Transverse Mercator and the series that computes it

Ordinary Mercator gets all of the argument and transverse Mercator carries all of the data. Every national grid, every UTM coordinate, essentially every survey measurement made in the last century is expressed in it, and it is the one projection in common use that cannot be written down.

The construction #

Take Mercator and rotate its axis by ninety degrees. That is the whole of the idea.

Ordinary Mercator wraps its cylinder about the rotation axis, so the equator is the line of contact and the distortion grows toward the poles. Transverse Mercator wraps it about an axis lying in the equatorial plane, so a meridian is the line of contact and the distortion grows east and west from it.

The aspect is a free choice and the property survives it exactly, because a rotation of the sphere is an isometry and an isometry changes none of the invariants. The site asserts that: an oblique projection’s principal scales, areal factor and angular deformation must equal its normal-aspect original’s at the corresponding rotated point, to a part in fifty million.

So on a sphere, transverse Mercator is Mercator, pointed elsewhere, and it needs no new mathematics whatsoever.

Why the ellipsoid ruins that #

On an ellipsoid the argument stops working, and the reason is worth being precise about because it is not obvious.

A sphere is symmetric under every rotation. An ellipsoid of revolution is symmetric only under rotations about its own axis. So rotating a spherical projection’s axis produces another projection of the same body, while rotating an ellipsoidal projection’s axis produces a projection of a different body — one whose flattening now runs the wrong way relative to the map.

There is no shortcut. Ellipsoidal transverse Mercator has to be derived from the conformality requirement directly, and when it is, the answer is not an elementary function.

What the exact projection is #

The requirement is stated in two lines. The projection must be conformal, and the central meridian must be at true scale.

The second condition fixes the northing on the central meridian: it must be the distance from the equator along the meridian, which is

$$M(\varphi) = a(1-e^2)\int_0^{\varphi}\left(1 - e^2\sin^2 t\right)^{-3/2}\,\mathrm{d}t$$

That integral is elliptic. It has no expression in elementary functions, and everything downstream inherits the difficulty: the conformal continuation of a boundary value off the central meridian is fixed uniquely by the Cauchy–Riemann equations, and it is not elementary either.

So the exact transverse Mercator exists, is unique, and cannot be written.

What is computed instead #

A power series in the third flattening $n = f/(2-f)$, which for the Earth is 0.001679.

That is a small number, and it is small because of the way it is constructed: the flattening itself is 0.00335 and $n$ is roughly half of it. A series in $n$ therefore converges extremely fast, and that speed is the entire reason a series is the practical answer rather than a numerical quadrature.

The standard form is Krüger’s, from 1912. It carries the geodetic latitude through the conformal latitude into a complex variable, applies four correction terms with coefficients that are themselves series in $n$, and returns easting and northing:

$$\xi = \xi' + \sum_{j=1}^{4}\alpha_j\sin 2j\xi'\cosh 2j\eta', \qquad \eta = \eta' + \sum_{j=1}^{4}\alpha_j\cos 2j\xi'\sinh 2j\eta'$$

with $\alpha_1 = n/2 - 2n^2/3 + 5n^3/16 + \ldots$ and so on. The inverse uses a second set of coefficients and a third set recovers geodetic latitude from conformal.

None of that is illuminating to read. What matters is the shape of the thing: the projection as specified is conformal and the projection as computed is not, by an amount that is known, bounded, and chosen.

What each term is worth #

The numbers are the argument for the whole approach. Across a zone extending three degrees either side of the central meridian:

• one term: worst error 4.9 metres
• two terms: 8.0 millimetres
• three terms: 17 micrometres

Each term buys between two and three decimal orders, and the fourth is below anything a survey can measure. A national grid formula written to four terms is exact for every purpose the grid has, and calling it an approximation is technically correct and practically misleading.

Widening the zone to six degrees barely moves the figures — 5.2 metres, 9.0 millimetres, 21 micrometres — which is the property that makes the series usable at all. What breaks it is going much further: the series is a local construction and it diverges, so a transverse Mercator computed twenty degrees from its central meridian is not a bad answer but a meaningless one.

That is why the zones exist, and it is a rather different reason from the one usually given.

The check that is not a longer series #

Comparing a truncated series against the same series with more terms is a weak test. It measures convergence and would pass just as well if every coefficient were wrong in the same way.

