Where two zones meet

A survey mark sits at 6°E, which is the boundary between UTM zones 31 and 32. Its coordinate in zone 31 is 705,929 metres east. Its coordinate in zone 32 is 294,071 metres east.

Both are correct. The mark has two coordinates, four hundred kilometres apart, and neither is more right than the other.

What a zone is #

Universal Transverse Mercator divides the world into sixty strips six degrees of longitude wide, and each strip is a separate projection about its own central meridian. Not a region of one map: a different map, with a different distortion pattern, whose coordinates happen to be expressed in the same numeric range as its neighbours’ because they all carry the same 500,000-metre false easting.

That is the source of everything in this essay. Two zones are not two halves of a coordinate system. They are two coordinate systems, and the boundary between them is not a line on a map but a place where one map stops and another begins.

Both zones are right, and by how much #

A point does not have to be inside a zone for that zone’s projection to give it a coordinate. The transverse Mercator formulae work everywhere — the distortion grows, and nothing breaks — so a point at 6°E is perfectly representable in zone 31, in zone 32, and in zone 25 if anybody insists.

The practical question is how much a job loses by using the neighbouring zone rather than its own, and the answer is: almost nothing, over a short distance. The twenty-kilometre baseline above comes out at 20,002.4 metres in zone 31 and 20,002.4 metres in zone 32, differing by 58 nanometres — which is not a geometric quantity at all but the resolution a double-precision float has left after carrying a six-figure easting.

That is why the specification publishes an overlap belt: forty kilometres either side of each boundary, within which either zone may be used. A job in the belt picks one zone and stays in it, and pays nothing measurable for the choice.

And mixing them is not a small error #

Taking one end of the baseline from zone 31 and the other from zone 32 gives 392 kilometres for a twenty-kilometre line.

The reason is arithmetic rather than geometric. The two eastings are measured from meridians six degrees apart, and subtracting them measures neither the ground distance nor anything else — it measures the gap between two conventions, with a bit of real geometry buried in it. The result is not an error in the sense of a measurement gone wrong; it is a calculation performed on quantities that do not belong in the same expression.

The assertion in this figure is the pair rather than either half. Both zones must agree to nanometres and mixing them must be wrong by more than the baseline is long. Checking only the second would pass on a system where the two zones happened to agree exactly, and a grid whose zones agreed would be a grid with no zones.

The belt is wider than the seam is deep #

It is worth putting a number on how much a job actually loses by staying in the wrong zone, because the answer changes what the belt is for.

The scale factor at the edge of a zone reaches about 1.00098 — 980 parts per million. Forty kilometres beyond the edge, in the belt, it reaches roughly 1.0018, or 1,800 parts per million. So a job that uses its neighbour’s zone throughout carries a scale error about twice the zone’s own worst case: real, systematic, and entirely computable.

That is not a small price and it is a known price, which is the distinction that matters. A job in the belt applies the scale factor its coordinates actually have, computed from the projection it actually used, and the reduction is correct. Nothing degrades except the size of a correction that was going to be applied anyway.

Compare that with the mixed computation, where no correction exists because no single projection was used. The belt is not a region of tolerable error; it is a region where a different correct answer is available, and the specification’s forty kilometres is a statement about how far that stays practical rather than about how far it stays valid.

What the belt is spending is the scale factor’s own growth. It keeps rising quadratically beyond the zone’s plotted edge, so a job forty kilometres outside its zone has a larger correction to apply and a perfectly well-defined one.

Why this one is dangerous #

Every other failure in this field produces a plausible number. This one produces an absurd number, which ought to make it the safest of the lot — and it is instead the most common serious error in practical work, because the absurd number is rarely computed.

What is usually computed is a coordinate, not a distance. A point read into a system as zone-32 coordinates when it is zone-31 data lands four hundred kilometres away, in another country, and somebody notices. But a point near the boundary read with the wrong zone number attached and the right coordinates is a different failure: the numbers are internally consistent, they are in the right numeric range, and they describe a place four hundred kilometres away that is nonetheless a perfectly ordinary place.

