UTM and the zone system

The Universal Transverse Mercator grid is the most used projected coordinate system in the world and it is not a projection. It is sixty of them.

The design, in five constants #

A UTM coordinate is produced by five decisions, and every one of them is a trade with a size.

The projection: transverse Mercator on the WGS84 ellipsoid, computed as a truncated series.

The zone width: six degrees, giving sixty zones.

The scale factor: 0.9996, applied to the whole projection.

The false easting: 500,000 metres added to every easting.

The false northing: zero in the northern hemisphere, 10,000,000 metres in the southern.

The last two are bookkeeping. The first three are engineering, and the numbers in them came from a requirement.

Why six degrees #

The usual explanation is that six divides 360 neatly. That is true and it is not the reason.

Distortion in a transverse Mercator grows as the square of the angular distance from the central meridian:

$$k \approx 1 + \frac{\lambda'^2\cos^2\varphi}{2}$$

At three degrees on the equator — the edge of a six-degree zone — that is 1,382 parts per million, or about 1.4 metres per kilometre. The scale factor of 0.9996 then brings the worst case down to 981 parts per million, or under a metre per kilometre.

Under a metre per kilometre is at the boundary of what ordinary survey work of the 1940s could ignore, and that is where the width came from. A wider zone would exceed the tolerance; a narrower one would mean more zones and more boundary problems for no gain.

So sixty zones of six degrees is not a convention. It is the width at which the error reaches the tolerance, and the tolerance came from the instruments of the decade in which the system was adopted.

Where 0.9996 comes from #

The constant is often described as “reducing distortion”, which is not quite what it does. It moves the distortion, and the two curves above are the whole argument.

Without it, the projection is exact on the central meridian and grows monotonically wrong away from it: worst case 1,382 ppm at the edge, best case zero in the middle.

With it, the projection is 400 ppm too small on the central meridian, reaches true scale at 1.61° out, and is 981 ppm too large at the zone edge. The worst case has fallen from 1,382 ppm to 981 — an improvement of a factor of 1.41, which the site measures rather than quotes.

That is the secant construction applied at large scale: convert one zero line with error growing away from it into two zero lines with error of opposite sign between them. The projection is now wrong everywhere and less wrong at its worst, which is the minimax trade this subject makes over and over.

The value 0.9996 is a round number near the optimum rather than the optimum itself. Balancing the two extremes exactly would put the scale factor at about 0.99931, giving a worst case of 690 ppm either side. Choosing 0.9996 instead accepts an asymmetric split — 400 ppm one way, 981 the other — in exchange for a constant that can be written down and remembered, which for a system operated by hand in the field was worth more than 300 parts per million.

Two assertions hold that argument up at the equator. The scaled projection’s worst case must be smaller than the tangent one’s, and it must actually reach true scale somewhere inside the zone. The second matters: a scale factor too far below one produces a map that is uniformly too small rather than a redistributed error, and the two look identical in any summary that reports only the worst case. Both assertions have a latitude at which they stop holding, which is the subject of the next section.

The scale factor is a latitude effect too #

One consequence of the $\cos^2\varphi$ in the formula is rarely mentioned and changes how the system behaves in practice.

The distortion at a zone edge shrinks as latitude increases, because the zone itself is narrower on the ground. At 60° north the zone edge is three degrees of longitude but only 1.5 degrees of great-circle distance, and the tangent projection’s error there is 343 ppm rather than 1,382.

So at 60° north the scale factor makes the map worse. Not marginally: the tangent construction’s worst case over the zone is 343 parts per million and the scaled one’s is 400, so applying the constant costs seventeen per cent rather than buying twenty-nine.

The figure refused to draw before that was written down. Its assertion said the scale factor reduces the worst-case error across the zone, which is true at the equator and false at 60°, and the build stopped rather than produce a picture contradicting its own caption.

The generator now predicts which regime it is in — the constant helps exactly where the tangent construction’s edge error exceeds the 400 ppm the constant subtracts — and asserts that the measured outcome matches the prediction. That is a check on where the crossover is, at $\cos^2\varphi = 400/1382$, or 57.5°, rather than a restatement of whichever number came out smaller.

A latitude-dependent scale factor would fix it: 0.99983 balances the error at 60°. Nobody has ever done that, because it would mean a coordinate system whose definition changes as one moves north, and the whole value of a grid is that its definition does not change at all.

