Where the series stops being the map

The projection most of the world’s survey data lives in cannot be written down. On a sphere the transverse Mercator has a closed form of three lines; on an ellipsoid the same specification — conformal, with the central meridian at true scale — has no elementary solution, and what every national grid actually computes is a truncated power series in the third flattening, on a body that is not a sphere and cannot be treated as one here. That much this collection has already taken apart: the series that computes it shows what each term is worth and how few of them a three-degree zone needs.

That essay asked how many terms. This one asks a different question, and the answers are not the same shape: does another term always help?

At 30° from the central meridian the second term is worth 21.5 metres, the third 10 centimetres and the fourth six tenths of a millimetre — each a two-hundredth of the one before. At 88° the second is worth 26 kilometres, the third 135 and the fourth 903. Somewhere between, the series stops being an approximation to anything.

The coefficients, computed rather than transcribed #

The whole content of the projection is one analytic function: the map carrying the conformal latitude — the one Mercator’s construction needs — to the rectifying latitude, the one that measures distance along a meridian. Krüger published it in 1912 as four polynomials in the third flattening n, and every grid formula in ordinary use is a transcription of them.

That function is odd and has period π, so it is a sine series in 2jχ and its coefficients are integrals rather than expansions:

$$\alpha_j = \frac{4}{\pi}\int_0^{\pi/2}\left[\mu(\chi) - \chi\right]\sin 2j\chi \; d\chi$$

Computing them that way uses no expansion in the flattening anywhere — only the two latitudes this site already computes by unrelated routes, and a quadrature.

The agreement is not a tolerance but a pattern, and the pattern is the check. Krüger’s polynomials stop at n⁴, so the jth coefficient — whose leading term is of order n^j — should be right to a relative n^(4−j): 8 × 10⁻¹², 4.7 × 10⁻⁹, 2.8 × 10⁻⁶ and 1.7 × 10⁻³. The measured disagreements are 7.1 × 10⁻¹², 7.9 × 10⁻⁹, 6.2 × 10⁻⁶ and 5.8 × 10⁻³, each within a factor of three and each in the right place. A transcription error in any one of the four would break it in a way no loose bound would notice.

The decay rate is the whole answer #

A sequence of coefficients that decays geometrically is the signature of a function analytic in a strip, and the width of that strip follows from the rate. The measured ratios are 0.541, 0.937, 1.208 and 1.387 times n, rising and converging; Aitken’s extrapolation puts the limit at 1.74 n, which is 2.92 × 10⁻³.

That number is the answer to the question this essay asks, by way of one observation about the projection’s own arithmetic. Every term in the series carries cosh 2jη′ for the transverse coordinate η′, which grows like e^{2jη′}. So the jth term behaves like

$$r^j e^{2j\eta'}$$

and the sum converges exactly while r·e^{2η′} < 1 — that is, while

$$\eta' < \tfrac{1}{2}\ln(1/r) = 2.9176$$

Where the coefficients run out is measured rather than assumed: the quadrature is run a second time at three fifths of the sample count, and a coefficient that moves by more than a tenth between the two is noise. The fifth is 5.7 × 10⁻¹⁵ and survives that test; the sixth is 4.6 × 10⁻¹⁷ and does not. The limit therefore comes from four ratios and an extrapolation, not from summing terms until they misbehave, and saying so is part of the measurement.

Where that is, on the ground #

The transverse coordinate is not a longitude, and converting the limit into one gives the shape of the domain rather than a number. On the equator, η′ = asinh(tan Δλ), so the boundary is at 83.81° from the central meridian. At 60° north the same coordinate is reached much further round, and at the pole the boundary closes: the region the series describes is a lens, not a strip.

The direct measurement agrees with it from an entirely different quantity. Taking the corrections in metres — the differences between successive orders of the projection’s own output — the third correction stops being smaller than the second at 85.46°, and the fourth stops being smaller than the third at 84.83°. Those crossings fall as the order rises, towards the 83.81° the coefficients predict, which is what a sequence of finite-order symptoms of an infinite-order limit should do.

What was computed, and how #

Three quantities, each by a route that does not use the others.

The coefficients, by Kahan-compensated quadrature over twenty thousand samples of the conformal latitude, with the rectifying latitude taken from a meridian-arc integration at 8,192 steps. The sum is of order 10⁻³ and the answer of order 10⁻¹⁵, which is why the compensation is there.

