Grid north is not north

A grid is a set of squares. Its north is the direction of increasing northing, it is the same direction at every point of the sheet, and that is the whole reason a grid is useful. It is also not north.

Three norths #

A survey has to keep three directions apart, and confusing any two of them is a mistake with a size.

True north is the direction of the meridian through the point: the way to the pole along the ground.

Grid north is the direction of the sheet’s own $y$ axis, which is the direction of the central meridian’s image, everywhere. It is a property of the coordinate system rather than of the ground, which puts it on the same side of the line as $h$ and $k$ in the invariants argument.

Magnetic north is where the compass points, which is a fact about the Earth’s field rather than about geometry and is not this essay’s subject. It is the one of the three most map readers have heard of, because it is the one printed in the margin — and it is the only one that changes from year to year.

The first two coincide on the central meridian and nowhere else. The angle between them is the meridian convergence, written $\gamma$, and it is a property of the projection that a bearing taken from a map and a bearing observed astronomically differ by.

The size of it #

Measured from the transverse Mercator’s own derivatives — the direction of the image of the meridian, which is the curve traced by increasing latitude:

Two to three degrees at the edge of a UTM zone. That is far larger than any error in the projection’s scale, larger than most survey tolerances by orders of magnitude, and it is not an error at all — it is a real angle between two real directions, and a survey that fails to apply it has simply used the wrong north.

The formula, and how much of it is a truncation #

Every survey manual gives

$$\gamma \approx \Delta\lambda \sin\varphi$$

which is a first-order truncation, and the amount it leaves out is measurable. Against the derivative-based measurement at the edge of a zone:

• at 20°, the manual’s form is 3.0 arcseconds short;
• at 45°, 3.5 arcseconds;
• at 60°, 2.2 arcseconds;
• at 75°, 0.6 arcseconds.

A few arcseconds is a real correction for precise work and negligible for anything else, which is why the first-order form survives. The next term is $O(\Delta\lambda^3)$, and the non-monotone pattern above is the product of a rising $\sin\varphi$ and a falling $\cos^2\varphi$ in that term.

There is a second, much smaller residual worth separating out. The exact spherical convergence is $\arctan(\tan\Delta\lambda \sin\varphi)$, and the difference between the measurement and that is the part the ellipsoid is responsible for: 13 milliarcseconds at 52°, and 31 at 20°.

Separating the two matters because they are different quantities. One is the price of a truncated series and is removed by using more terms; the other is the price of the Earth not being a sphere and is not. Both are asserted at their own size on every build, and a run in which either came out at the other’s magnitude would mean the derivative measurement was wrong.

The second correction: a straight line is not straight #

The convergence is about directions at a point. The other correction is about a line between two points, and it is the one that surprises people.

A geodesic on the ground — the line a theodolite sights along — is a straight line on the Earth and a curve on the grid. So a survey has two different angles available for the same observation: the bearing of the curve’s tangent at the standing point, which is what the instrument measures, and the bearing of the straight chord to the target, which is what a computation on grid coordinates uses.

The difference is the arc-to-chord correction, written $(t - T)$, and it is the second thing applied after the scale factor.

For a 67 km line at 52° north, the correction is 17.3 arcseconds at 1.5° from the central meridian — which over that distance is 5.6 metres of position. That is not a small quantity for a control survey.

Two routes to the same number #

The correction is computed here directly and checked against the classical approximation, which is the site’s ordinary habit and is worth spelling out because the two share no algebra.

Directly: take the geodesic’s azimuth at the standing point from the inverse problem, step one metre along it, project both points, and compare the direction that gives with the direction of the chord to the projected target.

Classically:

$$(t - T) \approx -\frac{(y_2 - y_1)(2x_1 + x_2)}{6R^2}$$

with $x$ measured from the central meridian and $R^2 = MN$ at the mean latitude.

The two agree to 0.013 arcseconds out of 17.3, which is 0.08%, and the assertion requires agreement within 5%. That the closed form is quite so good over a 67 km line is the useful part of the check: it means a survey can use the formula rather than the geodesic, and knows by how much.

