A straight stick tests the index only across the sight

A water surface’s height trades against its index gave three eyes a stick half under water and asked them for two things at once: the height of the surface the stick goes through, and the index of the water below it. The eyes found both, in principle, and badly in practice. The two trade along a valley — a surface assumed higher bends every ray sooner, and an index assumed weaker makes up for it — and fits from readings good to a quarter of a pixel do not scatter about the truth. They slide down the valley, so that the index comes back 0.076 low on average against water’s 1.333, the surface 69 millimetres high, and the metre-long stick 881 millimetres long.

That essay triangulated each of the stick’s eight points on its own. Every point carried three coordinates of its own for the fit to adjust, against two numbers — the index and the height — that all of them shared, and that is the shape of problem in which a few shared numbers are known to be pulled off by many private ones. It ended by naming the test: fit the same readings with every stick held to a straight line, which takes the private numbers from twenty-four down to twelve, and see whether the slide goes. If it does, any rig whose objects have known shapes should fit the shapes. If it stays, it belongs to the refraction.

It stays — for the stick that essay used. And the reason it stays says exactly when it would not.

A wrong index draws a straight stick straight #

Before running any noisy fit, there is a question that settles most of it: what does a wrong index do to a straight stick? A constraint can only help a fit if the wrong answers break it. If a straight stick read with the wrong index still comes back straight, then “the stick is straight” is something the wrong answers already satisfy, and adding it to the fit tells the fit nothing it can use.

The rig is the earlier essay’s, built on the one a third eye sees the water the pair cannot introduced. A level pair of eyes 65 millimetres apart, 1.6 metres above the water, with a third eye raised a hundred millimetres above their middle; a metre-long stick leaning thirty degrees under the surface two metres out, leaning straight away from them; eight points along it. Read without error, every direction each eye reports is exact. Then the eyes are told a wrong index and asked for the surface’s height that best suits it, and the points they place are compared with the stick.

The answer is the whole of the essay in one picture. Told 1.25, the eyes put the surface 32 millimetres above where it is, and they put the stick back shorter, shallower and leaning a little further over — 884 millimetres long at 32.2 degrees from the vertical. They also put it back straight. Its eight points lie on one line to within sixteen thousandths of a millimetre, which is a thousandth of the width of the line drawing it. Told 1.40 instead, they lower the surface sixteen millimetres and return a stick 1,090 millimetres long, leaning a little less steeply, and straight again to fourteen thousandths. Across every index from 1.2 to 1.45 the stick’s straightness never moves by more than a few hundredths of a millimetre, while its length runs from 811 to 1,156.

That is a different statement from the one a straight stick in water is a kink and a curve made about the stick’s image. The image, seen by one eye, is kinked at the surface and curved below it; what three eyes rebuild from their images, at the wrong index, is straight. So a wrong index does not break the stick. It moves it, shrinks it or stretches it, and tilts it, and leaves it exactly the kind of object a straight-line fit expects to find. A fit that holds every stick to a line is asking the wrong answers to pass a test they all pass.

The line constraint buys nothing here, and costs a little #

That makes a firm prediction for the noisy fits, and the prediction holds. With exact readings, the disagreement among the three eyes at a wrong index is the same whether each point is free or held to the stick’s line — the same to a part in ten thousand at 1.25 — because the free points were already on a line. Linearised at the truth, the fit’s own curvature promises the index to ±0.143 either way, to three decimal places. The line constraint takes twelve private numbers away from the fit and removes nothing it was using.

The slide survives the constraint at every reading error, and it grows a little. At a twentieth of a pixel the line-held fits come back 0.011 low, just as the free ones do; at a quarter of a pixel they come back 0.099 low, against 0.076 for the free points, and one of the forty rigs runs off down the valley altogether to an index in the thousands — the same runaway the earlier essay met at 0.3 pixels with the points free, arriving sooner. The spread barely changes, 0.115 against 0.127.

Taking freedom away from a fit is supposed to help, and the reason it does not is the previous section: the private numbers were never what was pulling the shared ones off. The earlier essay’s own diagnosis is the one that stands. The valley the index and the height lie in is curved, noise pushes each fit a random distance off the truth, and the fit then settles on the valley’s floor on the side where the floor is gentler. That is a property of what a flat refracting surface does to rays from one stick seen from nearly one direction, and no description of the stick’s shape that a wrong index already satisfies can reach it.

The lower curve on the same figure shows what does reach it, and the rest of the essay is about that curve. Before turning to it, though, the stick has to be turned round.

