The seven parameters, and what each one does

A datum transformation arrives as a row in a table: three translations in metres, three rotations in arcseconds, one scale change in parts per million. Seven numbers, three units, and an implicit ranking — the translations are hundreds, the rotations are fractions of one, so the translations are what matters and the rest is polish.

That reading is wrong, and it is wrong by a factor of thirty.

The transformation #

The Helmert transformation — a similarity transformation, in the general vocabulary — relates two Cartesian frames. Not two sets of latitudes and longitudes: the conversion to Cartesian coordinates has to happen first, on the source ellipsoid, and the conversion back has to happen afterwards, on the target one. Everything in this essay happens in between.

Given a point $(X, Y, Z)$ in the source frame:

$$\begin{pmatrix} X' \\ Y' \\ Z' \end{pmatrix} = \begin{pmatrix} t_x \\ t_y \\ t_z \end{pmatrix} + (1 + s)\begin{pmatrix} 1 & -r_z & r_y \\ r_z & 1 & -r_x \\ -r_y & r_x & 1 \end{pmatrix}\begin{pmatrix} X \\ Y \\ Z \end{pmatrix}$$

The rotation matrix is written to first order in the three angles, which is exact enough by a wide margin: the angles are of order a microradian, so the neglected second-order terms are tenths of a millimetre on the Earth’s radius. That linearisation is not a shortcut taken for convenience here — it is how the parameters are defined and published.

Seven degrees of freedom, and the seven divide into two groups that mean different things.

What each group means #

The three translations are the placement. They say where the source ellipsoid’s centre sits relative to the target’s. For a datum realised before satellites, that is several hundred metres, because there was no way to find the geocentre and no reason to try. OSGB36’s translations are 446, −125 and 542 metres; the vector’s length is 715 metres, which is how far the Airy ellipsoid sits from the centre of the Earth.

The three rotations and the scale are the realisation. They are not a statement about the ellipsoid at all. They come out of the fit between two sets of marker coordinates, and they absorb the systematic part of the difference between two networks — one observed with theodolites over eighty years, one observed from orbit. A rotation of 0.84 arcseconds does not mean the old surveyors had their axes crooked; it means the accumulated twist in the triangulation, spread over the whole country, came out looking like a rotation to the fit.

That distinction is why the second group cannot be dismissed as noise. It is small in its own units because arcseconds and parts per million are small units.

What was computed, and how #

The figure above is produced by a single technique: take the transformation, zero six of its seven parameters, apply it to a point, and measure how far the point moved. No approximation and no linearisation beyond the one already in the definition.

The per-unit numbers are the useful ones, because they let the seven be compared before the published values are even looked at. At a point in northern Britain:

• one metre of translation moves the mark 1.00 metre, by definition;
• one arcsecond of rotation moves it 18 to 31 metres, depending on which axis, because the lever arm is a substantial fraction of the Earth’s radius;
• one part per million of scale moves it 6.36 metres, because the point is 6.36 million metres from the origin.

So the three units are related by roughly $1 : 25 : 6$. An arcsecond is worth twenty-five metres and a part per million is worth six. Once that is in hand, the published row reads completely differently: OSGB36’s −20.489 ppm of scale is a 130-metre effect, larger than its 125-metre $t_y$, and its 0.842 arcseconds about the $Z$ axis is a 15-metre effect, which is a hundred times a survey tolerance.

The comparison between the two figures is the reason this generator takes the datum as a parameter rather than drawing one. ED50’s published transformation is a pure translation. That looks like a cleaner relationship and it is the opposite: a three-parameter fit has four fewer degrees of freedom with which to match two networks, so whatever the rotations and scale would have absorbed is still there, sitting in the residual, unmodelled.

The route the parameters take, and a shortcut around it #

The seven parameters act on Cartesian coordinates, and almost nobody has Cartesian coordinates. So the full journey from one datum’s latitude and longitude to another’s has four steps, and only the second involves the parameters at all:

1. geographic to Cartesian, on the source ellipsoid;
2. the seven-parameter transformation;
3. Cartesian to geographic, on the target ellipsoid;
4. and, if a grid coordinate is wanted, the projection — which is everything the rest of this collection is about, and is a separate question entirely.

