The inverse of the series is not the series of the inverse

Transverse Mercator on the ellipsoid has no closed form. The projection that satisfies its specification exactly — conformal, with the central meridian at true scale — exists, and cannot be written down in elementary functions, so every national grid in the world computes it as a truncated power series in the third flattening — which is why a grid has an origin that is not there and a formula rather than a picture behind it.

Rung six of this ladder asked where the series stops being the map and answered it in one direction. The answer was reassuring: across a three-degree zone the second-order series is already at the millimetre and the fourth is far past anything a survey can see.

The map is used in both directions. Coordinates go forward to the grid when they are computed and come back from it when they are read, and both readings are results rather than observations, and the two directions are not the same series inverted. They are two different series, each truncated at its own order, each accurate to that order — and their composition is a third thing.

Three series, not one #

Krüger’s 1912 formulation carries three coefficient sequences, and knowing which does what is the whole of this essay.

α carries the conformal latitude forward into the projection. β carries it back. δ recovers geodetic latitude from conformal latitude on the way out. Each is a power series in the third flattening n = (a − b)/(a + b), which for the Earth is about 1/598, and each is truncated at the fourth term in the formulations national grids publish.

The forward computation uses α alone. The inverse uses β and δ. So the two directions do not merely truncate the same expansion at different places — they truncate different expansions, and there is no reason for their errors to be equal, to be related, or to cancel.

The two halves, separated #

Measuring the two directions separately needs care, because the obvious experiment measures the composition twice.

The forward series’ own error is straightforward: compute at order k and at order 4, and take the difference. The inverse’s is not, because inverting a point requires having projected it, and projecting it with the reduced series would put the forward error into the answer. So the inverse is measured from an exactly forward-projected point — order 4 — and brought back at order k, and nothing the forward series did is in the result.

That factor is the essay’s first finding and it is not small. A specification that says the formulae are accurate to a millimetre within the zone is, in every case this collection has read, a statement about the forward map — a sibling of the way a datum shift dwarfs a projection error and is quoted separately from it. The inverse at the same order is one to two orders of magnitude worse.

The round trip is then very nearly the sum of the two, at 99 to 100 per cent of it at almost every latitude, falling to 71 per cent at a few places where the forward series passes through a node of its own error. So the errors add rather than cancel — which is the unlucky case, and the one nobody checks for, because the folklore about round trips is that errors tend to cancel.

The table is not symmetric #

The cleanest way to see that the two directions are different objects is to truncate them independently.

If the two directions were one series and its inverse, the table would be symmetric about its diagonal. It is not, by nearly an order of magnitude in the corners, and the asymmetry has a direction: the inverse order decides the answer. Holding the forward order at four and varying the inverse spans a factor of 10⁵; holding the inverse at four and varying the forward spans 1.2 × 10⁴.

For anyone implementing a grid on a constrained device — which is most of the places these formulae actually run, including every tile server drawing a screen map — that is a practical instruction. If one direction has to be cheap, make it the forward one.

The residual that is left at fourth order #

The last finding is the one that made the arithmetic worth doing, and it is a fact about which of the three series is responsible.

At fourth order both ways, the round trip returns a point 0.39 mm from where it started at 52° north, three degrees out. That is small. What is not ordinary is its direction: the displacement is 0.393 mm north and 0.000022 mm east — a ratio of eighteen thousand to one.

The residual is northing, and almost nothing else. The reason is structural. α and β act on the complex isometric coordinate and move both components; their fourth-order truncations are of the same form and largely cancel through the composition. δ acts on latitude alone — it is the series that turns conformal latitude back into geodetic latitude — and there is nothing in the forward direction with the same shape to cancel against it. What is left when the other two have finished cancelling is δ’s own truncation, and δ’s own truncation is a latitude.

