What a closed figure cannot see

Every survey specification leans on a closure check. Run a traverse round a loop, arrive back where it started, and the distance between where the arithmetic puts the finish and where the start actually is — the misclosure — is the number a job is accepted or rejected on.

A traverse must close establishes that this check is blind to a uniform scale error: multiply every leg by 400 parts per million and the figure closes to the last bit of a double while every dimension in it is wrong. That is one blind direction. This essay asks how many there are and how large they get.

Counting the conditions #

A closed traverse of $n$ legs carries $2n$ observations: a bearing and a length for each. It has to place $n-1$ intermediate stations, which is $2(n-1)$ unknowns. The difference is two.

Two conditions: the closing point must land back on the starting point in easting and in northing. Everything else the observations could be wrong about is unconstrained, and “everything else” is a space of dimension $2n - 2(n-1) - 2 = 0$ in the counting sense — every observation is used — but the check only tests two numbers.

That is the whole structure and it is worth stating in the abstract before the geometry. A check tests as many things as it has conditions, and a closure check has two. Any error in the observations that leaves those two numbers alone passes.

Which errors leave them alone #

Three, and they are a group rather than a list.

Scale. Multiplying every length by a constant produces a similar figure, and a similar figure that closed still closes. Measured: 400 parts per million leaves the misclosure below $10^{-9}$ metres while every dimension is wrong by 400 parts per million.

Rotation. Adding a constant to every bearing rotates the whole figure about its start, and a rotated closed figure is still closed.

Translation. Moving the start moves everything and closes as before, though translation is not an observation and so is not a blunder anybody can commit.

Those are the similarity transformations, and they are exactly the same group a datum fit absorbs: two parameter sets, one transformation finds a seven-parameter Helmert transformation with a near-null direction, and the direction is a translation traded against a rotation. A closed figure and a datum fit are blind to the same things, because in both cases the observations are invariant under the group.

The uneven part, which is the finding #

Blindness to the similarity group is exact and well known. What is not stated anywhere the site has found is that the sensitivity to everything else varies enormously around one loop.

An angle blunder at a station rotates everything downstream of it about that station, because a traverse’s bearings are carried forward from one angle to the next. So the closing point moves by

$$2D\sin(\delta/2)$$

where $D$ is the distance from the station to the closing point. That prediction is computed from the geometry alone and matched against the traverse’s own misclosure to four parts in $10^{11}$.

A factor of twenty-eight between the most and least visible station, and the ordering is entirely geometric: it is the order of the stations’ distances from the closing point.

The shape of the loop decides it #

This effect is invisible on the figure the site first used to measure it. A square traverse has every station at roughly the same distance from the close, so every station is roughly equally visible and the ratio is 1.6.

A real traverse runs round whatever the ground allows — a block, a field boundary, a road — and its last station before the close stands a few tens of metres from its first. That station is the blind one, and it is blind because of where the survey had to go rather than because of anything the surveyor decided.

So the rule is: the closure check is least sensitive to blunders at the stations nearest the closing point, and a loop whose last leg is short has a station that can hide almost anything.

A length blunder behaves differently #

Everything above is about angles, and the other kind of blunder — a length read wrongly, a tape misread, a prism constant forgotten — behaves in a way that is simpler and, for once, uniform.

A blunder of $\Delta$ in one leg’s length moves the closing point by exactly $\Delta$, whatever leg it is in and wherever that leg sits in the loop. The misclosure is the vector sum of the legs, so an error in one leg’s length adds to that sum directly.

So the closure check is uniformly sensitive to length blunders and unevenly sensitive to angular ones, and a specification’s single number therefore means one thing for lengths and seven different things for angles round a seven-station loop. That asymmetry is not in any specification the site has seen, and it follows from one line of vector arithmetic.

It also explains why the closure check’s blind directions are the ones they are. A uniform scale error is a length error in every leg, proportional to that leg — the one combination of length errors whose vector sum vanishes for a closed figure — and it is the exception to the uniformity above for exactly that reason.