Both corrections change sign #

The most useful single fact about both quantities is that they are antisymmetric about the central meridian.

The convergence at 2.5° east of the axis is $+1.9705°$ and at 2.5° west is $-1.9705°$, agreeing to a part in $10^6$. The arc-to-chord correction is $+17.326''$ on one side and $-17.326''$ on the other.

That is asserted on every build, and it is the check most likely to catch a sign error — which is the characteristic failure of this whole area of geodesy, since every formula involved has four sign conventions in circulation and the wrong choice produces a perfectly plausible number with the wrong sign. A quantity that is zero on the axis and antisymmetric about it cannot be produced by a formula that has lost a sign.

It is also the reason the corrections are easy to overlook. A survey confined to one side of a zone sees a systematic bias; a survey spanning the axis sees the corrections cancel in the mean and mislead in the detail.

What a survey does with them #

The three corrections, applied in order, are what turns an observation on the ground into a computation on the grid:

1. The line scale factor converts a measured distance to a grid distance. UTM’s 0.9996 is the constant part of it and the position-dependent part is the rest, running to 981 parts per million at the zone edge.
2. The convergence converts an observed azimuth to a grid bearing.
3. The arc-to-chord correction converts a grid bearing of the geodesic to the bearing of the straight line the coordinates imply.

After all three, plane trigonometry on grid coordinates gives the right answer, to the accuracy of the series the projection is computed by. That is the actual justification for using a projected coordinate system at all: not that the distortion is small, but that it is known and invertible.

That is the same bargain the zone system makes at a larger scale and the datum transformation makes at a different one: accept a known systematic difference and carry its inverse, rather than looking for a system without one. That distinction is the practical form of this site’s central claim. Measuring instead of naming is what makes a distortion usable, because a measured distortion is a transformation with an inverse and a described one is a warning.

The correction at other latitudes and other lengths #

Both quantities scale in ways worth knowing, because they decide when a survey can skip them.

The arc-to-chord correction is proportional to the length of the line and to its distance from the central meridian, so it is quadratic in the size of the job in the same sense every other quantity on this site is: a line half as long, half as far out, has a quarter of the correction.

The practical threshold follows. Below about 10 km of line length the correction is under an arcsecond anywhere in a zone, which is the reading precision of a good theodolite, so a local survey can ignore it. Above about 50 km it is tens of arcseconds and cannot be ignored by anything.

That is the same boundary the plane-survey limit draws for the scale factor, arriving from a different quantity, and the coincidence is not one: both are the curvature of the Earth multiplied by an area, and both cross the threshold of a survey instrument at about the same size of job.

Where the corrections come from geometrically #

Both are consequences of the same thing: the projection is conformal, so it preserves angles between curves at a point, and it does not preserve straightness.

Preserving angles is what makes the convergence well defined — the angle between the meridian’s image and the grid’s $y$ axis is the same as the angle between the meridian and the central meridian’s direction on the ground, so $\gamma$ is a genuine geometric quantity rather than an artefact of the drawing.

Not preserving straightness is what forces the arc-to-chord correction. A conformal map takes the geodesic to a curve whose curvature at each point is computable from the map’s scale factor, and integrating that curvature over the line gives the angle between the curve and its chord. Nothing about conformality helps here, which is the point the finite-angle essay makes in its own terms: conformality is a statement about infinitesimal angles, and a survey line is not infinitesimal.

The two corrections also behave differently under refinement, which is worth stating beside the point about the walker. Magnetic variation is measured and expires: it changes by several minutes a year, so a sheet’s printed value is wrong the moment it leaves the press and a reader has to age it. Convergence is computed and does not: it is a function of the position and the grid’s own parameters, both of which are fixed for the life of the grid, so the printed value is correct forever.

That is the opposite of how the two are treated. The variation, which decays, is the one a walker is taught to apply and to update; the convergence, which never decays, is the one nobody applies at all. The reason is only that one of them is larger today — and on a grid whose central meridian is far away, or at a high latitude, the ordering reverses and nothing in the training does.

