A length fixes the scale where it lies, and the worst camera nowhere

Two short bars are not one long one settled how long a measured length has to be: long enough that the pictures’ few millimetres of uncertainty about its two ends are a small share of it, and in one piece, since two short lengths with an unmeasured gap between them are two short lengths. It set aside the other half of the question. Every length in that essay lay on the wall the third camera faces, near the start of the walk. A length can be put anywhere, and the essay ended by asking whether it matters where.

It proposed a test with two outcomes. If moving one metre bar round the ring to the far side made the open walk’s worst camera fall sharply, a survey should measure its reference where the walk is least sure. If the closed loop’s curve was flat, a closure carries every position’s scale to every other and position does not matter there.

The second half came out as predicted. The first did not, and the reason is that the worst camera was the wrong thing to watch.

The worst camera does not care where the bar is #

The walk is the one this sequence has used throughout: twenty-four cameras on a circle eight metres from the centre of a ring of facades thirteen metres out, each looking outward through a hundred-degree field, marks read to a whole pixel, every figure a linearised covariance at the true configuration with the first camera held. The bar is one metre long, measured to a tenth of a millimetre, and laid level on the wall a stated camera faces. It is moved from the third camera’s wall to the twenty-third’s. The walls beside the start are left out, because a bar there is seen by the first pictures and the last and would close the open walk by itself.

On the open walk the worst camera, which is the last one, sits between 199 and 225 millimetres from certain along the line from the start wherever the bar is — least with the bar on the ninth camera’s wall, most with it on the twenty-third’s. With no bar at all it is 496. The bar takes more than half the worst camera’s uncertainty away and takes about the same share from any wall. On the closed loop the worst camera, now across the ring, sits between 80 and 103 millimetres against 411 with no bar; the bar takes three-quarters away, from anywhere.

So the prediction for the closed loop held, and the one for the open walk failed: moving the bar to where the walk is least sure does not rescue the least sure camera. A loop’s far side is a length found that the worst camera’s uncertainty is mostly along the line from the start and that only a measured length reaches it. That is still true here — the bar is what takes the 496 down to 210 — but once the bar has fixed the walk’s scale, what is left at the worst camera is the drift of directions round the ring, and a length says nothing about directions wherever it lies.

Where the bar moves the error to #

The worst camera is one number, and it hides what the bar’s position does to the others.

With the bar at the start the uncertainty grows steadily from the first camera: 56 millimetres at the seventh, 132 at the thirteenth, 214 at the last. With the bar across the ring the early cameras are less sure — 113 at the seventh — and the middle and late ones about as sure as before. With the bar near the end, the seventh camera is 153 millimetres out and the thirteenth 212, while the last is still at 215.

The shapes say what the bar is doing. Each camera’s uncertainty, measured from the first camera, is part scale — its distance from the start known only as well as the walk knows its own scale — and part drift. A bar near the start fixes the scale where the early cameras are, so their distances from the start are good. A bar near the end fixes the scale where the late cameras are, and the early cameras have to borrow it back along the walk, through every picture in between. The last camera’s uncertainty is dominated by the drift either way, so it barely moves. The bar changes where along the walk the error grows; it does not change how much the end has accumulated. Drag the bar round the ring and the dip in the profile follows it: laid on the ninth wall, the bar brings the thirteenth camera down to 101 millimetres, the best that camera does anywhere, while the last camera never leaves the band from 199 to 216.

The scale is local on an open walk #

The worst camera mixes scale and drift, and the question the earlier essays kept returning to was the scale alone. The way to ask it is the one two short bars are not one long one used: lay an unmeasured four-metre line on a wall and ask how well the walk knows its length. That question has no drift in it, only scale, and it can be asked on every wall.

The answer is plain. With no bar the walk knows lengths to about three per cent everywhere, a little worse as it goes. With the bar at the start it knows a length beside the bar to 0.41 per cent, a length across the ring to 1.08 and a length at the far end to 1.58. Move the bar across the ring and the curve moves with it, 0.43 per cent at the bar and about one per cent at both ends. Move it near the end and the best reading goes there too. The scale is good where the bar is and worse with every picture between the bar and the line being measured.

That is a local scale, and it is what an open walk is. Each picture shares points with its neighbours, and those shared points carry the scale from one pair to the next with a small loss at each hand-over, so a scale stated at one wall is known less and less well further along. A scale chain leans rather than wanders measured the same hand-over along a street and found that it leans rather than wanders; here the loss is steady too, from 0.41 to 1.58 per cent over twenty walls, rising a little with every wall and never recovering.

So the answer to “where is a length worth most” on an open walk is: where the lengths that matter will be measured. A surveyor who needs to measure a doorway on the far wall should put the reference there, and one who puts it at the start has measured the far doorway to a per cent and a half when a bar beside it would have given four tenths.

What carries a scale from one wall to the next #

It is worth being exact about what is handed over, because it explains both the loss and why a closure repairs it. Seven numbers no picture can name counts what a set of photographs leaves free: a position, an orientation and one scale, for the whole scene at once. The walk has one scale, not twenty-four. A bar anywhere fixes that one number exactly as well, in principle, as a bar anywhere else.

What differs is how well the pictures tie each wall’s lengths to that one number. Two views give shape and no size is the root of the tie: every ratio of distances in the scene is determined, including the ratio of a length on the far wall to the bar on the start wall, and that ratio is determined only as precisely as the chain of pictures between the two walls allows. Each picture shares points with the next, those points place the next picture relative to this one, and each placing is a little uncertain. The ratio between two lengths on walls twenty pictures apart inherits twenty placings’ worth of uncertainty; the ratio between two lengths on one wall inherits almost none.

