Two short bars are not one long one

A scale bar is worth its ends, not its tape found that a length measured at the start of a walk round a ring pins the walk’s scale only to the precision with which the pictures place the length’s two ends — about three and a half millimetres there, whatever the tape says. A short length measured superbly is therefore not a long length measured well: a bar under about thirteen centimetres tells the walk less about its scale than the walk already knows.

A long bar is awkward to carry and a short one is not, and that essay ended with the obvious workaround. Two short bars, set some distance apart on the same wall, each measured, state two lengths; the distance between them is not measured but is read by the pictures like any other. If two short bars four metres apart state the scale as well as one four-metre bar, a ruler can be replaced by a pair of cheap targets. If they do not, the reason says whether a scale is carried by a measured distance or by a measured set of marks.

They do not, by a wide margin, and the reason is exactly the one that essay’s closing sentence named: a scale is carried by a measured distance. The span of the marks carries nothing.

Two bars at every gap #

The walk is the earlier one, unchanged: 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 loop is closed — the points seen at both ends are recognised as the same points — because on the open walk no length at the start does better than the drift the walk accumulates on its way round, as the earlier essay found, and the comparison would be between two answers that are both the drift.

The bars here are stood on end, one above the other on the wall the third camera faces, so that both are seen by the same three pictures wherever the gap puts them. Laid side by side along the wall instead, a bar moved a metre sideways leaves one camera’s view and enters another’s, and the comparison stops being about the gap. Each bar is 25 centimetres long and measured to a tenth of a millimetre; the gap between them is not measured at all.

With no bar the worst camera — the one across the ring — is 411 millimetres from certain along the line from the start. One 25-centimetre bar brings it to 222. Two bars touching bring it to 175, which is what two independent statements of scale should do: better than one, by less than a factor of two.

Then the gap opens, and the two bars get worse: 176 millimetres at half a metre apart, 192 at two metres, 219 at three and a half. One bar spanning the same arrangement — from the lower bar’s foot to the upper bar’s top, measured as one length — falls the other way, from 136 millimetres at half a metre to 72 at four metres. At the widest gap the long bar is three times better than the two short ones, and the two short ones are no better than one short one was.

So the gap buys nothing, and the slight loss as it opens is a separate, smaller effect: the bars move toward the top and bottom of the three pictures that see them, where those pictures place their ends a little less well. The main result does not depend on it. Side by side or four metres apart, two short bars are two short lengths.

What the walk then knows about its own scale #

The worst camera mixes two things — the walk’s scale, and the drift the pictures add on the way round — and a length can reach only the first. The figure below separates them by asking a question that has only scale in it: how well does the walk know the length of a four-metre line laid level on the same wall, never measured?

With no bar at all the walk knows that line to 2.56 per cent: it has a scale, set by its own first steps and carried round, and it is not a good one. One 25-centimetre bar states it to 1.35 per cent. Two bars state it to 1.03 per cent touching and 1.33 per cent three and a half metres apart. One bar over the span states it to 0.226 per cent at four metres — six times better than the two short bars at their widest.

The arithmetic behind those numbers is simple enough to do by hand, and it is the whole argument. A measured bar tells the walk how long a line the pictures place is. The pictures place a bar’s two ends, relative to each other, to a few millimetres; the tape states the bar’s length to a tenth of a millimetre; so the bar states the scale to the placing error over its own length. A 25-centimetre bar with its ends placed to four millimetres states the scale to about 1.5 per cent. Two such bars, independent, state it to about 1.1 per cent, and combined with the walk’s own 2.56 per cent, to about 1.0 — which is the 1.03 the figure measures. Nothing in that calculation contains the gap. The distance between the bars is not measured, so it is a length the walk must scale, not one it can scale by.

Why the gap cannot be read as a length #

It is worth being exact about why the pictures cannot supply the missing length themselves, since they plainly see both bars and the wall between them. Two views give shape and no size is the root of it: any number of photographs determine every ratio of distances in a scene and no distance at all. Seven numbers no picture can name counts what is left free — a position, an orientation and one scale — and the scale is the one a measured length exists to fix.

The pictures therefore know the gap between the bars perfectly well as a ratio: so many times the length of the lower bar, to the precision with which they place the marks involved. That ratio is exactly as good at three and a half metres as at half a metre, and exactly as useless for scale, because it converts a measured length into an unmeasured one without adding a measurement. It is the same move the one thing a single view cannot give describes for a single photograph: every ratio recoverable, no size. Two measured bars and an unmeasured gap are two sizes and one ratio, and the ratio says nothing the sizes did not.

The contrast with the eighth held number bends the scene is instructive. There, holding one number too many forced the reconstruction to bend. Here the second bar is not one number too many — it is a second statement of the same one number, the scale, and the adjustment simply averages the two statements. It cannot do anything else with them.

A length is known to its ends #

The number that sets everything is how well the pictures place each bar’s two ends relative to each other, and it can be read off the adjustment directly.

With the bars’ own measurements weighted out of the way, the pictures know each bar’s length to 7.47 millimetres. Part of that is the walk’s scale, 2.56 per cent of a quarter of a metre; the rest is the placing of the ends, 3.84 millimetres with the bars side by side. As the gap opens the ends are placed less well — 4.88 millimetres for the lower bar and 6.07 for the upper at the widest gap — because the bars have moved toward the edges of the three pictures that see them, where those pictures agree about a point less closely than they do near the middle of the frame.

That is the slight loss the first figure showed, and it is the only thing the gap does. The earlier essay found the same few millimetres of end placement on a bar lying flat, seen by more pictures, and called it the property of the walk that decides the shortest useful bar. Here it decides something more: that a bar’s statement of scale is its own business, made over its own length, and cannot be lengthened by putting another bar somewhere else.

