A grid has an origin that is not there

The origin of the British National Grid is a point in the Atlantic Ocean, about 90 kilometres south-west of the Isles of Scilly. Nothing is there. No mark, no monument, no survey station — the seabed, two kilometres down.

Every coordinate in Britain is measured from it, and the choice of where to put it was made by an administrator rather than a geodesist.

What a false origin is #

A transverse Mercator has a natural zero: the point where its central meridian crosses whatever parallel the grid’s designers nominated as the true origin. Coordinates measured from that point are the projection’s own output, and they run in all four directions.

A false origin is a pair of constants added to those outputs before anybody sees them. The British grid adds 400,000 metres to every easting and subtracts 100,000 metres from every northing, which places the zero of the printed coordinate system at that spot in the sea. Universal Transverse Mercator adds 500,000 metres to the easting of every zone and, in the southern hemisphere, 10,000,000 metres to the northing.

The constants are called false because the point they measure from is not the point the projection is built about. That is the whole of it. The false origin does no geometry.

The reason, and it is not a geodetic one #

Negative numbers are transcribed wrongly.

A survey record is copied by hand, read aloud over a radio, typed into a form, and stored in a file whose format nobody documented. At every one of those steps a leading minus sign is the single character most likely to be lost, and losing it moves a point to the wrong side of the meridian rather than producing an obviously broken value. A coordinate of −12,431 that arrives as 12,431 is 25 kilometres out and looks entirely reasonable.

So the constants are chosen to make every coordinate in the grid’s working area positive, and large enough that no accident of rounding can push one below zero. This is a clerical decision about a human process, and it is the reason a reader who wants to understand what a grid does should ignore the first digit or two of every coordinate they meet.

The ten million, and the hemisphere it hides #

UTM’s southern zones carry a second constant: 10,000,000 metres added to the northing, so that a point on the equator reads 10,000,000 and a point near the southern limit of the system reads about 1,100,000. Northern zones add nothing, so the equator reads zero going north and ten million going south.

The consequence is that a UTM northing of 5,827,940 describes a point in the English Midlands or a point in the Southern Ocean south of Australia, and there is no arithmetic that distinguishes them. The hemisphere is a sixth declaration, carried alongside the zone number and absent from both coordinates.

This is the false-origin problem in its sharpest form. The constant was chosen to keep the number positive, it succeeded, and it did so by making two entirely different places share a numeric range. A designer optimising for a clerk copying digits produced a system in which the most common serious error — a point in the wrong hemisphere — is invisible to every check the digits themselves permit.

What survives the move, and what does not #

The claim that a false origin changes nothing needs a test, because a great many things about a grid do change its geometry. Here is the test: compute the distances and bearings between three points, move the origin a million metres east and a third of that south, and compute them again.

The bearings come back bit-for-bit identical. A bearing is an arctangent of two differences of coordinates, and a constant added to both terms of a difference cancels before the arctangent ever sees it.

The distances do not come back bit-for-bit identical, and the reason is worth more than a tolerance.

The one place a false origin is not free #

A double-precision float carries about sixteen significant decimal digits. A raw easting of 131,866.42 metres uses six of them before the decimal point, leaving ten for the fractional part — far more than any survey needs. Add 400,000 and it still uses six. Add a million and it uses seven, and one digit of precision has gone.

Subtracting two such numbers to get a distance then loses the leading digits they share, and what comes back is a few units in the last place away from the answer the unshifted coordinates give. On the test above the worst distance moves by about six hundredths of a nanometre.

That is not a geometric effect and it must not be reported as a tolerance, because “the distances agree to within a nanometre” would pass equally well on a grid where the origin had begun doing something real. What the assertion demands instead is that the disagreement is at the resolution of the shifted coordinate itself — below eight units in the last place of the largest number involved. A residual of that size can only be rounding. Any other cause would be larger, and the check would catch it.

This is the same discipline the collection applies to what survives a change of coordinates, where the meridian and parallel scale factors are shown to be properties of the graticule by swapping the output axes and watching them trade places. A quantity that survives an irrelevant change is describing the map; a quantity that does not is describing the description.

