Setting out runs the chain backwards

A survey turns observations into coordinates. Setting out does the reverse: it takes coordinates off a drawing and turns them into an instrument reading and a direction to point in.

The reverse of a chain is not the same chain with the operations inverted. It is the same chain with the operations inverted and the order reversed, and for two of the corrections here the order matters.

What setting out actually needs #

The reversal is not only about bearings. Every step of the reduction chain has to be undone, and the list is worth writing out because it is not symmetric with the forward list.

A drawing gives two grid coordinates. The instrument needs a distance to measure on the ground and a direction to point in.

The distance is the grid distance divided by the combined factor — one division, and the order does not matter because all three factors are multiplications. That is what a tape measures run backwards, and it is the easy half.

The direction is the grid bearing corrected to an observable azimuth, which is the hard half and is this essay. It is hard for a structural reason: the distance corrections are scalars applied to a scalar, and the direction corrections are angles evaluated at positions that the corrections themselves move.

There is also a third thing setting out needs that surveying does not: an orientation for the instrument, established by sighting a known point. Every error in that backsight rotates the whole set-out, which is a similarity and is therefore invisible to any check performed on the set-out points among themselves — the argument a traverse must close makes about scale, applied to rotation.

The distance half reverses easily: three multiplications applied in any order, undone by three divisions in any order. The bearing corrections in this essay have no such freedom, and the difference between the two halves is that one operates on a number and the other on a direction at a place.

The two corrections #

Meridian convergence, $\gamma$, is the angle between grid north and true north at a point. On a transverse Mercator it is roughly $\Delta\lambda \sin\varphi$ and reaches two degrees at the edge of a UTM zone. It is a property of where the observer is standing, and it is the subject of grid north is not north.

The arc-to-chord correction, conventionally written $(t - T)$, is the angle between the straight line on the grid joining two points and the image of the geodesic between them, which is a curve. It depends on both points and on their positions relative to the central meridian, and on this line it is 6.2 seconds of arc.

An instrument sights along a geodesic — light travels the shortest path — so the direction it points is the tangent to the geodesic’s image, not the chord. The chord is what the coordinates give.

The correct order #

Going from a grid bearing to an azimuth to observe:

1. compute the grid bearing from the two sets of coordinates;
2. add $(t - T)$, converting the chord direction into the tangent direction — a grid-to-grid correction, computed from grid positions;
3. add the convergence, converting the grid tangent into a true azimuth — a grid-to-geodetic correction, computed from geodetic positions.

Step 2 lives entirely on the grid and step 3 crosses from the grid to the ellipsoid. Doing them in the other order applies the arc-to-chord correction to a quantity that is already a geodetic azimuth, which is applying it at the wrong point.

What the wrong order costs #

Four thousandths of a second of arc, which is 1.1 millimetres of lateral offset at the far end of the 51.8-kilometre line.

That is small. It is also real, and it is worth having the number, because the alternative to having it is one of two errors: believing the order does not matter, or being unnecessarily careful about an effect that is a thousandth of the corrections it sits between.

The bar chart puts each correction’s lateral offset beside it for exactly this reason. The convergence is 783 metres sideways. The arc-to-chord is 1.57 metres. The ordering error is 1.1 millimetres. A bearing error is a number nobody can picture; a sideways miss is the thing that misses.

Why a millimetre is worth measuring #

A millimetre at fifty kilometres is far below any setting-out tolerance, and stating it is not an argument that it matters.

It is an argument that the size is known. The alternative to knowing it is one of two failures, both common. A practitioner who believes the order does not matter will one day apply the same reasoning to the datum transformation, where the same structure costs 1.3 centimetres, or to a case where the two evaluation points are far apart and the cost is metres. A practitioner who believes the order matters enormously will spend care on a millimetre while the backsight orientation is contributing centimetres.

The value of a computed error bound is that it stops the argument, and this collection’s habit of computing rather than describing exists mostly for that. An unmeasured effect is argued about by people reasoning from its plausibility; a measured one is a number in a table.

Why bearings are where non-commutation lives #

The three multiplications in the reduction chain — slope, elevation, grid — all commute, because multiplication commutes. A job may apply them in any order and get the same distance.

