle the numbers name #

There is a fifth way to get the same numbers wrong, on top of the four commitments above. Five auxiliary latitudes are each computable from the geodetic one exactly, each is used somewhere in the machinery, and none of them is what a published coordinate means.

Even inside one datum the word latitude names six different angles. Geodetic against geocentric latitude works through the largest of the gaps, which reaches 11.5 arcminutes — 21 kilometres on the ground, and larger than every datum shift in the hero figure put together.

Where the model stops #

Three honest limits on what has been shown here.

The transformation parameters are published inputs, not derived ones. This site computes rather than quotes wherever it can, and here it cannot: the seven parameters relating OSGB36 to WGS84 are the output of a least-squares fit over real stations that this site does not hold. What is computed is everything downstream of them — the ground displacement, the round trip, the vertical component, the residual structure. The parameters are treated the way $a$ and $f$ are treated: as defining inputs, quoted, with the arithmetic done here.

Different authorities publish different parameters for the same pair of datums, and all of them are correct. That sounds like a contradiction and is not; it is a consequence of fitting seven parameters to a network that has more structure than seven parameters can hold, and which set comes out depends on which stations were used. That mechanism is measured in where a fit leaves residuals, and it is one of the more surprising things in this field.

A datum is not a projection. Everything in the rest of this collection — Tissot’s indicatrix, the families of construction, what each projection optimises — happens after the question this essay asks has been answered. The projection takes a latitude and a longitude and produces a point on a flat sheet. It has no opinion about what the latitude and longitude referred to. That is why datum shifts dwarf projection errors: the projection is the part everybody argues about and the datum is the part that is actually wrong.

The generalisation #

The pattern here is not specific to geodesy, and naming it makes the four commitments easier to hold onto.

Any measurement is a number plus a statement of what it was measured against, and the second half is systematically dropped because it is boring, stable and usually agreed. A temperature is meaningless without a scale. A share price is meaningless without a currency and a date. A position on a chart is meaningless without a datum and an epoch.

What makes the geodetic case instructive is that the reference is not a convention — it is a physical object with a shape, a location, a set of marks and a history, and every one of those is measurable. So the cost of dropping it is not a philosophical problem. It is two hundred metres.

The three failure modes follow directly:

• A coordinate without a datum is out by up to a few hundred metres, silently, and looks completely normal.
• A coordinate without an epoch is out by centimetres a year, cumulatively, and looks completely normal for about a decade.
• Two coordinates from different datums compared directly produce a difference that is mostly the datum and not the thing being measured — which is the failure that gets built into a database and then reasoned from.

None of the three produces an error message.

The same numbers, read four ways #

This essay’s argument is that a coordinate is a statement about a model. The applied field turns that into a measurement by taking one pair of numbers and reading it as four different things.

At 2° west, 53° north the three published datums put the ground 103, 138 and 195 metres apart, and reversing the pair moves it 7,942 kilometres. The middle of that range is the dangerous part: a hundred metres is too large for any survey and too small to look wrong, so nothing catches it. The axis swap, which everybody calls obvious, is invisible along the line λ = φ and small within a band 0.013° wide about it.

One pair of numbers read four ways puts all of it in one place — a coordinate without its system is not a location: the datum readings are all plausible positions, and the swapped reading is off the frame by a factor of thirty thousand. Even that has an exception — on the diagonal a swap moves nothing at all, and the band inside which it stays under a kilometre is a few thousandths of a degree wide. Narrow, and not zero.

Who found it, and when #

The four commitments were separated slowly and in the order of how hard they are to see.

The shape came first: the eighteenth-century arc expeditions settled that the Earth is flattened, and the nineteenth century produced a succession of ellipsoids fitted to whatever ground the fitter had. The placement was implicit until satellite geodesy in the 1960s made the geocentre observable, at which point every national datum was revealed to be several hundred metres off-centre — not wrong, exactly, but local in a way nobody had had to state before.

The realisation was understood by the people doing the adjustments long before it was stated in textbooks, because they were the ones looking at the residuals. The epoch is the newest: space geodesy in the 1980s made plate motion a routine measurement rather than an inference, and the datums built since then carry a date in their names for that reason.

The word datum itself is doing badly. It suggests a fact — something given. What it names is a set of four engineering decisions, three of which are compromises and one of which is a table.

A name that suggests a fact where there are four decisions is not a small problem, since it is the word every specification, every file header and every conversation uses.

A better word would not fix anything by itself, but it would stop the vocabulary arguing against the content every time somebody explains what a datum is.

Where this goes next #

The ladder from here runs through the four commitments in the order of how much each costs to get wrong. Next is the transformation itself: the seven parameters and what each does, where the units are shown to be hiding the fact that a rotation of less than one arcsecond and a translation of a hundred metres are the same size of thing.