A grid scaled to the ground is not a map

A construction site cannot work in grid coordinates. A setting-out engineer holds an instrument, reads a distance, and needs that reading to be the number on the drawing — and on the British National Grid it is not, by about four hundred parts per million, which is forty centimetres in a kilometre.

The industry’s answer is to multiply the whole grid by a constant so that the two agree. The result is used on most large projects in Britain and on a great many elsewhere, it has a name — a surface or site coordinate system — and it is not a map projection.

The size of the problem it solves #

Four hundred parts per million sounds like nothing until it is put against what a setting-out job is held to.

A structural steel frame is set out to five millimetres. A tunnel drive is held to tens of millimetres over kilometres — and a drive from both ends meets in the middle, so the two headings’ coordinate systems must agree, which is the tolerance decides the model at its least forgiving. A piling rig is positioned to twenty-five. Against any of those, four hundred parts per million — forty centimetres per kilometre — is not a correction to be applied carefully but a disagreement large enough to make the drawing unusable.

The engineer’s options are three. Apply the combined factor to every dimension by hand, which is error-prone and is applied inconsistently the moment anybody is in a hurry. Work in ground distances and accept that the coordinates on the drawing do not match the national grid, which loses the tie to everything outside the site. Or rescale the coordinate system once, at the start, so that the two agree — which is what surface systems do and why they exist.

The third option is the only one that puts the correction in a single place where it can be got right once. That is a genuine engineering virtue and it is worth stating before the essay spends the rest of its length on what the option costs.

The construction #

At the project origin, compute the combined factor — the grid scale factor times the elevation factor, which is the quantity the ground is not the grid is about. On this example it comes to 0.9995964.

Multiply every grid coordinate by its reciprocal, 1.0004037, about that origin.

That is the whole of it. A grid distance computed from the new coordinates now equals a ground distance measured at the origin’s elevation, and the setting-out engineer’s tape reading is the drawing’s number.

Why it is not a projection #

A map projection is a function from the ellipsoid to the plane. This is a map projection composed with a similarity of the plane — a uniform scaling about a point — and the composition is not a projection of anything.

The distinction is not pedantry. It has three consequences that matter on site.

It has no inverse to latitude and longitude that any software knows. A surface coordinate is not in the EPSG registry, cannot be tagged with a code, and is meaningless to anybody who does not have the origin and the factor. It is a private coordinate system, and what a grid is made of applies with a sixth declaration added.

Its scale factor is not the projection’s. A surveyor who applies the national grid’s scale factor to surface coordinates applies the correction twice.

And it is not conformal, equal-area or anything else in the usual taxonomy — it inherits conformality from the transverse Mercator, because a similarity preserves angles, but the properties everybody quotes for the parent grid are quoted at the parent’s scale factor and no longer apply.

The anisotropy nobody mentions #

A surface system is invariably described as good “within a few kilometres” of its origin, which is a radius, and the error is not radially symmetric at all.

Sixty kilometres due east, the set-out distance is out by 5.6 metres. Sixty kilometres due north, it is out by 71 millimetres. A factor of eighty, in the same system, at the same distance.

The reason is the one that runs through this whole field: a transverse Mercator’s scale factor depends on the easting and almost not at all on the northing. The elevation factor is being held constant by the construction, so the only thing that varies with position is the grid factor, and the grid factor varies in one direction.

So the useful area of a surface system is a strip running north–south, not a circle. A project laid out along a motorway running east is in trouble at a fraction of the distance that the same project running north would tolerate, and no guidance note the collection has seen says so.

The strip has edges, and they are lines of constant easting #

The anisotropy is stated above as a ratio at one distance, and the shape behind it can be written down. A transverse Mercator’s point scale factor is k₀(1 + E²/2R²) to the order that matters here, with E the distance from the central meridian, so a surface system built at an origin whose easting is E₀ carries a relative error of

$$\frac{E^2 - E_0^2}{2R^2}$$

at any other point. That expression contains no northing at all. The region within a stated tolerance is therefore bounded by two lines of constant easting, exactly — a strip, of unlimited extent north and south, whose edges are decided entirely by how far the origin sits from the central meridian.