How the constants were chosen #

The British grid’s numbers look round — 400 kilometres and 100 kilometres — and the roundness is the last step rather than the reasoning.

Measured over the land the grid covers, the westernmost point sits 323 kilometres west of the central meridian, so a false easting of 400 kilometres clears it with 77 kilometres in hand. The southernmost point sits about 100 kilometres north of the true origin’s parallel at 49°N, so a false northing of −100 kilometres clears it with about fifty metres to spare.

Fifty metres, on a grid 1,300 kilometres tall. The eastern margin is generous and the southern one is a rounding away from failing, which says the constants were fitted to the country and then rounded to a whole number of hundred kilometres, in that order. A designer starting from roundness would not have landed that close to the edge by accident.

Why this matters more than it sounds #

The difference between two grid coordinate systems is almost always dominated by their false origins, and the false origin is the one term in the difference that means nothing.

A reader who subtracts a British grid coordinate from a UTM coordinate for the same point gets 5,544 kilometres and concludes that the datums are wildly inconsistent. The datum term in that number is 106 metres, the difference the two scale factors make is half a metre, and everything else — 5,543 kilometres of it — is the distance between two arbitrary decisions about where zero goes. Taking that apart is two grids over the same ground, and the reason it needs taking apart is this essay.

What a coordinate does not carry #

A grid coordinate is a pair of numbers. It does not say which grid it belongs to, and the false origin is why that is dangerous rather than merely untidy.

Because every grid puts its coordinates in the same broad numeric range — six figures for the easting, six or seven for the northing — a coordinate from one grid is a plausible coordinate on another. It will fall on land. It will be in roughly the right country. Nothing about the number betrays that it has been read against the wrong system, which is the failure mode what a grid is made of catalogues in full.

Contrast that with a latitude and longitude, where a value of 52.5 for the longitude of somewhere in Britain is instantly wrong to anybody looking. Making the numbers uniform made them uncheckable.

Two grids that share a scale factor and nothing else #

The false origin is easiest to see against a grid that differs in nothing else. Take the British grid and remove only its scale factor, leaving the same ellipsoid, the same central meridian, the same true origin and the same constants; the coordinates move by hundreds of metres because the projection has changed. Now put the scale factor back and move only the constants; the coordinates move by hundreds of kilometres and the geometry does not move at all.

The comparison is worth making because the two changes look identical in the printed numbers. Both produce a coordinate that differs from the published one, both differ by an amount too large to be a measurement error, and only one of them is a different map. Nothing in the coordinate distinguishes a grid whose zero was moved from a grid whose projection was changed — which is the argument for what a grid is made of being a list rather than a number.

The grid square, which hides the constants again #

Britain’s grid does not usually present its coordinates as metres from the false origin. It presents them as a two-letter square followed by digits — SU 387 145 — and the letters encode which 100-kilometre square the point falls in.

That is a third origin convention layered on the first two, and it exists for the same clerical reason: five-figure numbers are easier to read aloud and harder to mistranscribe than six-figure ones. The letters are assigned from a 5×5 lettering scheme laid over a 500-kilometre grid, which is why the sequence skips I.

The important consequence is that a grid reference and a grid coordinate are the same measurement in two encodings, and converting between them is arithmetic with no geometry in it whatever. A reference that has lost its letters has lost 100 kilometres of information, and there is no way to recover it from the digits.

It is also a precision convention, and an unusually honest one. SU 387 145 is a hundred-metre square, SU 3871 1450 a ten-metre square, and SU 38 14 a kilometre square; the number of digits is the claimed precision, so a reference cannot quietly overstate what it knows. A decimal coordinate has no such discipline — 429157.31 asserts a centimetre whether or not one was measured, and a coordinate that has been through a conversion or two usually asserts several digits nobody ever observed. That distinction returns in a published coordinate is a result, where the digits of a control coordinate turn out to describe a definition rather than an observation.

What was computed here #

Four things, and one of them is the assertion that could have failed.

The extent in the projection’s own coordinates. The country’s outline sampled at 441 points, projected through the transverse Mercator at the British grid’s stated parameters, with the false origin left off. That gives the fraction taking a negative coordinate — 52% — and the westernmost and southernmost reach.