Leaning across, the same index bows the stick #

The stick in the earlier essay leant straight away from the eyes. That is a special lean. A stick leaning away lies in the vertical plane through the eyes, and every ray from it to an eye bends within its own plane of incidence, the vertical plane containing the ray — which for this stick is nearly the same plane for every ray. The whole reading is close to a flat problem drawn on one sheet. A stick leaning across the line of sight leaves that sheet: the rays to its upper and lower ends cross the surface in different directions from the eyes and bend in different planes.

The difference is not subtle once it is looked for. At the same wrong index of 1.25 the stick leaning away comes back bowed by 0.016 millimetres, as before. Turned thirty degrees round it bows by 0.44 millimetres, at sixty degrees by 1.14, and leaning straight across the line of sight by 2.22 — a hundred and forty times as much, and in a smooth arc along its length, flat at the ends and furthest out in the middle. The length, meanwhile, is short by nearly the same amount at every lean: 116 millimetres leaning away and 95 leaning across.

So the measure of how much a straight-line constraint can tell a fit is the ratio between what a wrong index does to the stick’s shape and what it does to everything else. Leaning away, the wrong index changes the length seven thousand times more than it changes the straightness; leaning across, forty-three times more. That still makes the length the louder signal by far. But two millimetres is a bow a fit can see, where sixteen thousandths of one is not, and the question is what the fit does with it.

Why the bow appears in that direction and grows the way it does was not derived here. It was measured, and what matters for the fit is that it is there.

Across the sight, the line constraint works #

Turning the slider on the noisy fits to the stick leaning across gives a different figure, not a different shade of the same one. With every point free, the fits slide much as before: 0.010 low at a twentieth of a pixel and 0.058 low at a quarter, with a spread of 0.130. Held to a straight line, the slide is gone — 0.0004 low at a twentieth of a pixel and 0.004 high at a quarter, which is well inside the forty rigs’ own standard error — and the spread falls to 0.060. The fit’s curvature agrees: it promises the index to ±0.142 with the points free and ±0.052 with the stick held straight. No rig runs off.

This is the outcome the earlier essay’s question was hoping for, and it arrives only on the stick that was not used. Held to a line, a stick leaning across the sight is an object a wrong index cannot reproduce: every wrong answer bows it, the constraint refuses every bowed answer, and what is left is the truth. The private numbers were never the cause of the slide, but the shape is a cure for it — exactly where the shape is something a wrong index changes.

That gives a rule that is easy to apply and easy to get wrong. A known shape helps a refraction fit in proportion to how much the wrong refraction distorts that shape. Straightness in the plane the rays bend in is not distorted, and is worth nothing; straightness across it is distorted by millimetres, and is worth a factor of two in the index’s spread and the whole of its bias. A rig that wants to use the straightness of what it sees through water should look at objects leaning across its line of sight, and should expect nothing from objects leaning away.

What the length does #

The bow is a weak signal and the shortening a strong one. Leaning either way, a wrong index changes the stick’s length by about a hundred millimetres for a tenth of index, and a length is the easiest thing in the scene to know. It is one number a tape measure gives.

The tape enters the fit as one more reading beside the twenty-four rays: the stick’s length as the fit places it, compared with the length the tape says, weighed by how well the tape is read. The stick’s length is taken from all eight points rather than its two ends — their places along the stick, regressed on their number — so that no one point’s depth error carries straight into it; how each point is placed from its three bent rays is the angular choice the midpoint is a choice of ruler examined.

On the stick leaning away, where the line constraint did nothing, the tape does everything. The forty fits that slid down the valley without it gather on the truth with it: the index averages 1.337 against water’s 1.333, half a standard error away, with a spread of 0.049 rather than 0.127; the surface averages 8 millimetres high, ±27, rather than 69 ± 87. The valley is still there, and the tape crosses it. A fit sliding down the valley towards a weaker index shortens the stick, the tape says the stick is a metre, and the slide is refused at its first step. On the stick leaning across, the tape does as well as the line constraint did and a little better — a slide of 0.006 high and a spread of 0.039.

Exact readings with the tape come back to water’s index from a start at 1.25 to seven decimal places, and a tape that reads wrong is followed: told a stick is an eighth longer than it is, a fit moves the index strong to make the stick that long. The tape is trusted, which is what makes it useful and what makes it a thing that must be right.

A rough tape is nearly as good as a fine one #

How well does the tape have to be read? The natural expectation is that a length good to a millimetre would fix the index far better than one good to five centimetres. It does not, and a scale bar is worth its ends, not its tape found the same thing for a measured bar on a ring of cameras: the bar’s information is limited by how well the pictures place its two ends, not by how well the tape was read.