Step 3 is the awkward one. There is no closed form for latitude given a Cartesian triple on an ellipsoid: the standard route iterates, converging in a handful of passes, and a good starting value comes from the parametric latitude. That was expensive on the machines this work was first mechanised for, and the response was a family of formulae that skip Cartesian coordinates altogether and apply the shift directly in latitude, longitude and height.

The shortcut is a first-order expansion, and being able to say of what, and how much is dropped is the whole difference between a shortcut and a mistake. Molodensky’s shortcut does that arithmetic. What matters here is a limit on the shortcut rather than on the route. Molodensky’s formulae carry the three translations and nothing else — no rotations, no scale — so they can only be used on a datum whose transformation is a pure translation. Applied to OSGB36, whose other four parameters are worth 150 metres, they are out by tens of metres and give no sign of it.

What the seven actually do to a coordinate #

Two readings of the same numbers, on two national datums, and they land on opposite sides of the global answer. This is worth stating because it is the reason a datum cannot be guessed after the fact. If every old datum were offset the same way, a file of undocumented coordinates could be repaired by pattern-matching against the coastline. They are not, because each was fitted to its own ground, so an undocumented file is a file whose error is unknown in both size and direction.

The same point in datum shifts dwarf projection errors is made against the other half of the pipeline: the projection error at this scale is metres and the datum error is hundreds of metres, and the projection is the part that gets argued about.

The inverse is not the negated forward map #

Here is the part that has to be got right and usually is not.

The obvious inverse of the transformation is to negate all seven parameters and apply the same formula. It is wrong, and requiring the round trip to close is what catches it.

The forward map does two things in a fixed order: it scales and rotates the point, and then it translates. Undoing that requires the translation to come off first, before the scale is divided out. Negating the parameters and re-applying the forward formula does them the other way round, and since scaling and translating do not commute, the result is off by the product of the scale change and the translation — about $2 \times 10^{-5} \times 715$ metres, or 1.3 centimetres.

A centimetre and a third is a wonderful size of error. It is small enough to look like rounding in any single conversion, and large enough to be a legal problem in a land registry, where boundaries are recorded to the centimetre and disputes are conducted in millimetres. It survives every test that only checks the forward direction against published values, because the forward direction is right.

The check that finds it is one line of intent: convert a point, convert it back, and require it to land where it started to well under a millimetre. Which is the general form of a habit worth having — an operation with an inverse should be tested against its inverse, not against a table.

Where the model stops #

Three limits, and the third is the interesting one.

Seven parameters are a rigid body plus a size. A similarity transformation can translate, rotate and scale. It cannot bend. So it can carry the entire difference between two definitions exactly — different ellipsoid, different placement — and it cannot carry any part of the difference between two realisations that is not a rigid motion.

That figure is the control, and it is worth pausing on because it is what makes every other residual in this collection a measurement. With no network strain the fit recovers the transformation to a third of a millimetre — and that third of a millimetre is not slop, it is exactly the term the linearised rotation matrix drops. When the same machinery reports 1.6 metres for a strained network, the 1.6 metres is not the method’s error. The method’s error is known and is three orders of magnitude smaller.

The parameters depend on which stations were used. Two authorities can publish different seven-parameter sets for the same pair of datums and both be right. This is not sloppiness; it is what happens when seven parameters are fitted to something with more structure than seven parameters, and where a fit leaves residuals measures the mechanism directly.

The transformation says nothing about accuracy. A published parameter set has a stated fit region and a stated RMS, and using it outside that region produces numbers with no error bar rather than large errors. The formula does not know where it is. This is the same failure the site records for regional distortion measures: a quantity computed over one region and quoted over another is not a worse measurement, it is a different one.

The generalisation #

The lesson generalises past geodesy and it is about units rather than about datums.