What that means for a specification #

A specification stating “closure better than 1 in 20,000” is stating a tolerance on one number, and the table above says that tolerance corresponds to different angular tolerances at different stations — 2.1 arcseconds at one and 57 at another.

Nobody writes it that way, and the honest reading of a passed closure is therefore: no blunder large enough to move the closing point past the tolerance is present. That is a weaker statement than “no blunder is present”, and how much weaker depends on the loop’s shape.

The practical corollary a surveyor can act on: check the angles at the stations nearest the close by some other means, because the closure will not. A repeated round of angles there is worth more than at any other station in the loop, and the usual practice of observing every station equally is spending effort where the check is already strong.

What a second condition would buy #

The obvious fix is more conditions, and the classical figures are exactly that.

A traverse closed on azimuth — one that starts and ends on known bearings — has a third condition: the sum of the angles must equal the change in bearing plus a multiple of a straight angle. That condition is equally sensitive at every station, because every angle enters the sum with coefficient one, and it catches precisely the blunder the coordinate closure cannot.

A braced quadrilateral, observing all eight angles of a four-sided figure with both diagonals, has four conditions on eight observations. A triangle has one condition on three angles, and it is blind to everything about size and orientation — which is why a triangulation network is a chain of triangles and not a single one.

The pattern in the last column is the one worth carrying: a figure is blind to whatever its observations do not determine, and the classical figures were designed by working out what a given set of observations leaves free.

The projection’s own contribution to the misclosure #

A traverse computed on a grid carries one more term that is not a blunder at all, and it is worth separating out because it is the largest thing in the list on a long loop.

The grid’s scale factor varies across a zone, so the legs of a traverse are reduced by slightly different factors, and a loop that is exactly closed on the ground does not close exactly on the grid unless every leg’s own factor is used. On a British loop 20 kilometres across, the variation in $k$ across the loop is 4.9 parts per million, which over a 60-kilometre perimeter is 292 millimetres — fifteen times the tolerance this essay has been using.

There is a diagnostic in that. A misclosure that is proportional to the loop’s perimeter and in a consistent direction is a reduction error; one that is not is a blunder. The largest of the chain’s steps is the one nobody names — using a height above sea level where a height above the ellipsoid is wanted — which what a tape measures puts at 583 millimetres on a 77-kilometre line. The closure check reports a single number and throws that distinction away, which is one more thing it cannot see.

Why this is not least squares #

Nothing here fits anything. The quantities are geometric — a rotation about a point, and the displacement it produces — and the detectability threshold is a division: the tolerance over the misclosure a unit blunder produces.

That boundary is one this field holds strictly, and it is recorded in the site’s claim register. Network adjustment, weighting, variance and the whole apparatus of estimation belong to a numerical-methods subject and to a statistics subject respectively; what belongs here is the geometry of what a figure constrains, which is a question about ranks and distances rather than about distributions.

The distinction has a concrete edge. A least-squares adjustment would produce a test for a blunder at each station, with a power that depends on the same geometry — and the geometry is what this essay computes. The estimation theory sits on top of it and is somebody else’s.

Where the same counting appears elsewhere #

The structure — count the observations, count the unknowns, and the difference is what the check tests — is not particular to surveying, and the site has met it twice already in different clothes.

A grid has an origin that is not there is the one-parameter case: a false origin is a translation, no geometric measurement sees it, and it is therefore a convention rather than a quantity. What a grid is made of counts five declarations a coordinate does not carry and finds four of them invisible in the numbers.

In each case the question is the same: given these observations, which transformations leave them all unchanged? The answer is the blind set, and the blind set is what has to be supplied by declaration rather than by measurement. A survey supplies it with a starting coordinate and a starting bearing; a grid supplies it with a false origin; a datum supplies it with a constraint on the adjustment.

The check on the check #

Three clauses, and the first is the one that makes the rest a measurement.