The example origin at 1.5° west sits 32.9 kilometres east of the British grid’s central meridian, and the arithmetic recovers the figure’s own number: at sixty kilometres further east the expression gives 93 parts per million, which is the 5.6 metres the caption reports.

Turned round, it gives the strip. At a tolerance of ten parts per million the condition is |E² − 1,082| ≤ 812 in square kilometres, so E runs from 16.4 to 43.5 kilometres — a band 27 kilometres wide that does not contain the central meridian. Standing on the central meridian itself is 13 parts per million out from this origin, which is outside the tolerance, and the error there is not large because the site is far away but because the site is not far enough.

That is the shape of the failure that no radius can express. A surface system’s good ground is a band at a fixed distance from a line the project has never heard of, and moving towards the middle of the parent grid leaves it just as surely as moving away.

The assertion, and what it had to reject #

Two things are asserted about this construction and the second is the essay.

It is exact at its own origin, to better than a millionth of a part per million — which it must be, because that is what the construction does, and a check that failed here would be reporting a bug in the arithmetic rather than a property of the design.

And the east–west degradation exceeds the north–south degradation by more than a factor of ten. That is the check with content. Asserting only a ceiling on the north–south rate would pass on a grid whose scale factor had stopped varying at all — which is the failure mode that produces a beautiful figure showing nothing. The floor on the east–west rate is what forces the picture to contain the effect it claims to.

The measured ratio at sixty kilometres is eighty, comfortably above the ten the assertion demands, and the margin is itself informative: an assertion that had to be loosened toward ten would be telling a reader that the anisotropy had weakened, which would mean the origin had drifted toward the central meridian where the scale factor is flat.

Where the elevation goes #

A surface system fixes the elevation factor at the origin’s height and then carries that value everywhere, which is a second source of error that the figure above deliberately does not show.

A project whose ground rises two hundred metres across its extent has an elevation factor varying by 31 parts per million from end to end, and the surface system knows nothing about it. On a five-kilometre site that is 16 centimetres.

That is usually smaller than the grid-factor term and it behaves entirely differently — it follows the topography rather than the easting, so it has no direction and cannot be reasoned about geometrically at all. A site on a hillside and a site on a plain of the same extent have different amounts of it, and the only way to know is to look at the ground.

The name is the problem #

Almost every difficulty in this essay traces to one thing: a surface coordinate looks exactly like a grid coordinate.

Same magnitude, same units, same two-number format, often the same false origin because the scaling is applied about a project point rather than about the grid’s zero. A file of surface coordinates dropped into a national-grid workflow is accepted without complaint and is wrong by decimetres — the fifth entry in the family of errors the units are part of the coordinate catalogues, all of which survive every internal check because all of them are similarities.

Some projects mitigate this by giving their surface origin a deliberately implausible offset — adding a hundred thousand to every northing, say, so that a surface coordinate is obviously not a grid one. That is an ugly solution to a real problem and it works, in the same way and for the same reason that the false origin’s own clerical reasoning works: make the wrong reading produce an obviously wrong number, since no amount of care produces a check that a similarity cannot pass.

What it is really competing with #

The alternative to a surface system is a low-distortion projection: a real map projection whose scale factor has been chosen so that the combined factor is one at the region’s mean elevation. Several American states publish dozens of them, and the design is set out in designing a grid for one region.

The comparison is instructive because the two achieve the same thing by different means and have different failure modes.

A low-distortion projection is a projection. It has an EPSG code, an inverse, a published specification, and a scale factor that varies correctly across its own zone — so a distance far from the centre is still computed right, merely with a larger correction. Its cost is administrative: fifty zones for a state, and a seam wherever two meet.

A surface system is a private scaling. It has no code, no published specification, and a scale factor that is a constant when it should be a function — so a distance far from the origin is computed wrong, and by an amount nobody has calculated. Its benefit is that it takes five minutes and needs nobody’s permission.

The first is better engineering and the second is what most projects use, for the obvious reason.

Two sites, and the coordinate that belongs to neither #

The failure a surface system produces at scale is not a distance error. It is two projects.