A long length measured in pieces #

The same arithmetic makes a prediction about a long length that is not measured in one go — and a prediction that separates two ways of measuring it that look alike.

Measure the four-metre length as sixteen separate bars laid end to end, each 25 centimetres long with its own two marks — two marks a hair apart at every joint — and the worst camera is 95.7 millimetres out, a third worse than the 71.7 one four-metre bar gives. Measure it as sixteen pieces of one tape, each piece running from one mark to the next and every interior mark shared by two pieces, and the worst camera is 68.9 millimetres, slightly better than the single bar.

The difference is whether the pieces add up. The tape’s sixteen measured pieces sum to the whole length, and in the sum every interior mark appears twice with opposite signs: its placing error cancels, and what is left is the placing of the two outermost marks against a measured four metres. The tape is the long bar, plus fifteen more points for the pictures to agree about, which is where its slight gain comes from. The separate bars do not sum to anything. Each has its own two ends placed afresh by the pictures, and sixteen independent statements each good to four millimetres over 25 centimetres state the scale as one bar a metre long would — four metres divided by the square root of sixteen.

So the rule the earlier essay reached for is sharper than “a scale is carried by a measured distance”. It is carried by a measured distance between two marks the pictures place, and pieces are worth their sum only if their marks are shared. A surveyor’s tape read at every graduation is one long length. The same tape cut into sixteen bars and scattered is sixteen short ones.

The shape each leaves on the ring #

The figure below shows the whole ring, camera by camera, for the four cases that matter.

All four curves have the shape a closed loop always has — pinned at the start, rising to the far side, falling back — and differ only in how high the far side rises. The two short bars sit between no bar and a long bar, and much nearer no bar: 219 millimetres across the ring against 411 and 69. The tape read at every mark lies just under the long bar, at 65. Closing a loop mends its ends is the reason the curve comes back down at all: the closure carries the start’s scale round from both directions, and a loop’s far side is a length is the reason it is a length, not a picture, that lowers the far side’s peak. What these curves add is that the length has to be one length.

Which way the far side is uncertain #

A plan of the ring shows the same result as a shape rather than a height, and the shape says which direction of the far cameras’ uncertainty a length reaches.

With the two short bars, every camera across the ring carries an ellipse stretched along the line from the start: the worst is 219 millimetres along that line and 63 across it. The walk knows the far side’s bearing from the start and not its distance, which is what an uncertain scale looks like from the start’s point of view — every length from the start is uncertain in the same proportion, and a camera sixteen metres away, across the ring, is uncertain along its sixteen metres.

With one four-metre bar the ellipses are nearly round: the worst camera is 71 millimetres along the line and 64 across it. The scale is fixed, and what is left is the drift of directions round the ring, which has no preferred way to point. That is what a long length buys and two short ones do not: not a smaller error of the same shape, but a different shape, the stretched component along the line from the start removed. An uncertainty is quoted from something is the reminder that every ellipse here is drawn with the first camera held; held elsewhere they would sit elsewhere, and the contrast between the two figures would not change.

What a surveyor would take from it #

The practical content is short. A pair of cheap targets on a wall is not a scale bar, however far apart they are set, unless the distance between them is measured — and measured between the targets themselves, not between two bars that happen to carry them. A long reference is best carried as a tape with marks the cameras can see along its length: it is as good as a rigid bar of the same length and slightly better, and it rolls up. Several short bars are worth having only as several short bars, each stating the scale to its own end placement over its own length, and they add up in the way independent measurements do, as the square root of their number.

That last point bounds the substitute exactly. Sixteen separate 25-centimetre bars state the scale as one one-metre bar would. To match one four-metre bar by independent short bars would take the square of the ratio of the lengths — 256 bars of 25 centimetres — which is a strong argument for the tape.

A scale chain leans rather than wanders found that a scale passed from pair to pair down a street does not spread like a random walk; here scale is stated rather than passed, and it does combine like one. The difference is whether the statements share anything. Pieces of a tape share their marks and their sum is exact; bars share nothing and their statements merely average.

What was assumed #

The bars are seen by the same pictures. Stood on end on one wall, both bars are seen by the same three cameras wherever the gap puts them, which is what isolates the gap. Two bars on different walls are seen by different cameras and bring the walk’s structure into the comparison — a bar across the ring, for instance, restates the scale where the drift has accumulated, and that is the question of a length’s position rather than of its span.

Every end is a point the pictures read to a pixel. A bar’s end marks are targets as good as any scene point. Real targets are often better — coded, circular, read to a tenth of a pixel — and every placement error here would shrink with them, and every short bar’s statement of scale would improve in proportion. The comparison between short and long would not change, because both are limited by the same placing.

The walk’s own marks are unchanged. Adding a bar adds two points and a measured length and nothing else; the walk’s pictures and the scene they see are the same in every case. A tape read at every mark adds more points than two bars do, and part of its small advantage over a single bar is those points.

Still open: whether a length across the ring is worth more than a longer one at the start #

The first assumption set aside the other half of the question this sequence has been circling: not how long a measured length should be, but where it should be. The earlier essays found that on the open walk no length at the start helps past a certain size, because what remains is the drift, and that a tape across the ring and a length at the start did about equally well on the closed loop.

The measurement that settles where a length is worth most fixes its size — one metre, measured to a tenth of a millimetre — and moves it round the ring from the start wall to the wall across from it, on the open walk and the closed loop, recording the worst camera at each position. If the open walk’s worst camera falls sharply once the length is moved to the far side, a length is best placed where the drift has accumulated, and a survey should measure its reference where it is least sure rather than where it began. If the curve is flat on the closed loop, the closure has already carried every position’s scale to every other, and a length is worth the same wherever it lies — which would make the rule for a closed loop only a rule about length, and the rule for an open walk only a rule about place.