A scale bar is worth its ends, not its tape

A loop’s far side is a length found that twenty-four cameras walked round a ring of walls are uncertain, on the far side of the ring, almost entirely along the line from the start: the walk knows the far side’s bearing and not its distance. A picture across the ring barely helped. A single measured length did more than recognising the loop’s own closure — a four-metre length on the start wall took the worst camera from 520 mm to 325, and a tape across the ring did slightly better. The essay noted that the two lengths, one near the start and one across the ring, gave almost the same result on the open walk, and read that as a sign that what matters is a length’s size against the short first step that set the walk’s scale, not where the length lies.

That invited an exact question. If a length’s worth is its size against its accuracy — how many parts per thousand it states the scale to — then a ruler measured very well at the start is as good as a tape measured roughly across the ring, and a walk’s scale can be fixed with a ruler. If instead a short length is too local for the pictures to carry, there is a shortest useful length, and it is a property of the walk rather than of the ruler.

It is the second, and the property turns out to be one number: how well the pictures place the two ends of the length relative to each other.

A bar of any length, at the start #

The walk is the earlier one, unchanged: twenty-four cameras on a circle 8 m from the centre of a ring of facades 13 m out, give or take two, each looking out through a 100° field, marks read to a whole pixel, every figure a linearised covariance at the true configuration in the gauge that holds the first camera. The only addition is a scale bar: two new points laid level on the wall the third camera faces, a stated length apart, each read in every picture that sees it, with their separation measured to a stated accuracy. The bar sits a little way along the walk so that both its ends are seen only by cameras near the start; a bar on the first camera’s own wall is also seen by the last cameras, and on an open walk that makes it half a loop closure.

The hero figure sweeps the bar’s length from 5 cm to 6 m with the length measured to a tenth of a millimetre — far better than any tape — and reports the worst camera’s uncertainty along the line from the start. With no bar, the worst camera is 496 mm out on the open walk and 411 mm with the loop closed.

Closed, the bar helps at every length and keeps helping. A 5 cm bar brings the worst camera to 386 mm; 25 cm, to 204; a metre, to 89; four metres, to 68; six, to 66. The last few metres buy little, because by then the bar has stated the scale about as well as the rest of the loop can use it, and what remains is the direction a length cannot reach — the across component, 64 mm, set by the walk’s own geometry.

Open, the bar helps and then stops. A 5 cm bar gives 458 mm, 25 cm gives 247, and from half a metre on every bar gives 214 mm. A metre, two, six: the same. The slider degrades the measurement from a tenth of a millimetre to 2 cm, and the shape survives until the tape itself becomes the weak part.

So there are two questions in the earlier one, and they have different answers. How short can a length be and still state the scale? That depends on the length and not on the ruler, and the next two figures say why. How much can a length at the start do? On a closed loop, nearly everything the far side lacks; on an open walk, a fixed amount and no more, for a reason the last figure shows.

A better tape buys nothing past a few millimetres #

The ratio hypothesis makes a clean prediction: halve the measurement’s error and the bar is worth twice its length.

It fails at once. Measured to 0.03 mm or to 0.3 mm, a one-metre bar leaves the worst camera at 89 mm either way; measured to 1 mm, at 90. A 25 cm bar gives 204 mm at 0.03 mm and 209 at 1 mm. The curves are flat until the measurement’s error reaches about 3 mm, and only then does the tape start to matter. For the 4 m bar the flat stretch runs out to 10 mm.

Each flat stretch sits at its own height, set by the length: 332 mm for a 10 cm bar, 204 for 25 cm, 89 for a metre. So at a fixed ratio of accuracy to length the worst camera is not fixed. A 25 cm bar measured to 1 mm — four parts in a thousand — leaves 209 mm; a metre bar measured to 4 mm, the same four parts in a thousand, leaves 108; a four-metre bar measured to 16 mm, 91. The length itself carries information the ratio does not.

The pictures place the ends to a few millimetres #

What sets the floor is visible if the bar’s own measurement is taken away and the pictures are asked how well they know its length.

The pictures know a bar’s length to $\sqrt{(0.0255\,L)^2 + (3.4\ \text{mm})^2}$, to within two per cent across lengths from 5 cm to 6 m. The first term is a fixed share of the length — 2.55 per cent — and it is the walk’s own knowledge of its scale near the start: a bar’s length is only as well known as the ruler the walk is measuring it with. The second term is a fixed amount, 3.4 mm, and it is how well the pictures place the bar’s two ends relative to each other. Each end is a mark read to a whole pixel in the handful of pictures that see it, at five metres through a 100° lens, and that is what a pixel is worth there.

Now the floor explains itself. A bar measured to an accuracy $\sigma$ tells the walk its scale to about $\sqrt{\sigma^2 + (3.4\ \text{mm})^2}/L$ — the measurement’s error and the ends’ placing, over the length. Once $\sigma$ is well under 3.4 mm the placing dominates, and the tape stops mattering; that is the knee of every curve in the previous figure. And the bar’s statement of the scale improves the walk’s own 2.55 per cent only when $3.4\ \text{mm}/L$ is smaller than it, which is at lengths above 13 cm.

So the shortest useful length is 13 cm, and it is a property of the walk: the placing error of a mark over the walk’s own scale uncertainty. A 5 cm bar measured perfectly states the scale to 7 per cent, worse than the walk already knew it; it still helps a little, because two independent statements of a quantity are better than one, but it is not a ruler in any useful sense. A walk whose marks were read ten times more finely would place the ends to 0.34 mm and could use a bar a tenth as long.