Which is the recurring theme of grid design: once published, a constant cannot be improved, because everything downstream is expressed in it. UTM’s 0.9996 was chosen for the worst case at the equator and applied everywhere, and above 57.5° north or south it is doing harm — across Scandinavia, most of Canada, most of Russia and all of Alaska, which is a good deal of the world’s surveyed land.

What a false easting is for #

Not a distortion correction. A formatting decision, and a good one.

The projection puts zero easting on the central meridian, so half of every zone has negative coordinates. Negative numbers in a coordinate are a source of transcription errors, sign errors and sort-order surprises, and none of that has anything to do with geometry.

Adding 500,000 metres puts the whole zone in the range roughly 160,000 to 840,000, positive throughout, six digits everywhere. The false northing of 10,000,000 in the southern hemisphere does the same for the other axis.

The consequence worth knowing is that a UTM easting near 500,000 means the point is near its zone’s central meridian, which is where the map is most accurate. That is a useful thing to be able to read off a number, and it is an accident of the choice.

What happens at a zone boundary #

The thing the system does not handle, and it is structural rather than an oversight.

Two points a metre apart across a zone boundary have completely different eastings — one near 840,000 in its zone and one near 160,000 in the next. The distance between them cannot be computed from their grid coordinates by any formula. Neither can a bearing.

There is no continuity to recover, because the two coordinates are in different projections. The usual workarounds are to extend one zone past its nominal edge — which is legitimate, since the series is still accurate a few degrees further out, though it degrades — or to convert both points to geographic coordinates and back into a single zone.

A coordinate is therefore incomplete without its zone, and that is a genuine defect in the design of the notation rather than of the projection. The zone number is metadata, it travels separately from the numbers, and it is lost exactly as often as any other metadata is.

The failure is worse than it sounds because a UTM pair without its zone is not obviously broken. Every zone produces eastings in the same range and northings in the same range, so a point from zone 31 read as though it were in zone 33 is a perfectly ordinary-looking coordinate about four hundred kilometres from where it should be. There is no digit out of range, no negative number, and nothing for a validator to catch. The only check available is whether the resulting position falls where the data is supposed to be about, which is a judgement rather than a test.

Reading a reference #

The notation is worth a paragraph, because it encodes the design and it is read by people who never see a formula.

A full UTM reference is a zone number, a hemisphere, an easting and a northing: 31N 630084 5812456. The zone fixes which of the sixty projections applies, the hemisphere fixes which false northing was used, and the two long numbers are metres.

The military variant, MGRS, replaces the leading digits with letters. 31U DQ 30084 12456 names the zone, a latitude band letter, a 100-kilometre square within the zone by a pair of letters, and then the position within that square to whatever precision is quoted. Dropping digits from the right coarsens the reference symmetrically: eight digits is a metre, six is ten metres, four is a hundred.

That truncation property is why the notation survives. A grid reference degrades gracefully — a shorter one is a larger square containing the same point — where a truncated latitude and longitude degrades into a rectangle whose shape depends on latitude. It is a small design decision and it is the reason grid references are read aloud over radios and geographic coordinates are not.

The exceptions #

Three zones are not six degrees wide, and the reasons are worth knowing because they show what the system is for.

Zone 32 is widened at the latitude of southern Norway, so that the whole of the country’s south-west coast falls in one zone. Zones 31, 33, 35 and 37 are widened around Svalbard and zones 32, 34 and 36 correspondingly do not exist there.

Both exceptions exist so that a piece of land stays in one zone rather than being cut. That is the priority the system reveals under pressure: a coordinate system is for computing on a region, and cutting a region in half costs more than a few extra parts per million.

The polar regions above 84° north and below 80° south are not covered at all. Transverse Mercator’s distortion in a zone is fine at high latitude but the zones themselves become absurdly narrow on the ground, and the system switches to a Universal Polar Stereographic grid instead — a different projection, with its own scale factor of 0.994, chosen by the same minimax reasoning.

It is worth keeping in view how little of the underlying projection a zone actually uses. Drawn whole, a transverse Mercator covers the sphere; a UTM zone is the three degrees either side of the vertical line down its middle, and everything else in that picture is territory no grid is defined over.

National grids are the same thing with better parameters #

UTM’s constants are universal, which is another way of saying they are optimised for nowhere in particular. A country that wants better can have it, and most do.