Bearings are different because each correction is evaluated at a position, and the corrections change which position is meant. That is the general reason a chain of corrections has an order: not because the operations fail to commute as arithmetic, but because their arguments do.

The same structure produced a much larger surprise one field along. The seven parameters of a Helmert datum transformation do not invert by negation, because scale and translation do not commute, and undoing them in the wrong order costs 1.3 centimetres — too small to see, too large for a land registry, and caught only by requiring a round trip.

The third north, and the one that is not a correction #

A setting-out operation meets three norths and only two of them appear above.

Grid north is the direction of the northing axis, parallel everywhere on the sheet by construction. True north is the direction of the meridian, different at every point, and the convergence is the angle between them.

Magnetic north is the third, it differs from true north by the declination, and the declination is a property of the Earth’s magnetic field rather than of any coordinate system. It is tens of degrees in places, it drifts by fractions of a degree a year, and no projection has anything to say about it.

It is worth naming precisely because it is not a correction of the kind this essay is about. The convergence and the arc-to-chord correction are computed from the projection’s own derivatives and are exact; the declination is read from a model of a physical field that changes. A practitioner who groups all three as “the corrections to north” has put a measurement of the Earth’s core in the same list as two pieces of geometry, and only one of those has an error bar.

Two routes to the arc-to-chord correction #

The library computes $(t - T)$ two ways and requires them to agree.

The exact route projects a point one metre along the geodesic from the start, takes the direction of that step on the grid, and compares it with the direction of the chord to the far end. That is a measurement of the picture rather than an evaluation of a formula.

The classical route is the expression every survey manual gives:

$$(t - T) \approx -\frac{(N_B - N_A)(2E_A + E_B)}{6R^2}$$

with eastings measured from the central meridian. It is a truncation of a series and it is what a computation actually uses.

They come out at 6.2418 and 6.2374 seconds. The difference — 4.4 thousandths of a second — is the classical formula’s truncation error, and it is the same quantity as the ordering error above, because reversing the order is equivalent to evaluating the approximation at the other end of the line.

The assertion, bounded both ways #

The check requires the two routes to differ by more than 10⁻⁴ seconds and less than five seconds.

The upper bound catches the two routes having stopped being the same quantity — a sign error, a wrong radius, an easting measured from the false origin instead of the meridian.

The lower bound catches something a ceiling cannot: somebody quietly replacing one route with the other. If the exact computation were reimplemented as a call to the classical formula, the two would agree to the last bit and the check would report perfect agreement, which would prove nothing at all.

That trap is not hypothetical in this collection. The essays on datums found it in Clairaut’s residual, where an assertion bounded only above would have passed on a substitution of the theorem’s own prediction for the measured gravity flattening — and the fix was the same one, a floor as well as a ceiling.

Where the corrections change sign #

Both corrections change sign across the central meridian, and both are zero on it.

That has a practical consequence worth stating: a job entirely on one side of the meridian carries corrections of one sign throughout, and a job that straddles it carries both. The first case is a systematic offset that any external check will reveal; the second is a pattern that varies across the site and is much harder to spot, because it looks like random scatter with a structure nobody is looking for.

The same structure turns up wherever the corrections are large rather than small. Molodensky’s formulae evaluate a three-parameter datum method against the exact route, and their residual grows with latitude — the cost of an ordering choice scales with the size of the corrections, not with the size of the thing being corrected.

When the order stops being a footnote #

The millimetre here is small because the two evaluation points are close. It is worth asking when they are not, because the answer marks the boundary of the essay’s own reassurance.

The gap between the two orderings grows with the difference between the positions the two corrections are evaluated at, and on a bearing computation that difference is the correction itself — an angle of a few seconds. So the effect is second order in a small quantity, and it stays a millimetre for any line a setting-out operation would ever use.

It stops being second order when a correction is large. The datum transformation is the standing example: there the two operations move a point by hundreds of metres, so their arguments differ by hundreds of metres rather than by seconds of arc, and the non-commutation is 1.3 centimetres rather than a millimetre — a hundred times worse, from the same structure.

The rule that falls out is a useful one to carry: the cost of the wrong order scales with the size of the corrections, not with the size of the thing being corrected. A chain of tiny corrections may be applied in any order; a chain containing one large one may not.

