A traverse must close

A traverse walks from a known point through a series of legs and returns to where it started. If every leg were measured perfectly the figure would close exactly, and it never does. The gap is the misclosure, and it is the check every survey specification in the world is built on.

It cannot see a scale error. Not approximately, not with difficulty — not at all.

Why the scale error vanishes #

Scaling every leg of a closed figure by the same factor produces a similar figure, and a similar figure is still closed.

The proof is one line. A closed traverse is a set of vectors summing to zero:

$$\sum_i \mathbf{v}_i = \mathbf{0}$$

Multiply every leg by $1 + \varepsilon$ and the sum is $(1+\varepsilon)\sum \mathbf{v}_i = \mathbf{0}$ still. The misclosure is exactly zero, for any $\varepsilon$ whatever.

There is nothing subtle here and nothing approximate. A closure test is a test on the shape of a figure, and a scale error does not change a shape.

The other errors that leave a figure closed #

A scale error is the clearest case and it is not the only one. Any transformation that maps a closed figure to a closed figure is invisible, and there are four of them.

A translation. Adding a constant to every coordinate leaves every difference unchanged, which is the false origin’s whole argument, and leaves the misclosure exactly zero.

A rotation. Turning the whole figure — which is what a wrong orientation at the starting station does — preserves every length and every internal angle.

A uniform scaling, as above.

A reflection. Swapping the two coordinates transposes the figure and preserves every distance, which is why the axis-order error in what a grid is made of is so hard to catch.

Those four together are the similarity group of the plane, and a closure test is precisely a test of the figure’s geometry modulo that group. Everything the group contains is invisible; everything outside it shows up. Stated that way the check’s competence is completely specified, and it is a more useful thing to know than any rule of thumb about acceptable misclosures.

What that means in practice #

A job run entirely in ground distances, on a grid that expects grid distances, has every leg wrong by the combined factor — around 400 to 600 parts per million on a typical British site. The traverse closes perfectly. The misclosure is reported as one part in several million, the specification is satisfied, and the coordinates are the right shape and the wrong size.

The perimeter of the figure in the hero above is out by 1.17 metres on a 2,920-metre traverse. The misclosure is two ten-thousand-billionths of a metre.

This is the same failure as the units are part of the coordinate describes for the US survey foot, and the same as a grid scaled to the ground is not a map describes for a surface system used where a national one was expected. All three are similarities, all three preserve every ratio and every angle, and all three are invisible to a closure by exactly the argument above.

What the check does see #

The pairing is what makes the figure honest. Alongside the scale error, the same traverse with one angle disturbed by twenty seconds of arc — a much smaller disturbance in its own units — fails to close by 62 millimetres, or one part in 47,054.

Most specifications would reject that. So the closure test is not broken and is not weak; it is a sensitive and useful check that answers a different question from the one it is usually credited with. It tests the angles and the relative lengths. It says nothing about the absolute scale.

That distinction has to be asserted in both directions, which is what the machinery does: the scaled figure must close to below a nanometre, and the bent figure must fail. A check demanding only the first would pass on a traverse routine that had stopped computing anything at all.

Where a misclosure comes from #

If the misclosure is not measuring scale, what is it measuring?

Everything that is not a similarity. Angular errors, both random and systematic. Individual leg lengths that are wrong — a mistranscribed number, a target set on the wrong mark, a prism constant applied twice. Instrument maladjustment that biases some observations and not others. And, importantly, corrections applied to some legs and not others, which is the failure mode of a job interrupted and resumed.

What unites them is that they are not uniform. A misclosure is a measure of the inconsistency between the observations, and a uniform error is perfectly consistent.

The precision figure, and what it is not #

A misclosure is conventionally quoted as a ratio: the linear misclosure over the perimeter, expressed as one part in something. One in 8,344 for the blundered figure above; one in 47,054 for the angular error; effectively one in infinity for the scale error.

That number is often read as an accuracy statement, and it is not one. It is a consistency statement, and the two differ precisely by the class of errors that leave a figure consistent.

A traverse closing at one in fifty thousand has demonstrated that its observations agree with each other to one part in fifty thousand. It has demonstrated nothing about whether all of them share a bias, and the biases that matter most in this field — scale factor, elevation factor, unit, datum — are all of exactly that kind.

How large a scale error can hide #

It is worth asking whether the blindness has any limit at all, and over a real traverse it very nearly does not.

The argument above is exact for a plane figure, so the only thing that could break it is the plane approximation itself. A traverse computed in grid coordinates is a plane figure by construction, so a uniform scaling of its legs is exactly a similarity and the misclosure is exactly zero — the 2×10⁻¹³ metres measured is floating-point residue rather than a small real effect.

A traverse computed on the ellipsoid is a different matter, because scaling the legs of a figure on a curved surface does not produce a similar figure. But no traverse is computed that way: the reason the reduction chain exists is precisely to get the observations onto a plane where a traverse can be computed, which is what a tape measures. So the blindness is not an artefact of an approximation but a consequence of the computation being done in the place it has to be done.

What does catch a scale error #

Three things, and none of them is internal to the traverse.

A tie to external control. Two known points fix the scale, because a similarity has one scale parameter and two points constrain it. This is the standard remedy and it is why specifications require traverses to start and finish on published marks rather than merely to close on themselves.

A measured baseline. A single distance between two marks whose separation is known independently does the same job with one observation.

Computing the same figure two ways. A distance derived from grid coordinates and a distance derived from latitude and longitude are independent routes to the same quantity, and a scale error affects the first and not the second.

All three work for the same reason: they bring in information from outside the figure, and the errors this essay is about are exactly the errors that are invisible from inside it. An error a survey can find on its own is an error that breaks its own consistency, which is the general statement and is why the remedy is always independence rather than care.