A tape read to ±5 millimetres and one read to ±50 give the same spread of 0.050 and slides of less than 0.004 either way. The slide begins to come back only past ±100 millimetres, where the index is 0.020 low, and at ±200 millimetres it is 0.053 low with a spread of 0.080 — still better on both counts than no tape at all. The reason is how well the three eyes themselves read the stick’s length when nothing is wrong. At the true index and the true surface, a quarter of a pixel of reading error leaves the read length scattered by ±66 millimetres from rig to rig, because each point’s depth is the least well measured of its three coordinates — the same weakness a point under water has two depths found in a single point’s image before any surface was fitted. A tape better than that adds nothing the eyes can use. A tape near that tells the fit which end of the valley it is at, and that is all the slide needed.

The other end of the scale is a ruler: a stick whose every mark is known, so that the fit can compare each point with where it truly is. With all eight points’ places known, the index comes back to ±0.0012, a hundred times tighter than without and forty times tighter than with a tape. That is a calibration target lowered into the water, and it removes the problem rather than reducing it. Between the ruler and nothing lies a single length, read roughly, which takes away the bias entirely and two-thirds of the spread.

A tape on one stick measures the others #

The use of all this is not the stick that was taped. It is everything else in the water, which the rig could not reach with a tape: a second stick further out, a stone on the bed, a fish. The earlier essay found that a second stick four metres out, added to break the valley, came back badly short when the index and the surface were both fitted, because its shallow rays pulled the fit to a weaker index still.

The tape on the near stick fixes the far one. With both the index and the surface fitted and nothing taped, the far stick comes back 687 millimetres long, a third short. With the near stick’s metre read off a tape and nothing else known, the far stick comes back 1,016 ± 236 millimetres, beside 1,007 ± 184 when the index is simply held at water’s value. It is the same move a length fixes the scale where it lies made for a ring of cameras in air, where one measured metre pinned a whole walk’s scale. The tape cannot do better than knowing the index — it is a way of learning the index, and it learns it to ±0.050 — but it does as well as knowing it on average, and it works for a liquid whose index nobody knows. The far stick’s spread is wide because it is four metres out and seen through the surface at a shallow angle; its bias is gone.

That is the practical form of the result. The stick a stereo pair puts back found a pair alone 285 millimetres short with nothing in its pictures to say so. A rig looking into a liquid it cannot identify, through a surface it cannot see, needs one object of known size in the near water. A stick with two marks a metre apart, a float of known length, a tile of known width on the bottom: any of them does what holding the index did, without the index.

What was assumed #

The stick is straight. The line constraint across the sight works because a wrong index bows the stick by about two millimetres, and it can only work if the stick itself is straighter than that. A real pole, a branch or a length of pipe may well bow by more.

The tape is right. The tape is weighed as a reading of known precision and is followed when it is wrong, as the test with a tape an eighth too long showed. A length mismeasured by more than the eyes’ own ±66 millimetres moves the index with it, without the fit saying so; the residuals cannot warn of a wrong tape any more than the residual does not warn found them warning of a wrong model in the pair.

The surface is flat and level, and the eyes know where they are. Both assumptions are the earlier essay’s and both are as strong here; a rig that does not know its own cameras carries their uncertainty into everything it measures, which is what an uncertainty is quoted from something found for a height read from a rig. A tape fixes where on the valley the fit lies; it does nothing for a surface that ripples or a rig that has to find its own cameras.

Each lean was tried at one distance. Every measurement of the lean was made at two metres out with a thirty-degree tilt. On other rigs tried for the same check — sticks further out, steeper and shallower — the bow under a wrong index ranged from a few hundredths of a millimetre to two millimetres and the length’s change always dwarfed it, but how the line constraint’s value varies across all of them was not mapped.

Still open: whether a stick that is not quite straight fools the line constraint #

The line constraint across the sight works on a signal of two millimetres. That is the bow a wrong index of 1.25 puts into a metre-long stick leaning across the line of sight two metres out, and it is the only thing the constraint has to work with. A real stick is never perfectly straight: a wooden pole sags, a branch curves, a steel rod is straight to a fraction of a millimetre over a metre but a plastic one is not. A stick that is itself bowed by a millimetre or two in the right direction would be read as a straight stick seen through the wrong index, and the fit would move the index to straighten it.

The measurement that settles it bends the stick on purpose — an arc with a stated sag, across the sight and along it — and fits it held to a straight line, to find the sag at which the constraint’s bias exceeds the slide it was removing. If that sag is well above what ordinary poles carry, the line constraint is a usable tool for a stick leaning across; if it is within a millimetre, a fit should hold a stick to a line only when the stick is known to be machined straight, and otherwise trust a tape.