What was computed, and how #

The grid bearing, from the two points’ grid coordinates.

The convergence at the observer’s station, by differentiating the transverse Mercator numerically — not from the $\Delta\lambda \sin\varphi$ approximation, which is a first-order truncation and is the quantity the exact computation is checked against.

The arc-to-chord correction, both ways.

The ordering difference, and its lateral offset at the line’s far end.

Two assertions, bounded above and below.

Where the model stops #

The line is short by geodetic standards. At 51.8 kilometres the arc-to-chord correction is seconds; at 300 kilometres it is minutes, and the classical formula’s truncation grows faster than the correction does. Nothing here explores that regime, because a setting-out line of 300 kilometres is not a thing anybody does.

The instrument is assumed perfect. A real setting-out operation carries pointing error, target centring error, and the accumulated orientation error of whatever backsight established the instrument’s zero — all of which are larger than the millimetre this essay measures. The ordering error is worth knowing about not because it dominates but because it is the one term that is a choice rather than an instrument limitation.

And the vertical is absent. Setting out in three dimensions adds the deflection of the vertical, which relates an observed azimuth to a geodetic one and is the Laplace correction. That is a relation rather than a value under this collection’s own ruling on the geoid — the plumb line is not the normal sets out why — so it is named and not computed.

The generalisation #

Inverting a chain reverses its order, and an operation whose argument the other operations change does not commute with them.

The useful form of that is a diagnostic rather than a rule. To find out whether two corrections commute, ask what each is evaluated at. Two multiplications applied to a scalar commute. Two corrections applied to a direction, one computed from grid coordinates and the other from geodetic ones, do not — and the size of the non-commutation is the difference between evaluating one of them at the other’s input and at its output.

That is why the number here is small: the two evaluation points are close together, so the difference is second order. It also explains why the datum case is larger — there, the two operations move a point by hundreds of metres, so their arguments differ by hundreds of metres rather than by a few seconds of arc.

Who worked it out, and when #

The arc-to-chord correction is as old as the transverse Mercator’s use as a survey grid, which in Britain means the 1930s and in Germany somewhat earlier — the Gauss–Krüger tradition had it first. The name $(t - T)$ comes from the German literature, where $t$ is the grid bearing and $T$ the geodetic azimuth reduced to the grid.

The formula every manual gives is a two-term truncation chosen because it could be evaluated on a slide rule from quantities already on the computation sheet. That constraint is long gone and the formula has stayed, which is the same persistence the scale factor of a line finds in Simpson’s rule — and with the same justification, in that the truncation’s error is genuinely below what matters for the lines it is applied to.

The difference is that Simpson’s rule is exact for its integrand, so refining it buys nothing, while the arc-to-chord formula really is an approximation with a measurable error. The four thousandths of a second in this essay is that error, and it is the reason the exact route exists in the library at all.

What a satellite receiver did to the order #

The chain above is stated for a total station and a computation sheet, which is where the corrections were worked out and where they are still taught. Most setting out today begins with a satellite receiver, and it is worth being clear about what that changed, because the answer is not the corrections went away.

A real-time kinematic receiver reports grid coordinates directly. The surveyor stakes a point by walking until the handheld says the designed easting and northing, and no bearing is computed, no arc-to-chord correction is applied by anybody on site, and the order this essay is about is never chosen by the person doing the work.

The chain is still there. It has moved inside the firmware. The receiver observes in a global frame, applies a datum transformation, projects, and applies whatever scale treatment the job’s coordinate system specifies — the same operations in the same sequence, and with the same non-commutation between them. What has changed is who decides the order and whether the decision is recorded.

Usually it is not. A receiver’s coordinate configuration is a menu, the menu’s items are named after outputs rather than after operations, and two receivers configured to the same named system by two surveyors can differ in exactly the way this essay measures. The millimetres are the same millimetres; the difference is that they are now attributable to nobody.

Which is why the hand computation is still worth being able to do. Not to do it, but to have a number to check against — the one circumstance in which a discrepancy between two instruments becomes a question with an answer rather than a disagreement between two black boxes.

Where this goes next #

The corrections so far have been within one coordinate system. What happens when a job meets a second one is two grids over the same ground, and which of all these corrections a job may skip is the tolerance decides the model.