The limit, from the ratios of those coefficients by Aitken extrapolation, and separately from the arithmetic of cosh 2jη′.

The corrections, by evaluating the site’s own transverse Mercator at orders one to four and taking the distances between consecutive answers, in metres. This route knows nothing about Fourier coefficients: it is the projection as a grid would compute it, differenced.

The refusal is the one that makes the claim a measurement. Inside the computed domain — at 71.8° from the central meridian — each correction is 0.061 and 0.080 of the one before, so the series converges. Outside it, at 87.8°, they are 4.31 and 5.59, so it does not, and the gate requires both.

What a working zone actually uses #

The practical answer is that nothing in use is anywhere near the limit. A three-degree zone reaches 0.9 per cent of the domain and a six-degree zone 1.8; the fourth term contributes 6.7 micrometres at the edge of the latter. This is why the boundary has never had to be found by anybody computing a grid coordinate: the series is not merely convergent in the working range, it is convergent by a factor of fifty in the coordinate that decides it.

The zone system is not a convergence measure. It is a distortion measure: at 3° from the central meridian the scale factor is already 1.00037, and by 40° it would be 1.09 — nine per cent, which no survey grid could carry. The truncation and the distortion both grow with distance from the central meridian, and they are separated here by two orders of magnitude in where they become intolerable.

The two latitudes the series is made of #

It is worth being explicit about what the analytic function actually is, because the domain follows from it. The transverse Mercator’s specification names two things: the map is conformal, and the central meridian is at true scale. The first fixes what the coordinate must be — the conformal latitude, which is the ellipsoid’s own isometric coordinate and is why the ellipsoid’s several latitudes are different angles rather than one. The second fixes what the map must do on one line — put a point at its meridian-arc distance from the equator, which is the rectifying latitude.

So the projection is the analytic continuation of the map between those two latitudes, from the central meridian outward. That is not a metaphor: the series ξ = ξ′ + Σ αⱼ sin 2jξ′ cosh 2jη′ is precisely the function μ(χ) evaluated at the complex argument χ + iη′, with the real and imaginary parts separated. The whole projection is one real function of one real variable, continued.

Which is why a question about the domain of the projection turns into a question about the strip in which a one-dimensional function is analytic — and why the answer is a property of the ellipsoid’s own flattening rather than of anything cartographic.

What the same distance costs in scale #

Put the two limits side by side. At the edge of a six-degree zone the fourth term of the series contributes seven millionths of a metre and the scale factor departs from unity by four parts in ten thousand. At forty degrees the series is still converging comfortably and the scale factor is 1.09, which no survey grid could carry and no zone system would tolerate.

That gap is why the convergence limit has never been part of anybody’s practice, and it is also why it is worth computing: the two constraints on a formula’s range of use can be separated by a hundredfold, and the one that bites is not always the one that is easiest to check. Britain’s grid runs a single zone across ten degrees of longitude, which makes its scale factor a design decision and its truncation a non-issue — and which is why the seam between two zones is a question about coordinates rather than about convergence.

Where the model stops #

The limit computed here is a statement about the series, not about the projection. The exact transverse Mercator exists past 83.81° — it is a conformal map of the ellipsoid and does not stop being one — and it can be computed there by other means, at the cost of elliptic functions. What stops is this particular representation of it.

The number itself carries three assumptions worth stating. It is for WGS84’s flattening: n is 1/595.5, and a body with a different flattening has a different strip and therefore a different boundary — a more flattened body has a narrower domain, since the coefficients decay more slowly. It is computed on the equator, where the domain is narrowest. And it rests on an extrapolation of four ratios: the limit is 1.74n from Aitken’s Δ², and were the true limiting ratio 2n instead, the boundary would move to 82.3°, which is a difference of a degree and a half rather than of a kilometre.

What the measurement cannot see is where the exact map’s own singularities are. A geometric decay of the coefficients means some singularity sits at that distance in the complex plane, and where it sits on the ellipsoid is a question the arithmetic here does not answer.

The generalisation #

The argument has nothing to do with maps. A function’s Fourier coefficients decay geometrically at a rate set by the nearest singularity in the complex plane, and a series evaluated at a complex argument converges only inside the strip that rate defines. Every truncated trigonometric series in every subject has such a domain, and it is almost never stated because it is almost never reached.