The site checks the series against something with no algebra in common with it.

On the central meridian the projection’s own specification says what the northing must be: the meridian arc, the elliptic integral above. That integral is evaluated independently by Simpson’s rule with four thousand intervals — a computation that shares nothing with Krüger’s coefficients — and the two must agree.

They do, to 1.9×10⁻⁷ metres across every latitude from the equator to 84°.

That is the assertion worth having. The series could be wrong in a hundred ways that a self-comparison would not see, and only one of them survives a comparison with the integral it is approximating.

The meridian arc has its own story. Its series was originally written here with the leading coefficient applied to every term rather than to the first — the obvious way to write it, since it looks like a common factor — and that is wrong by 11 millimetres at 45°. Eleven millimetres is small enough to read as arithmetic noise and large enough to matter, and nothing but the independent integral would have caught it.

The scale factor, and where it goes #

A tangent transverse Mercator is exact on its central meridian and increasingly too large away from it. The growth is quadratic to leading order:

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

where $\lambda'$ is the angular distance from the central meridian.

At three degrees from the axis on the equator that is 981 parts per million, or about a metre per kilometre — which is far too much for survey work, and is why every practical grid applies a constant scale factor slightly below one. That is the secant construction in its large-scale form, and its arithmetic is the subject of the next essay.

The scale factor is also isotropic, which is worth checking rather than assuming. A conformal projection has one scale factor per point, the same in every direction, and the site measures the meridian and parallel scales separately and requires them to agree — they do, to 1.1×10⁻¹¹. If they ever diverged, the projection would have stopped being conformal and every grid computation resting on it would be quietly wrong.

What conformality is doing for the surveyor #

Not an aesthetic property. A working requirement, and naming it explains the whole design.

A survey measures angles. A theodolite reads the horizontal angle between two directions, and the entire structure of a triangulation network is angles with a small number of measured baselines. If the map preserves angles, an angle measured on the ground can be plotted directly and an angle read off the map can be set out on the ground, with no correction.

If the map does not preserve angles, every observation needs a correction that depends on position and direction, and the correction has to be computed before the measurement can be used. That is not impossible; it is a permanent tax on every operation.

So conformality is the property that makes a grid usable as a computing instrument rather than as a picture. Naming the purpose before the property is the site’s general rule, and this is the clearest case of it in practice: the projection was not chosen because conformality is elegant but because a theodolite measures angles.

Grid north is not north #

One consequence of the construction is the quantity a surveyor thinks about most and a map reader never hears of.

On the map, the grid lines run straight up and down. On the ground, the meridians converge toward the pole. So except on the central meridian itself, the direction “up the grid” and the direction “toward the pole” are different, and the angle between them is the grid convergence:

$$\gamma \approx \lambda'\sin\varphi$$

to leading order, with $\lambda'$ the angular distance from the central meridian. At three degrees out and 55° north that is 2.46 degrees — nearly two and a half degrees between grid north and true north, at the edge of an ordinary zone.

That is not a distortion and it is not an error. It is a consequence of drawing converging lines as parallel ones, it is exactly computable, and every bearing taken from a map has to be corrected by it before it can be used against a compass. Ordnance Survey sheets print the convergence for the sheet, alongside the magnetic variation, which is a separate and larger correction of an entirely different kind.

The convergence is also a case of a quantity that depends on the coordinate system rather than on the map. It says nothing about how distorted the projection is — a conformal projection has zero angular deformation and a large convergence at the same point — and it is exactly what the reader needs, because the question mentions a direction.

Why it cannot be drawn whole #

A practical asymmetry with ordinary Mercator, which shows what the rotation has actually done.

Ordinary Mercator sends both poles to infinity and shows everything else. Transverse Mercator sends two points on the equator to infinity — the two 90° from the central meridian — and shows both poles perfectly well. The singularity has moved with the axis, as everything else has.

The consequence is that a transverse Mercator of the whole world is a strange object: the poles are ordinary interior points, and the map runs away to infinity in two places on the equator that have nothing geographically remarkable about them. It is drawable on a sphere, and the hero figure above is one.

On an ellipsoid it is not drawable at all past a few degrees, because the series diverges long before the singularity is approached. The exact ellipsoidal projection exists out there and no ordinary formulation computes it, so the projection that carries the world’s survey data is one nobody has ever seen a complete picture of.