The needles become discs #

The earlier essay drew each camera’s uncertainty as the ellipse it is and found every one of them a needle pointing at the start: the walk knows its far side’s bearing and not its distance. A length is a statement about distance, so it should shorten the needles and leave their width, and the two bars on either side of the useful threshold show how far that goes.

With a 25 cm bar the needles are shorter — the worst camera is 204 mm along the line from the start where the closure alone left 411 — and they are still needles, three times as long as they are wide, still pointing at the start. The bar has stated the scale to about three per cent, a little better than the walk knew it, and the far side’s distance is correspondingly a little better known. The across component, 64 mm, has not moved at all: a length says nothing about bearing.

With a 4 m bar the needles become discs. The worst camera is 67 mm along the line from the start and 64 mm across it, and the other far cameras are the same shape. The length has done everything a length can: it has stated the scale well enough that the far side’s distance is no longer its weakest property, and what remains is the error in bearing and position that the closure and the pictures leave, which is about the same in every direction. A longer bar would shorten the along component by a few more millimetres and could never go below the across one, since nothing about a length reaches a direction.

That is the clearest way to see what the shortest useful length means. Below it, a bar leaves the needles as they were, because it says less about the scale than the walk already knew. Above it, the needles shorten toward the width they always had. And the width is where the far side’s uncertainty comes to rest once the scale is fixed.

On an open walk a length at the start runs out #

The open walk’s plateau at 214 mm is a different limit, and the profile round the ring shows what it is.

Without a bar, the open walk’s uncertainty along the line from the start is an arch: zero at the first camera, 496 mm across the ring, and back down toward the last camera, which stands beside the start. That arch is the walk’s scale, set by the first step and carried round: a scale error makes every camera too far or too near the start in proportion to its distance from it, and the far side is furthest.

A metre bar at the start removes the arch. What is left is a different curve: it grows steadily from the start, reaching 132 mm across the ring and 214 mm at the last camera, and it does not come back down. That is drift — the error each step of the walk adds to the next, accumulating in order round the ring — and it is exactly what a scale chain leans rather than wanders measured along a street and a chain and an adjustment singled out as where a sequence genuinely loses ground. No length at the start can reach it, because it is not an error in the start’s scale; it is error added after the start. Every bar longer than half a metre removes the arch completely and leaves the drift untouched, which is why the plateau is flat.

Closing the loop is what reaches drift, since the closure ties the end of the walk to its start and the drift has to fit between them — the mechanism closing a loop mends its ends measured camera by camera, each helped in proportion to how much the closure shortened its way back. With both, the worst camera is 87 mm across the ring, and the drift and the scale are both mostly gone. That is the result the earlier essay reached by a different route — the closure and a length together did far more than either — and the profile says why: the length removes the arch, the closure pins the ends of the drift, and neither can do the other’s job.

What a ruler at the start is for #

The practical answer to the earlier essay’s question is now short, and it is less convenient than the ratio would have been.

A walk’s scale can be fixed with a ruler at the start, but the ruler has to be long enough that its ends, placed by the pictures, state the scale better than the walk already does. On this walk that is anything over about 13 cm for a start, and in practice a metre or more, where the length’s own share of the error dominates the placing of its ends. Measuring the ruler to better than a few millimetres is wasted effort, because the pictures cannot place its ends that well. A survey is trusted at its own accuracy found the adjustment’s side of the same bargain for surveyed points: a reference entered with an accuracy it does not have costs the shape, and one entered more loosely than it deserves costs almost nothing. A scale bar measured to a micron and entered as such is safe, since its ends’ placing, not its stated accuracy, is what limits it; the effort spent on the micron is simply not repaid.

And on an open walk the ruler at the start fixes the start and nothing after it. An uncertainty is quoted from something found that the gauge holding the first camera is the right report for exactly the question asked here, how far each camera is from where the walk began; with the start’s scale fixed by a bar, the answer to that question for the far side is the drift, and only a closure or a second length further round can shorten it.

What the bar leaves out #

Where along the walk the bar lies. The bar here sits near the start. A bar across the ring, or half-way round, would fix the scale where it lies and leave drift on both sides of it; a loop’s far side is a length found a tape across the ring worth slightly more than a length at the start, and how the two compare once both are measured well was not re-measured here.

Marks read more finely. Every number assumes whole-pixel reading. The placing error of a bar’s ends scales with the reading error, so a matcher reading to a tenth of a pixel places them to about a third of a millimetre, and the shortest useful bar falls with it. Whole pixels cut space into shells found the same currency deciding a stereo pair’s depths: what a reading is worth is set by how finely its marks are read, and a better instrument elsewhere in the chain does not buy it back.

A bar that is not level. The bar lies level on the wall. A bar set vertically or along the line of sight is placed by the pictures with a different error in each direction, and its ends may be much worse placed along the line of sight, where a mark’s depth is the weakest thing a picture reads.

Still open: whether two short bars are one long one #

A long bar is awkward to carry and a short one is not. 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.

The measurement that settles whether they substitute for one long bar sweeps the gap between two 25 cm bars from zero to four metres and compares the worst camera with a single bar as long as the whole arrangement. If two short bars four metres apart state the scale as well as one four-metre bar, the length that matters is the span of the measured marks rather than any one of them, and a ruler can be replaced by a pair of cheap targets. If they do not, the shortest useful length found here applies to each bar separately and the span buys nothing — and the reason would say whether a scale is carried by a measured distance or by a measured set of marks.