What a national grid does instead #

The seam is the strongest practical argument for a country publishing its own grid, and it is stronger than the distortion argument that designing a grid for one region measures at a factor of 1.51.

Britain’s grid is a single zone 700 kilometres wide. That costs distortion — the scale factor has to work harder over a wider strip — and it buys the complete absence of this failure mode for every domestic job. There is no boundary, no belt, no zone number to lose, and no possibility of subtracting two eastings that were never in the same system.

The trade is explicit: a wider zone is worse everywhere and has no seam. For a country that fits in one, that is usually the right trade, and it is why the countries with the most emphatic national grids are the ones the international scheme would otherwise cut in half.

The other seam, which nobody calls one #

Zone boundaries are the visible seams. There is an invisible one in every grid and it is worth naming beside them.

A datum has a seam too. Two countries whose national grids abut use different datums, different ellipsoids and different realisations, so a point on their shared border has two coordinates that differ by a hundred metres or more — the term what a grid is made of measures at 106 metres for Britain against the global system. That seam runs along political boundaries rather than meridians, it is not documented in any projection specification, and there is no overlap belt for it.

Set beside that, the zone seam looks well engineered. It is documented, it has a belt, it has a rule for which side a point belongs to, and the failure it permits is enormous enough to be obvious once anybody computes a distance. The datum seam has none of those properties and produces errors of exactly the size that survives a plausibility check, which is the argument what a coordinate refers to is built on.

The seam a country cannot avoid #

A country too wide for one zone has no good option and must choose which bad one.

Two zones gives a seam through the middle of the country, in the place where the most economic activity usually is, with the failure above available at every job that crosses it.

One overwide zone gives distortion growing as the square of the excess width — a zone twelve degrees wide has four times the worst case of a six-degree one — which is survivable for mapping and not for engineering.

Many narrow zones gives many seams, and is the choice several large countries make for exactly the reason that many small errors at known places are easier to manage than one large error everywhere. The trade appears again in the American low-distortion projections, where a state may carry fifty zones.

There is no fourth option, because the difficulty is not a shortcoming of the transverse Mercator. It is the impossibility, showing up as an engineering constraint: a strip can be flattened with bounded distortion and a continent cannot, which is what can be unrolled applied to a working coordinate system.

The check that catches it #

A mixed-zone computation is trivially detectable and the check is worth stating because almost nobody runs it.

Compute the same distance from the latitudes and longitudes. A geodesic distance on the ellipsoid needs no zone at all, so it is immune to this failure by construction, and comparing it against the grid distance catches a zone mismatch instantly — the discrepancy is hundreds of kilometres rather than the hundreds of parts per million a real reduction produces.

This is the same defence as the one the units are part of the coordinate recommends against a wrong foot, and it works for the same reason: an independent route to the same quantity catches errors that are invisible to any amount of internal consistency checking. The difference is that a wrong foot needs two external control points to detect, and a wrong zone needs only the geographic coordinates the grid values came from.

Which zone a point belongs to, and which zone a job should use #

The specification assigns every point to a zone by its longitude, and that assignment is about the point rather than about the work.

A job whose extent lies mostly in zone 31 with a few marks over the boundary should use zone 31 throughout, and the belt exists to make that legitimate. Assigning each mark to its own zone by the rule — which is what a naive automatic conversion does — produces exactly the mixed computation this essay is about, and produces it while following the specification correctly.

That is a genuinely awkward property of the design. The zone rule is a function of position, jobs are not points, and the reconciliation is a human decision that no amount of correct per-point processing can make for itself. Software that converts a file of coordinates “to UTM” without being told which zone has already made the wrong choice, and the file it produces is internally inconsistent in a way that no subsequent check on that file can find.

There are three exceptions to the longitude rule in the specification itself — an enlarged zone 32 off south-west Norway, and the Svalbard zones — which exist so that particular pieces of land are not cut in half. Their existence is an admission that the rule is about convenience rather than geometry, and that where convenience and the rule conflict, the rule gives way.