The misclosure must be the rotation it is: $2D\sin(\delta/2)$, computed from the stations’ coordinates rather than from the traverse, matching to four parts in $10^{11}$. Without that the account of the mechanism would be a story that happens to fit.

A rotation about the closing point must produce nothing, exactly. Station 0 is the close, and rotating the whole figure about it is the rotation blindness, so its misclosure must be zero rather than small.

The stations must not be equally visible, by an order of magnitude. That is the refusal: on a square figure the ratio is 1.6 and the essay would have nothing to say, and requiring 10 forces the measurement onto a figure where the effect is real.

What to do about it, in four lines #

The measurement suggests a procedure, and the procedure is cheap.

Compute the sensitivity before the survey, not after. The distances from each planned station to the closing point are known from the plan, so the table in this essay can be written for a loop that has not yet been observed. It takes the station coordinates and one multiplication.

Re-observe the angles at the blind stations. The two or three stations nearest the close are where the check is weakest, and a second round there costs an hour.

Close on azimuth wherever a known bearing exists. The third condition is equally sensitive everywhere and is free if the endpoints are on control.

State the tolerance in the units the job is sold in. A closure in millimetres is a statement about positions; a client buying an area is buying a quantity whose error is twice the linear one, which an area on the grid is not an area on the ground measures.

Do not read a passed closure as a passed survey. It is a passed test of two numbers, with a sensitivity that varies twenty-eightfold around the loop, and a published coordinate is a result is the essay about what a passed survey actually asserts.

Why a check with uneven sensitivity is worse than a weak one #

The twenty-eightfold variation around the loop is the finding with the most practical bite, and it is worth separating from the fact that the check is weak.

A uniformly weak check is manageable. If a closure detected every blunder with the same low probability, a surveyor could reason about it: run more loops, tighten the tolerance, accept a known detection rate. The weakness would be a number, and numbers can be designed against.

An unevenly sensitive check cannot be reasoned about that way. A blunder in one leg moves the closure a great deal and a blunder of the same size in another barely moves it, so the check’s power depends on where the error is — which is precisely the thing nobody knows. The overall detection rate is a mixture over a distribution nobody has.

And the insensitive legs are not random. They are decided by the figure’s geometry, so they are the same legs every time the same shape of traverse is run, and an office with a house style has a systematic blind spot rather than an occasional one.

There is also a reading for whoever receives the survey. A closure figure quoted in a report is a single number, and its meaning depends on a geometry the report may not include. Two jobs quoting the same closure over the same total length can have detection powers differing by more than an order of magnitude, and nothing in the quoted number distinguishes them — so a closure is a claim about a particular figure rather than a comparable measure of quality.

Which is the same complaint this collection keeps making in different currencies. A summary statistic is quoted, it is portable, and the thing that decides what it means travels separately or not at all. Here the missing companion is the geometry, and unlike most of the others it is already in the job file — the coordinates of the stations are the whole of it, so the sensitivity could be printed beside the closure at no cost whatever.

Which turns the finding into something checkable in advance. The sensitivity of each leg comes from the geometry alone, before any observation, so a surveyor can compute which legs their closure will barely see and observe those more carefully — or add the tie that makes them visible. That is the same move the design analysis makes for a network, arriving in the simplest figure there is.

Where this ladder goes #

The reduction ladder has taken a survey from what a tape measures, through the traverse that must close, the two conventions for spreading a misclosure, the chain run backwards for setting out, the tolerance that decides which corrections matter, two grids over one piece of ground, and the published coordinate that is a result rather than a fact.

This adds what the closure check is actually testing, which turns out to be two numbers with an unevenly distributed sensitivity, and it does so without importing an estimator.

What is left is the figure that has more conditions than a traverse and fewer than a network — the classical braced figures, whose condition equations are geometric and whose design was the whole art of triangulation before least squares absorbed it. That is a subject with its own literature and this site has one essay’s worth of it: the counting above.