A quantity published in a small unit looks small. Parts per million, arcseconds, basis points, decibels — each of them exists because the quantity it measures is usually a tiny fraction of something, and each of them therefore invites the reader to treat a value near one as negligible. The correction is always the same: convert every term into the units of the thing being predicted, and compare there.

This site does the same conversion everywhere else for the same reason. Scale distortion is the third failure converts an areal factor and an angular deformation into a common currency before ranking them; the trade-off is two lines puts conformality and equal-area on one axis so that “both at once” can be seen to be impossible rather than merely hard.

In this case the thing being predicted is a distance on the ground, so all seven parameters become metres, and the ranking changes. In a financial model the thing being predicted is money, and a basis point on a large notional is not small. In an error budget for an instrument the thing being predicted is the measurement, and a part per million of scale on a ten-kilometre baseline is a centimetre.

The failure mode has a shape worth recognising: a term is dropped because its published number is small, and the number is small because of the unit it is published in. The scale parameter here is the standing example. It reads as −20, next to translations that read as 500, and it does more work than either of the smaller two.

Reading a parameter set critically #

Four things are worth checking before a published row is used, and none of them requires any tooling.

Which direction is it? A set labelled “to WGS84” and one labelled “from WGS84” differ by more than a sign, as the previous section shows. Applying one in place of the other is wrong by twice the translation, which for OSGB36 is 1.4 kilometres — an error large enough to be obvious, which is the only good thing about it.

Which rotation convention? The two in circulation — often called position vector and coordinate frame — differ by the sign of all three rotations. Applying a set from one under software written for the other is wrong by twice the rotation term, which for OSGB36 is thirty metres and for ED50 is nothing at all, because its rotations are zero. That is the dangerous case: the convention error is invisible on the datums where it does not matter and silent on the ones where it does.

What region was it fitted over? A national parameter set is a fit over that nation’s stations. Nothing in the arithmetic stops it being applied on the other side of the world, and nothing in the output indicates that it has been.

What residual came with it? A parameter set published without an RMS is a claim with no error bar. The residual is not a footnote to the fit; it is the part of the answer the fit could not express, and its size is what says whether seven parameters were enough.

The seven at a different point #

Comparing this with the mid-west figure above makes the point that no single number describes a datum shift. The parameters are fixed; what they do depends on where the point is, because a translation in a geocentric frame resolves differently into local east, north and up at every latitude and longitude.

Who found it, and when #

Friedrich Robert Helmert set out the general problem — relating two coordinate frames from observations at common points — in the 1880s, as part of the work that made geodesy a subject about adjustment rather than about instruments. The seven-parameter similarity transformation carries his name because he wrote down the estimation problem, not because he invented the algebra, which is older and belongs to mechanics.

The transformation became routine only when there were two frames worth relating. Before satellite geodesy there was no global frame, so there was nothing for a national datum to be transformed to — the question of where the ellipsoid’s centre actually sat was unanswerable and therefore not asked. The parameter sets that every geographic information system now carries date from the 1960s onwards, and the ones for older datums are still being revised, because the fit depends on which stations are included and new stations keep being observed.

The ordering of the operations — scale and rotate, then translate — is a convention, and one that has bitten people. Some published parameter sets use the opposite rotation sign convention, so applying a set from one authority with software written to another’s convention produces an error of exactly twice the rotation term. At OSGB36’s rotations that is thirty metres, in a conversion that otherwise looks completely correct.

Thirty metres is worth dwelling on for a moment: it is larger than the difference between many pairs of datums, so a sign error in the rotations can move a coordinate further than the transformation itself was correcting for, and it does so without producing anything that looks wrong.

The general lesson is the one the ordering convention illustrates: a transformation is a sequence of operations, and a sequence is not recoverable from the numbers it operates with.

Where this goes next #

The seven parameters are a fit, and this essay has taken the fit’s output as given. Two essays follow from asking where it came from.

A datum is fitted to a region asks what the ellipsoid was fitted to and what fitting locally actually buys, which turns out to be a removed bias rather than a removed error. And where a fit leaves residuals asks what the fit could not reach, which is the reason a national mapping agency ships a table rather than a formula.