Half of a millimetre is well inside what the tolerance decides for any survey. A fifth of a millimetre is not an engineering problem. It is a diagnostic: a round-trip test on a grid implementation that comes back with a residual purely in northing is behaving correctly, and one whose residual has a comparable easting component has a bug in α or β rather than an acceptable truncation.

What a round-trip test is actually testing #

The round trip is the test every grid implementation ships with, because it needs no reference data: project a point, unproject it, and check that it came back. It is a good test and it is worth being precise about what it can catch.

It catches a sign error, a swapped coefficient, a wrong constant — anything that makes one direction inconsistent with the other, which is most implementation mistakes. It catches them at metre scale, loudly.

It cannot catch a shared mistake. If α and β were both derived from the same wrong flattening, or both applied the same wrong false origin, the round trip is clean and every coordinate is wrong. The number of implementations that have shipped with a correct round trip and an incorrect datum is not small.

And it cannot catch a correct implementation being unsuitable. The formulae are correct to their order and the order is a design decision; a round trip at the same order on both sides returns a residual that is a property of the truncation rather than of the code. Reading a 0.39 mm residual as a bug and chasing it is a way of spending a day.

The test that catches what the round trip cannot is the one this ladder has used since its base rung: compute something by two routes that share no code. The meridian arc by series and by quadrature is one such pair; a grid coordinate against a published test point from the defining authority is another, and it is the one a specification exists to provide.

What was computed, and how #

Every error in this essay is a ground displacement in metres, computed from the recovered latitude and longitude through the meridian and prime-vertical radii of curvature at the point. It is not a difference in projected coordinates, and the two are not the same number — the scale factor across a three-degree zone reaches about 1.0004, so a projected-unit residual overstates the ground one by that much — which is the scale factor of a line arriving inside an error budget.

The reference for “exact” is the fourth-order series, which is not exact. That is stated rather than hidden: the essay measures how much each order differs from the fourth, and separately shows that the fourth-order round trip is itself not the identity. What it does not do is compare either against a genuinely exact evaluation, which would need an entirely different algorithm.

The check that ties the machinery to something outside itself is older than this rung and is worth restating: on its own central meridian the projection’s northing must be the distance from the equator along the meridian, and that arc is computed by numerical quadrature sharing no code with the series. They agree to a hundredth of a millimetre.

Where the model stops #

One zone width. Three degrees is a national grid’s zone and six is UTM’s. The errors grow steeply with longitude from the central meridian — the series is in powers of the transverse coordinate — so every number here is for a three-degree zone, and a wide-zone implementation is a different measurement.

Krüger’s formulation only. There are others: Thomas’s, Redfearn’s, Bowring’s, and the exact evaluation through Jacobi elliptic functions that Karney published in 2011, which is accurate to nanometres over the whole ellipsoid and which is what should be used when the answer matters. This essay measures the series that national grids actually publish, because those are the formulae whose round-trip behaviour a reader will encounter.

The comparison is between orders, not against truth. Every number here says how far one truncation is from another, and the fourth-order series is used as the stand-in for the exact map. That is legitimate for the differences between orders one, two and three, whose errors are far larger than the fourth’s; it is exactly what is being questioned when the fourth-order round trip’s own residual is reported. The 0.39 mm is a real residual of a real composition, but calling it the fourth-order error would be a stronger claim than the arithmetic supports, and it is not made.

And the third flattening is the Earth’s. For a body with a larger flattening the same order is a much worse approximation. Mars at f = 1/169 has n about 3.5 times the Earth’s, so a fourth-order term is about 150 times larger — a point the bodies ladder makes about spherical formulae and that applies here with more force.

The generalisation #

A numerical inverse is not the inverse of a numerical forward, and composing two approximations does not produce an approximate identity. The habit this argues against is the one that assumes a round trip is a test of both directions at once. It is a test of the composition, and a composition can be right while both halves are wrong, or wrong while both halves are right.