The other way to compute it #

Krüger’s series is not the only method and it is worth naming the alternative, because it changes what “accurate” means.

Karney’s 2011 formulation carries the same series to eighth order and evaluates the conformal latitude by a different route, reaching nanometre accuracy over a zone and remaining usable much further from the central meridian. It is what the modern libraries use.

There is also an exact method, due to Lee, that evaluates the elliptic integrals directly rather than expanding them. It is slower, it is accurate everywhere rather than locally, and almost nobody uses it — because a fourth-order series is already exact at the scale of the thing being surveyed, and the exact method’s advantage only appears in a region where the projection would not be used anyway.

Which is the honest summary of the situation: the exact projection is available and nobody wants it, because the approximation’s error is smaller than the ground it describes moves in a year.

The series gives the coordinates. What a survey also needs from it is the direction of the meridian, which the same derivatives supply.

What was computed here #

The forward and inverse series are Krüger’s, to fourth order in the third flattening, with the coefficients written out rather than tabulated.

Three assertions hold them. The northing on the central meridian must equal the meridian arc obtained by numerical quadrature of the defining elliptic integral, which it does to 1.9×10⁻⁷ metres — the check that shares no algebra with the thing being checked. The series must converge, each order gaining at least a decade over the last across a three-degree zone. And the projection must pass the site’s ordinary conformality test against the WGS84 metric, which it does at 2.1×10⁻⁶ degrees, on the same noise floor as every other conformal projection in the library — which measure between 1.2 and 1.7×10⁻⁶.

The scale factor is measured from the projection’s own Jacobian rather than from the usual series formula, and the meridian and parallel scales are required to agree to 1.1×10⁻¹¹ — a conformal projection has one scale factor per point, and measuring it twice is the cheapest available check that it still does.

The projection is registered in units of the semi-major axis rather than metres. In metres its areal factor against a unit-normalised metric measures 4×10¹³, which is arithmetically correct, entirely a statement about units, and would have placed the most carefully-built projection in the library off the top of the audit plot.

What the pictures cannot show #

The zone. Every figure of a transverse Mercator on this page shows the whole world, and no grid uses more than a six-degree strip of it — the rest of the picture is a region where the series has no meaning at all.

The hero figure is drawn on a sphere, because the spherical transverse Mercator is a rotation and can be drawn exactly. The ellipsoidal one differs from it by amounts that are invisible at map scale and are the entire subject of the essay, which is the usual difficulty here: the thing being discussed is smaller than a line width and larger than the tolerance.

A different series, held to the same standard #

Molodensky’s formulae for a datum shift are the same discipline applied to a different expansion, and measuring them the same way produces the same kind of answer: an expansion is entitled to an error of its own order, and the way to find out what it is entitled to is to implement the exact route beside it.

The comparison is worth making because the two series fail differently. The Krüger series’ truncation error grows with distance from the central meridian and is bounded by the next term; Molodensky’s grows with the size of the shift and with the difference between the two ellipsoids, and its published abridged variant is a different formula rather than a shorter one — worse by a factor of fifty, as Molodensky’s shortcut measures.

Who found it, and when #

Lambert constructed the transverse Mercator in 1772, on a sphere, in the treatise that also produced the conformal conic and the cylindrical equal-area.

Gauss derived the ellipsoidal version during the survey of Hanover in the 1820s — the same survey that produced the Theorema Egregium — and did not publish the formulae. They were reconstructed and published by Schreiber in 1866, and the working series form is Louis Krüger’s, from 1912, which is why the projection is called Gauss–Krüger across most of Europe.

Charles Karney’s 2011 paper extended the series to eighth order and gave error bounds, which is the version in current software. Two hundred and forty years separate Lambert’s construction from a formulation accurate to a nanometre, and the projection is the same projection throughout.

Two hundred and forty years is worth pausing on, because the thing that changed over them was not the projection. Lambert’s construction, Gauss’s ellipsoidal derivation, Krüger’s series and Karney’s eighth-order extension with error bounds are four descriptions of one object, each accurate enough for the work of its period and each superseded by the arrival of a use that needed more digits. The projection did not improve; what improved is the arithmetic available to evaluate it, and the standard the arithmetic is held to.

Where this goes next #

The grid system built on it is UTM and the zone system. The rotation it depends on is the aspect is a free choice. And the body it is defined on is the Earth is a sphere, and when it is not.