The cartographic version of the lesson is narrower and more useful: a formula published as a series has a region of validity that is not the region its author was interested in, and the two can be discovered independently. Krüger was interested in a zone a few degrees wide. The series he wrote is good to eighty-three, which nobody had cause to check for a century.

Who found it, and when #

Gauss developed the ellipsoidal transverse Mercator in the 1820s for the Hanover survey and published nothing; Krüger reconstructed and extended it in 1912, which is where the four coefficients in every national grid formula come from. Lee gave an exact treatment in terms of elliptic functions in 1976 — the same kind of closed form that a conformal map onto a polyhedral face turned out not to need, and Karney’s 2011 analysis extends Krüger’s series to eighth order and gives its accuracy at nanometres within about 3,900 kilometres of the central meridian — roughly 35 degrees — while stating that the series does not converge everywhere.

What this rung adds to that record is not the fact but the route: the coefficients as a quadrature rather than an expansion, the decay rate measured from them, and the boundary that follows — with a second, independent confirmation from the projection’s own output in metres.

What a series is and is not entitled to #

The rung’s boundary has a consequence for how a national grid’s formula should be read, and it is worth stating because the formula is usually presented as though it were the projection.

A series is a representation, not a definition. The transverse Mercator is defined by a condition — conformality on an ellipsoid, with a stated central meridian and scale factor — and that condition determines the map everywhere the map exists. Krüger’s four coefficients are a way of evaluating it, accurate to nanometres over a region and useless outside one.

The two are routinely conflated because the formula is what anybody sees. A grid specification prints the coefficients; software implements the coefficients; a practitioner asked what the projection is will produce the series. Nothing in that chain mentions the condition, so the region of validity has nowhere to be stated and generally is not.

And the failure outside it is not a gradual loss of accuracy. A truncated series does not degrade politely past its radius of convergence; it departs, and the departure grows with the order rather than shrinking — so adding terms makes a distant evaluation worse. That inverts the ordinary intuition about series and is the reason the boundary has to be known rather than felt.

The confusion also explains a class of software failure that looks mysterious. A library implementing a national grid’s series will happily accept coordinates from the other side of the world and return numbers, because a polynomial evaluates anywhere. The numbers are wrong by amounts that grow without bound and they carry no marker, so a pipeline that transforms a global dataset through a national grid produces plausible coordinates for the country and nonsense elsewhere — with nothing in between to signal where the transition happened.

The remedy is a bounds check the specification never asked for. Every national grid has a region of validity, it is usually published as a bounding box for administrative reasons, and it is very nearly the region where the series is honest. Refusing to evaluate outside it costs one comparison and converts a silent failure into an error.

Which is what makes this rung’s second confirmation worth having. A decay rate measured from the coefficients is a statement about the series; a departure measured in metres from the projection’s own output is a statement anybody can check against a requirement. The two agree, and the second is the one that belongs in a specification.

And the check has a second use for anybody assembling a pipeline. The validity region is a property of the coordinate system, so a dataset spanning several national grids has several regions, and the transformation that is honest for one part of it is dishonest for another. That is not a case for finding a single grid to use; it is the reason global work is done in a global system and reprojected locally rather than the other way round — a practice usually justified by convention and justified here by where a series stops being the map.

The practice is right, and it is usually taught as a habit rather than as a consequence of anything, so a reader who meets a case the habit does not cover has nothing to reason from. A convention that turns out to have a derivation is worth more than one that does not, because it survives the next person who asks why.

One thing the number is not, and the distinction is easy to lose. The distance at which the truncation reaches a stated tolerance is a property of the series and of the ellipsoid, not of the grid it is used in. A national grid that happens to be narrow is not thereby safer; it is simply not exercising the part of the series where the error lives. Widening a zone by decree, which is a thing that has been done, moves a coordinate system into territory the series was never checked over without changing a line of the specification that defines it.

Where the ladder goes next #

A series is one way to answer a specification that has no closed form, and solving for the map instead of choosing it is the other. The other is to solve for the map directly, which is what a map with no formula does with a least-squares fit over a region’s boundary, and what the condition-solving ladder does throughout. The two meet on the question this rung leaves open: given a specification, when is a series the right representation of the answer, and when is it a formula being asked to work outside the region anybody meant it for?