What was computed, and how #

A baseline straddling the boundary, with its half-length converted from kilometres to degrees using the local radius of curvature in the prime vertical rather than a spherical approximation — because the whole essay is about a boundary and a boundary is where an approximation would put the endpoints on the wrong side.

Three lengths: both ends in zone 31, both in zone 32, one from each.

The double coordinate, being the same ground point projected through both zones’ parameters.

Two assertions, in opposite directions. The consistent answers must agree to within twenty millimetres over twenty kilometres — they agree to 58 nanometres. The mixed answer must be wrong by more than a hundred kilometres.

Where the model stops #

The overlap belt is a specification, not a geometric fact. Forty kilometres is what the UTM specification says; the geometry would permit rather more, and the number is a convention chosen so that the extra distortion at the belt’s outer edge stays inside the system’s own tolerance. Nothing here derives it.

Nothing here handles the polar zones. UTM stops at 84°N and 80°S, and beyond those the Universal Polar Stereographic system takes over with an entirely different projection and its own conventions. The seam between UTM and UPS is a harder case than the seam between two UTM zones, because the two systems do not merely have different parameters but different shapes, and it is not treated here.

And the 58 nanometres is a property of double-precision arithmetic, not of the projections. Computed in higher precision the two zones would agree more closely still; computed in single precision they would differ by centimetres, and a reader working in a system that stores coordinates as 32-bit floats should expect that rather than this.

The generalisation #

Two representations that agree to arithmetic noise are interchangeable; two that agree to nothing are not comparable; and the dangerous case is the one that looks like the first and is the second.

What makes the zone seam a good teaching case is that it is unambiguously the second and is routinely treated as the first. The numbers look alike, the ranges overlap, the software accepts them, and the only thing distinguishing zone 31’s easting from zone 32’s is a piece of metadata that the coordinate does not carry — which is the argument of what a grid is made of in its most extreme form.

The general defence is the same one that essay recommends: attach the declaration to the data, not to the folder, the filename or the convention. Everything else is a promise that somebody will remember.

The other overlap belt #

A zone boundary is a seam placed where a projection’s error would otherwise grow, with an overlap belt in which a point carries two coordinates. The applied field meets two more seams and one of them is handled the same way.

A tile is rendered from the geometry inside it, so anything crossing an edge is drawn twice by two renderers that cannot see each other — and the repair is an overlap belt: each tile is rendered from a neighbourhood of itself, 23 per cent of a tile wide for a label, which costs 116 per cent in duplicated geometry. The same solution, for the same reason, at a completely different scale.

Two other seams are worth holding beside this one. A delivery scheme’s tile boundaries are a seam with an overlap: the pieces are exact and their labels are not the whole feature’s, by up to 1,863 kilometres, which is the price of independence between tiles. And longitude’s own cut is a seam that cannot be moved away at all — a bounding box across it becomes the whole world and a midpoint lands at the antipode. A zone belt is chosen; that one is forced by the topology of a circle.

Who set it, and when #

The UTM zoning was fixed by the US Army in 1947, adapting a scheme the German artillery had used from the 1930s. Six degrees was chosen as the widest strip whose scale error stays under one part in 2,500 at the equator with a central scale factor of 0.9996 — a bound rather than an optimum, as UTM and the zone system records.

The overlap belt was part of the specification from the start, which says the designers understood the seam perfectly well. What they could not anticipate was software: a system designed for a soldier reading a printed map, who could see which zone the sheet was for, became a system for files that carry numbers and not sheets. The failure this essay describes is not a flaw in the design but a consequence of the design outliving its medium.

The general form of that observation is worth keeping, because this collection meets it repeatedly. A design’s safeguards are addressed to the failure modes of its medium, and a medium change can remove a safeguard’s effectiveness without removing the safeguard. The overlap belt is still specified, still documented and still correct; what has gone is the reader who could see which zone their sheet was for, and nothing in the specification could have anticipated that the reader would be replaced by a file.

Where this goes next #

The last of the grid’s parameters is the one a job may override entirely, by scaling the whole system so that a set-out distance equals a ground distance: a grid scaled to the ground is not a map. What every grid then requires of a measurement is what a tape measures.