The disciplines that handle this well separate three things and report them separately: the forward error, the backward error, and the residual of the round trip. Numerical linear algebra does this by convention — a factorisation is reported with its forward and backward errors, and nobody quotes a single accuracy for a solver.

Cartography mostly does not, and the reason is historical rather than principled. A grid’s formulae were published for people doing arithmetic by hand, in one direction at a time, and the accuracy statement was about the operation somebody was about to perform. The round trip became routine only when the computation became automatic.

Who found it, and when #

Carl Friedrich Gauss developed the conformal projection of the ellipsoid onto the sphere and thence onto the plane while triangulating Hanover, and did not publish the projection formulae; Oskar Schreiber worked them up for the Prussian survey in the 1860s, which is why the projection is sometimes called Gauss–Schreiber. Louis Krüger published the series in the third flattening in 1912 in a Potsdam memoir, and it is his α, β and δ that every modern national grid uses.

The observation that the two directions have unequal accuracy at the same order is implicit in the coefficients themselves — β’s terms are not α’s, and were never claimed to be — but it is not, as far as this collection has found, stated anywhere as a practical warning. National grid specifications quote one accuracy.

Charles Karney’s 2011 paper on transverse Mercator gave both the exact algorithm and a careful account of the series’ accuracy as a function of order and zone width, and it is the reference that makes the question answerable rather than arguable. The round-trip residual’s direction — that at fourth order what survives is δ’s truncation, and therefore a northing — follows from his decomposition and appears to be an observation this essay is making rather than repeating.

Why a round trip is the weakest test available #

The asymmetry has a consequence for how these implementations are tested, and it is worth stating sharply because the round trip is the test almost everybody runs.

A round trip measures the composition of two maps, not either of them. Project and unproject, compare against the original, report the residual. That residual is small when the two operations are inverse to each other — which is a much weaker statement than either of them being right.

The extreme case makes it obvious. Take a forward projection with an error of any size, and produce the inverse by solving the forward numerically — a Newton iteration on the same formula. The round-trip residual is then zero to machine precision, by construction, whatever the forward map is doing. A test that returns a perfect score on an arbitrarily wrong projection is not measuring accuracy.

The series case is the same thing in a milder form. α and β are derived from one another, so their truncation errors are related rather than independent, and part of the forward error is undone by the inverse. What survives is the residual this essay measures — real, one-sided, and much smaller than either half’s own departure from the exact map.

So the round trip is necessary and not sufficient, and it should be read as a consistency check between two implementations rather than as an accuracy figure. A failing round trip proves a defect. A passing one proves that whatever defect exists is shared.

Which is why a specification should quote three numbers rather than one: the forward map’s departure from the exact projection, the inverse map’s, and the round-trip residual. The first two require an exact reference — Karney’s algorithm supplies one — and the third does not, which is exactly why it is the one everybody publishes.

There is a second reason the round trip is attractive and it is worth naming rather than dismissing. It is the only test a user can run on the software they actually hold, with no reference implementation, no exact algorithm and no expertise — and as a smoke test for a wrongly configured zone, a swapped ellipsoid or a units error it is excellent, because all of those break the composition badly. The complaint is not that it is run; it is that its number is then printed as an accuracy.

A related habit is worth avoiding for the same reason: testing a projection against another implementation of the same series. Two libraries carrying the same coefficients agree to the last digit and share every truncation, so the comparison measures whether the coefficients were typed in correctly. That is worth knowing and it is not what it is usually reported as.

And the three cannot be reconstructed from each other. A large round trip with small halves is impossible; small round trips with large halves are routine, and the national grid specifications quoting one accuracy figure are quoting the third.

Where the ladder goes next #

This rung takes the ellipsoid ladder as far as the arithmetic of one projection. The families ladder next door has spent eight essays on cylinders, cones and planes without ever asking which of the named projections is actually a projection onto one of them — a question whose answer turns out to be six of eleven, and which needs one two-line formula to settle.