What a specification should require instead #

If a closure cannot see a scale error, and a scale error is the most common serious failure in the field, a specification that requires only a closure is requiring the wrong thing.

The remedy is already in most specifications and is usually presented as a matter of good practice rather than as the load-bearing requirement it is: begin and end on published control. A traverse between two known marks is checked in position, orientation and scale simultaneously, because the two endpoints constrain all three. A loop closing on its own start is checked in none of them.

The difference is not a matter of degree. A loop traverse and a link traverse are checks on different things, and a specification that treats them as interchangeable at the same misclosure ratio is treating a consistency test as an accuracy test.

Worth stating because the loop is the easier field procedure and is what gets done when control is far away or hard to reach. The saving is real; what is given up is the entire class of errors this essay is about, and it is given up silently.

What was computed, and how #

A closed quadrilateral with sides 820 and 640 metres, bearings at right angles, so that the true figure closes exactly by construction — which is necessary, because a figure that did not close before the error was applied would make the measurement meaningless.

The same figure with every leg multiplied by 1.0004.

The same figure with one bearing disturbed by twenty seconds of arc.

Three assertions. The scaled figure must close to below a nanometre; its perimeter must nonetheless be wrong by more than half a metre; and the bent figure must produce a misclosure large enough that a specification would reject it.

The assertion that produced this essay #

The traverse machinery was written expecting a scale error to show up as a misclosure, because that is what a misclosure is for. The assertion said so and it failed, reporting a misclosure of 2×10⁻¹³ metres.

That is not a bug and the check was not mis-tuned. It was the measurement contradicting the intended claim, which is the outcome this collection’s habit exists to produce — and the finding was interesting enough that the whole essay was rebuilt around it rather than the check being softened.

The repaired assertion says the opposite of the original and is stated as three parts, because a check that merely demanded blindness would pass on a traverse routine that had stopped working. Requiring the blindness, requiring the size error to be real, and requiring the angular case to be caught is what pins the behaviour down.

Where the model stops #

This is a plane traverse. Everything here is computed in grid coordinates on a flat sheet, which is what a traverse computation actually is and is an approximation whose limits are set out in how small is flat enough. Over a few kilometres the error is far below the misclosures being discussed.

No angular closure is computed. A real traverse has an angular misclosure as well as a linear one, checked against the sum of interior angles, and the two are usually reported separately. The figures here disturb a bearing directly rather than an angle at a station, which is the same effect with a simpler bookkeeping.

The traverse is a quadrilateral. Real traverses have twenty or fifty legs, and the arithmetic of the misclosure changes in one respect: with more legs, random angular errors accumulate rather than cancelling, so the misclosure a specification permits is usually stated as a function of the number of stations. None of that touches the scale argument, which holds for a figure of any number of sides — the sum of the vectors is zero or it is not, and multiplying them all by a constant does not change which.

And nothing here is statistical. How large a misclosure ought to be, given the number of legs and the instrument’s precision, is a question about estimation and belongs to a different collection — normaldistribution.xyz owns it. What is owned here is the geometric fact that a class of errors produces no misclosure at all, which is true regardless of any distribution.

The difference between the two traverses can be counted rather than described, and the counting comes out exactly even. The similarity group of the plane has four parameters — two of translation, one of rotation, one of scale — and those four are precisely what a closure cannot see. A link traverse that begins on one published mark and ends on another supplies four equations: two coordinates at each end. So the two numbers match, and they match for a reason rather than by luck: two known points is the smallest amount of external information that pins a similarity, and it is exactly the amount a link traverse brings.

One known point supplies two equations and fixes only the translation, which is why starting on control and closing on yourself is not half a check but a quarter of one — it catches a position error and leaves the rotation and the scale exactly as free as the loop did.

The generalisation #

A self-consistency check cannot detect an error that acts on the whole system uniformly. Stated that way it is nearly a tautology, and it is violated constantly, because self-consistency checks are cheap and independent checks are not.

The cartographic case is a good teaching example because the uniform errors are not exotic. They are the ordinary corrections — a scale factor, an elevation factor, a unit — which is to say the failure mode is not some unlikely conspiracy but simply forgetting a step, and forgetting a step is the most common thing that happens on any job.

The word that carries the confusion #

Much of the trouble is in the word closure itself, which is used for two different things.

A traverse closes when its figure returns to its start. A survey closes on control when it agrees with independently known coordinates at its end. The first is a statement about the observations’ agreement with each other; the second is a statement about their agreement with the world.

They are reported the same way — a distance, converted to a ratio — and the ratio for the second is usually worse, because it includes everything the first cannot see. A crew reporting a closure of one in fifty thousand has said something quite different depending on which of the two is meant, and the two numbers are not comparable at all.

This is a smaller instance of a pattern that runs through the whole collection: a word that names a property, used as though it named a measurement. A projection is called conformal because that is its name; a traverse is called closed because it returned to its start. Both readings substitute a label for a quantity, and the remedy in both cases is to say which quantity was computed.

Who established it, and when #

Traverse closure as a formal check dates from the eighteenth century and the ratio convention from the nineteenth, when it entered the specifications of the national surveys. Its blindness to scale was understood from the beginning — which is precisely why those specifications require a traverse to begin and end on established control rather than on itself.

The requirement is often read as being about propagating coordinates from the control network, and it is at least as much about checking them. A closed loop that starts and ends on the same unknown point is a check on consistency; a traverse that starts on one known mark and ends on another is a check on scale, orientation and position as well, and the difference between them is the whole of this essay.

Where this goes next #

Given a misclosure, something has to be done with it, and there are two classical rules that do different things and cannot both be describing what happened: the two ways to spread a misclosure.