So “the walk’s scale” is a single number whose value is known well, but “the length of a line on the far wall” is that number times a ratio, and the ratio is where the loss lives. A chain and an adjustment found that composing poses link by link is not what a joint adjustment does, and the figures here are joint adjustments throughout; even so, a joint adjustment of an open walk has only the one route from the bar to the far wall, and every link on it costs something. A closed loop has two routes, and the shorter one is never longer than half the ring. That is the whole difference between the open walk’s figure and the closed loop’s.

The closed loop carries the scale both ways #

Closing the loop changes the picture completely, and it is the case the earlier essay predicted correctly.

On the closed loop, a bar anywhere gives a length beside it to about 0.4 per cent and a length anywhere else to at most 0.72. The bar’s own wall is still the best place to measure, but no wall is more than a factor of about 1.7 worse. Closing a loop mends its ends found that recognising the points seen at both ends carries the start’s geometry round the ring from both directions; here it carries the bar’s scale the same way, so the walls furthest from the bar along the walk are reached from the other side, and none is more than half the ring’s hand-overs away.

This is the sense in which a closed loop’s rule is a rule about length and an open walk’s rule is a rule about place. On the loop, where a single reference lies hardly matters, to the worst camera or to any length; how long it is and how well its ends are placed decides everything, which is the whole of the two essays before this one. On the open walk the reference’s position decides which lengths it serves, and its length decides how well it serves them.

Several references spread along an open walk #

If the scale is local, one reference cannot serve a whole open walk, and the natural remedy is several.

One bar at the start leaves the worst wall at 1.58 per cent. Moving it to the middle of the walk helps, to 1.21, because no wall is then more than half the walk from it. Two bars, one at each end, bring every wall under 0.80 per cent, and three — at the start, across and at the end — under 0.65. The scale is held everywhere once references are spread so that no wall is far from one.

The worst camera barely moves through all of it: 214 millimetres with one bar at the start, 214 with two at the ends, 180 with three. Spreading lengths holds the open walk’s scale; it does not hold the walk’s shape, whose error at the far end is directions accumulated round the ring. Only a closure — a recognition that the last pictures see what the first ones saw — or a measured direction reaches that, which is the division a loop’s far side is a length drew between what a picture across the ring buys and what a tape buys, read now from the other side.

A longer bar cannot reach the far end #

One more possibility would make the position of a bar matter less: a longer bar at the start, stating the scale so well that even after twenty walls of hand-overs it is still good. The figure below tries it.

Beside the bar, length pays as the earlier essays found: a line on the bar’s own wall is known to 1.27 per cent with a 25-centimetre bar and to 0.21 per cent with a four-metre one, each doubling of the bar cutting the error by nearly half at first and by a quarter by the end, as the ends’ placing stops being the larger part of it. Twenty walls away it barely pays at all. The far end’s line is known to 1.98 per cent with the short bar and 1.53 with the long one, and the curve is flattening towards a floor. The floor is the hand-over: whatever scale the start states, twenty walls of shared points carry it to the far end with about one and a half per cent of loss, and no bar at the start can state it better than the walk can carry it.

A bar of the same length at the far end gives the far line exactly what the start’s bar gave the start’s line — 0.41 per cent for a metre — because to the scale, the two ends of an open walk are the same kind of place. The best a single reference can do for a length is to lie beside it.

What the measurement says to a surveyor #

On a closed loop, put a reference anywhere convenient and make it long — long enough, as a scale bar is worth its ends, not its tape priced it, that the pictures’ few millimetres of uncertainty about its two ends are a small share of it; the loop will carry it. On an open walk, the reference’s position decides which lengths it serves: put it beside what will be measured, and if lengths will be measured along the whole walk, put several along it, spaced so that no measured thing is more than a few pictures from one. Neither choice does much for the camera furthest from the start on an open walk, whose error is the walk’s turning, and which only a closure fixes.

An uncertainty is quoted from something is the standing caution for all of these numbers: every camera’s uncertainty here is quoted from the first camera, which is the question a surveyor usually has, and the lengths are quoted as percentages of themselves, which is the question a builder has. The first moves with the gauge and the second does not. The division this essay draws — shape along the walk, scale beside the bar — is the division between those two questions.

What was assumed #

The bar is level on a wall and seen by three or four pictures. A bar seen by more pictures, or by pictures from both halves of the walk, is placed better and hands its scale over more widely; the walls beside the start were excluded for exactly that reason, and a bar on them would have closed the open walk.

The walk’s pictures and scene are the same in every case. The bar adds two points and a measured length and nothing else. A real survey that carries references along a walk usually also carries targets, which add points and strengthen the hand-overs between pictures, so the local character of the open walk’s scale would be less marked.

The lengths measured are on the walls. A four-metre line on a wall is seen by the same pictures that see the wall; a length across the ring, between two walls, is known through the drift as well as the scale, and would behave like the worst camera rather than like the lines here.

Still open: whether a bar’s direction on the wall decides what it fixes #

Every bar here lies level along the wall. A bar stood on end fixes the same scale but is seen differently — its two ends one above the other, in the same pictures — and two short bars are not one long one stood its bars on end for exactly that reason. A level bar’s ends, a metre apart along the wall, may fall in different pictures near a wall’s edge, which ties those pictures to each other through a measured distance as well as a scale.

The measurement that settles whether that matters lays the same metre bar level and upright on each wall, moves it from the middle of a picture to the seam between two, and asks two things: whether a level bar across a seam between pictures holds the open walk’s scale further along than an upright one on the same wall, because it also ties the two pictures’ relative position, and whether it moves the worst camera at all — which would make a measured length a partial witness of direction after all, provided it spans the join between two pictures rather than lying inside one.