Site A and site B are twenty kilometres apart, each with its own surface system about its own origin. Both are internally exact. A point on the boundary between them has two surface coordinates that differ by centimetres to decimetres, and neither system is wrong — the same structure as the double coordinate in where two zones meet, at a thousandth of the size and with none of the specification, belt or documentation that makes the zone case manageable.

Zone boundaries are published. Surface-system origins are in a project file. When the project ends, the file goes into an archive, and the coordinates in the as-built drawings become uninterpretable in a way that no amount of care about the original arithmetic prevents.

What to record so it survives #

The remedy is three numbers and a sentence, and it costs nothing at the time.

Record the origin’s grid coordinates, the scale factor applied, and the parent grid it was applied to, on the drawings rather than in a project file. With those, any surface coordinate can be returned to the national grid by anybody, forever, with arithmetic a spreadsheet can do.

Without them, a surface coordinate is a number in a private system whose parameters were known to a handful of people on one project. That is a strictly worse position than the legacy cadastral records this collection complains about elsewhere, because those at least used a published system whose conventions can be reconstructed — the argument a published coordinate is a result makes about why an external reference is worth more than an internal one.

What was computed, and how #

The combined factor at the origin, from the projection’s own derivatives and the Gaussian mean radius of curvature at that latitude, with the height taken as ellipsoidal.

The surface scale, its reciprocal.

The residual error in a set-out distance, at thirteen distances out to sixty kilometres, in two directions — due east and due north from the origin, with each direction’s step converted to degrees through the appropriate radius of curvature rather than through a single spherical one.

Two assertions, one confirming the construction and one requiring the anisotropy.

Where the model stops #

Only two directions are computed. The error in an intermediate direction is between the two and closer to the east–west curve than a naive interpolation suggests, because the easting changes as the sine of the bearing and the scale factor as its square. A figure showing the full two-dimensional error field would be more complete and would obscure the finding, which is the contrast between the axes.

The terrain is flat. Every number here holds the elevation at the origin’s value, which is what the construction does and is not what the ground does. The section above puts a number on that and it is a different quantity with different behaviour — and the height it uses must be the ellipsoidal one, for the reason height above what? sets out, which on a British site is fifty metres away from the number on the drawings.

And nothing here treats the legal question, which on a real project is the serious one: a surface coordinate system is not a recognised reference system, so a boundary or an easement described in one is described in terms nobody outside the project can reconstruct once the origin and the factor are lost. That has happened, it is not rare, and no amount of measuring the anisotropy addresses it.

The generalisation #

A local correction applied globally is exact where it was computed and wrong in proportion to the gradient of the thing it corrected. That is the whole content of the anisotropy: the error away from the origin is the gradient of the combined factor times the distance, and the combined factor has a gradient in one direction only.

Stated that way it stops being cartographic. It is the same structure as any linearisation about an operating point, and the practical lesson is the same one: the useful neighbourhood of such an approximation has the shape of the function’s level sets, not the shape of a circle, and describing its validity as a radius is describing the wrong thing.

The corollary is a usable test, and it is not the one anybody applies. A site system’s validity is not a radius to be quoted; it is a tolerance on the change in the combined factor, which converts into a distance in easting and into an unlimited distance in northing. A project that knows its own tolerance can compute that distance once, in one line, and write it beside the origin — and a project that instead writes down “good to five kilometres” has recorded the right order of magnitude in the wrong shape, which is the version that fails when the site is long.

Who does it, and when #

Surface coordinate systems have no single inventor and are not a published method. They are what site engineers arrived at independently, in several countries, once total stations made it routine to measure a distance to a millimetre and immediately obvious that the grid disagreed.

The practice is old enough that British guidance now addresses it directly, and the guidance says what this essay says: use one, keep the origin and the factor with the drawings, and do not carry the system beyond a few kilometres. What the guidance does not say is which few kilometres, and that is the gap this essay fills — the answer is a strip about ten kilometres wide and fifty long, rather than a circle of any radius.

Where this goes next #

That completes the grid as an object with declarations and parameters. What has to happen to a measurement before it can become one of its coordinates is the other ladder in this field, and it starts at what a tape measures.