The British national grid is a transverse Mercator on the Airy 1830 ellipsoid, central meridian 2° west, scale factor 0.9996012717, false easting 400,000 and false northing −100,000. Every one of those differs from UTM’s, and every one is fitted to Britain: the central meridian runs down the middle of the country, so the whole island sits within about 5° of it and no zone boundary crosses it at all.

The scale factor’s ten digits are not a measurement. They are a definition, fixed once, and unchangeable thereafter because every map, land title, survey mark and dataset in the country is expressed in it. The precision is administrative rather than physical — a number is as exact as it is written when it is written into a legal instrument.

That is the general shape of a grid system: a projection, an aspect, a scale factor and two offsets, of which two are geometry and three are administration.

What the grid does not record #

Two things, and both are outside the numbers.

Which ellipsoid. A UTM easting on WGS84 and the same easting on ED50 are different places, by a hundred and forty metres in western Europe. The datum is not in the coordinate and it dominates every distortion the projection introduces.

Which zone. Already noted, and worth repeating because it is the more common loss in practice. A dataset of eastings and northings with the zone in a filename is one rename away from being unusable.

The scale factor is the first of three corrections a survey applies. The second is a direction rather than a distance.

The third correction is the one that surprises people: a straight line on the grid is not the line a theodolite sighted along.

What was computed here #

The scale factors on this page are measured from the projection’s own Jacobian rather than taken from the usual series formula for $k$. That is deliberate: the series formula and the projection are two expressions of the same design, and comparing one with the other would restate rather than test.

The measurement route requires the projection to be conformal, since it reads the scale from the meridian direction, so the parallel direction is measured too and the two are required to agree — they do, to 1.1×10⁻¹¹, which is the check that the isotropy the method assumes is still there.

Two assertions hold the scale-factor argument. The scaled projection’s worst-case departure from true scale over the zone must be smaller than the tangent projection’s, which it is by a factor of 1.41. And the scaled projection must actually cross true scale inside the zone, which it does at 1.61° — a secant construction that never reaches unity would be a uniformly shrunken map rather than a redistribution, and the check exists to tell the two apart.

The zone lookup and the central-meridian formula are separate pieces of arithmetic and are required to agree, which is the sort of thing that is obviously right and is wrong about a tenth of the time near the ±180° wrap.

What the pictures cannot show #

A zone at its own scale. The zone figure draws all sixty on a world map, where each is a narrow strip; the grid-scale figure draws one zone’s scale profile, where the horizontal axis is three degrees and the vertical axis spans two parts in a thousand. There is no single picture in which both the extent and the accuracy are visible, because their ratio is about 10⁵.

The figures also cannot show the discontinuity at a zone boundary, because the two sides are drawn in different coordinate systems and a picture showing both would have to choose one of them — which is the problem itself.

What the scale factor means on the ground #

Set against the other reduction a measured distance needs, the central scale factor turns out to reinforce rather than cancel: both are below one, so at 1,500 metres of elevation the combined factor is 635 parts per million, which is six metres on a ten-kilometre baseline.

That is the constant’s cost to somebody who is actually measuring rather than drawing. The 0.9996 was chosen to halve the projection’s worst case across a zone, and it was not chosen against the elevation of the ground — so on high terrain the two corrections add. The ground is not the grid works through the alternative, which is a grid whose scale factor is picked so that the two cancel at the mean height of the country it covers.

Who found it, and when #

The zone principle predates UTM. The Gauss–Krüger system used in Germany from the early twentieth century divides the country into three-degree strips on exactly the same reasoning, and several national grids of the 1920s and 1930s do the same.

UTM in its current form was adopted by the US Army in 1947 and became a NATO standard shortly after, which is how a set of constants chosen for artillery survey came to underlie civilian mapping worldwide. The military requirement explains several of the choices: the zone width follows from a stated accuracy, the false easting exists so that a grid reference can be read aloud without sign confusion, and the polar exception exists because the alternative was a system that failed in the Arctic.

The ellipsoid has changed since — UTM was originally on the International 1924 ellipsoid and is now on WGS84 — and every other constant has not, which is the durability that made the false easting worth getting right in the first place.

Where this goes next #

The projection underneath is transverse Mercator and the series that computes it. The general form of the scale-factor trade is what a standard parallel buys. And the quantity that dwarfs everything measured here is datum shifts.