The margins the published origin leaves. The same sample with the constants applied, giving 77 kilometres in the west and 50 metres in the south. Both are measured rather than looked up, so a change to the sampled extent moves them and the essay’s claim about how the constants were chosen would have to move with it.

The invariance test. Three points, two origins a million metres apart, distances and bearings both ways.

The refusal. The bearings must agree exactly and the distances must not — and must miss by less than eight units in the last place. Demanding exact agreement on the distances fails, and demanding merely “small” agreement would pass on a grid where the origin was doing real work. The check is written to fail in both directions.

Where the model stops #

The invariance here is exact for a planar distance computed from grid coordinates, which is not the same thing as a distance on the ground. The chain between them has four steps and is what a tape measures; every one of those steps is untouched by the false origin, which is why the argument is clean, but none of them may be skipped because of it.

Nor does the argument extend to the other four things a grid declares. Change the central meridian, the scale factor, the ellipsoid or the unit and the geometry moves — sometimes by centimetres, sometimes by hundreds of metres. The false origin is the only parameter in the list that a reader may safely ignore, and the reason it is worth an essay is that it is the largest number in the list and therefore the one they cannot.

The generalisation #

The pattern is not confined to cartography. A false origin is an offset applied to make a representation convenient, chosen so that it cancels in every operation anybody performs on it — and the test for whether such an offset is safe is exactly the test run here: apply the operations, move the offset, and check that the results are unchanged to the precision the representation has left.

Where that test fails, the offset is not a convention. The clearest case in this collection is the epoch, which looks like a bookkeeping detail attached to a coordinate and is nothing of the kind: change it and the ground the coordinate names has moved.

A tiling’s own false origin #

A national grid’s origin is moved so that every coordinate comes out positive, and the essay’s argument is that the move is a convention with no geometry in it. Every screen map does the same thing and almost nobody notices, because the offset is hidden inside an index.

A tile scheme’s origin is the north-west corner of the projected world, so the row index counts down from 85.0511° and the column from 180° west. That is a false origin in exactly this essay’s sense — chosen so the indices are non-negative integers — and the symmetry of the cut about the equator is a second such choice, made so the equator lands at the middle of the root tile rather than at an offset.

The origin those indices count from is a corner of a rectangle rather than a place: quadkey 000 is the north-western tile at every level, and the ground it names is Arctic ocean — a screen map is a pyramid of tiles.

And the same choice forces a second convention with a measurable price. A one-tile root requires a square world and therefore a cut at 85.0511°, which drops 1,901,487 square kilometres; a two-tile root would have reached 89.786° and dropped 3,558. The convention that makes the indices tidy is the convention that decides how much of the Earth the scheme can address.

Who chose it, and when #

The British grid’s parameters were settled in the 1930s, in the retriangulation that also produced the OSGB36 datum. The true origin at 49°N, 2°W was chosen so that the central meridian ran up the middle of the country and the parallel sat below its southern tip; the false origin was then chosen so that no coordinate on land came out negative.

Universal Transverse Mercator’s 500,000-metre false easting arrived later and from a different direction — it is a military specification, designed so that a zone’s eastings run from about 166,000 to 834,000 and can therefore always be written in six digits, in any zone, anywhere in the world, with no sign and no ambiguity about how many characters to expect. The uniformity is the feature. It is also, exactly, the property that makes a UTM coordinate impossible to tell from another UTM coordinate sixty degrees away.

One last asymmetry is worth recording, because it is the reason the southern margin above is so tight. A false easting has a whole hemisphere of room either side of its central meridian and is chosen to clear a country’s width; a false northing has to clear a country’s height, which for a grid designed as a single zone is the larger of the two. Britain is 1,300 kilometres tall and 600 wide, so the constant with the smaller margin is the one working against the larger extent, and fifty metres of clearance on the shorter number is the arithmetic saying exactly that.

Where this goes next #

The false origin is the first of the five things a grid declares and the only one that is free. The other four are what a grid is made of, and the one that costs the most is the scale factor, which is a number somebody chose and which was chosen by an argument that can be reconstructed and checked.