The convergence is not an error, and the difference matters #

One framing point, because it decides how a survey treats the quantity.

The scale factor of a projection is a distortion: the map’s distance is not the ground’s distance, and the difference is something to be corrected away. The convergence is not like that. It is the correct answer to a well-posed question — what angle does the meridian make with the grid’s axis here — and there is no sense in which the map is wrong about it.

The consequence is that the convergence never goes away as the projection improves. A better projection has a smaller scale error; it has exactly the same convergence, because the convergence is a property of the grid rather than of how well the grid approximates the ground. Any coordinate system with straight parallel axes on a curved Earth has one, and the only way to have none is to have a separate grid at every point, which is what a geographic coordinate is.

That is the honest form of the trade a grid makes. A grid buys plane trigonometry, and it pays for it in three corrections — one that shrinks as the projection improves, and two that do not.

What a reader of a paper map does about it #

Nothing, usually, and the reason is worth recording since it is the commonest encounter anybody has with this quantity.

An Ordnance Survey sheet prints a small diagram in the margin showing the three norths and their separations at the centre of the sheet: grid north, true north, magnetic north, with the angles. For most British sheets grid north is within a degree of true north, because the national grid’s central meridian at 2° west keeps the whole country inside about three degrees of longitude either side.

A walker taking a bearing off the sheet and setting it on a compass applies the magnetic variation, which is several degrees and changes annually, and ignores the convergence, which is a fraction of a degree and does not. That is the right decision at walking accuracy, and it is the reason the smaller and more permanent of the two corrections is the one nobody has heard of.

The same picture nearer the equator, where the correction is smallest and still not negligible.

What was computed here #

The convergence is the direction of $(\partial x/\partial\varphi, \partial y/\partial\varphi)$ measured against the grid’s $y$ axis, computed by central differences on the Krüger series at $10^{-7}$ radians. It is compared with two closed forms — the manual’s first-order one and the exact spherical one — and the two residuals are asserted separately, at 0.5 to 60 arcseconds and 1 to 500 milliarcseconds respectively.

The arc-to-chord correction takes the geodesic’s initial azimuth from Vincenty’s inverse formula, steps one metre along it by Runge–Kutta integration of the geodesic equations, projects both points, and differences the resulting direction against the chord. It is checked against the classical closed form to 5% and required to change sign across the central meridian.

The convergence is asserted to be exactly zero on the central meridian, to within $10^{-9}$ degrees, and antisymmetric about it to within $10^{-6}$.

What the pictures cannot show #

The convergence figure draws meridians in grid coordinates over an eight-degree span of latitude, which exaggerates the fanning relative to a real map sheet — a survey sheet covers a few tens of kilometres, over which the fan is nearly a set of parallel lines two degrees off vertical.

The arc-to-chord figure plots a correction of a few arcseconds, which is far below the resolution of any drawing of the lines themselves. Nothing about the curvature of a projected geodesic is visible at map scale, which is exactly why the correction has to be computed rather than seen.

A fourth north #

So the count of norths at a point is not three but four: magnetic, grid, geodetic and astronomic. The first is a measurement of the field, the second a property of the projection, the third a property of the ellipsoid, and the fourth a property of where the rock is. The plumb line is not the normal takes the last of them, and the Laplace stations that studded every national triangulation exist because the correction between the last two accumulates along a chain.

Who found it, and when #

Both corrections are as old as national triangulation. Gauss’s Hanover survey of the 1820s needed them and Gauss derived them; the arc-to-chord correction appears in his work on the conformal projection of one surface onto another, which is the paper the transverse Mercator’s series comes from.

What has changed since is only who applies them. For a century and a half they were tabulated and applied by hand, which is why the first-order forms are the ones in the manuals; now they are applied by software, which is why a great many practitioners have never seen them and occasionally discover them the hard way — as a survey that closes several centimetres out on one side of a zone and not the other.

Where this goes next #

The ellipsoid ladder now runs from the shape of the Earth through the series that maps it to the corrections a survey applies to the result. What remains is the audit: a projection whose name promises something about angles, examined at the size angles are actually measured at.