Crooked sticks cancel each other's bows slowly, and eight of them match a tape

A bowed stick is read as a wrong index found the price of the cure an earlier essay had found for a refraction fit’s slide. Three eyes 1.6 metres above a pool read a submerged stick and fit the water’s index and its surface’s height together; left free, the fit slides down a valley towards a weaker index, and holding the stick’s points to one straight line stops it — for a stick leaning across the line of sight, where a wrong index bows the read stick visibly. But a stick that is itself bowed by a millimetre, in the plane where a wrong index would bow it, is to the line constraint a straight stick seen through the wrong index. Read without error, it moved the fitted index by 0.074 one way or 0.062 the other. A tape on the stick’s length did not care about the bow at all.

That essay ended on the obvious repair for a rig that cannot find a straight stick: use several crooked ones. Each is bowed by a millimetre or so in a direction of its own, so each pulls the index by an amount and a sign of its own, and a fit that holds every stick to a line combines them. The question was whether the pulls average away as one over the square root of the count — eight crooked poles doing what one straight one did — or whether sticks at different distances and leans respond so differently that one dominates; and how many crooked sticks it takes to match a tape, which the essay expected to be a large number.

The pulls cancel, and the bias is gone #

The rig is the earlier essays’: three eyes 1.6 metres up, a stereo pair 65 millimetres apart and a third eye 100 millimetres above them, cameras of 900 pixels’ focal length, eight points read on each stick. The sticks are a metre long and lean thirty degrees, standing between 1.6 and 3.2 metres out and up to 0.8 metres either side, and each is bowed a millimetre at its middle in a direction round its own axis drawn at random. The index and the height are fitted with every stick held to its own straight line.

Read without error, one crooked stick leaning across the sight leaves the index ±0.037 from water’s 1.333 over thirty draws of its bow’s direction — the earlier essay’s 0.07 for a bow in the worst direction, softened by every direction that is less bad. Two sticks leave it ±0.029, four ±0.024, eight ±0.016. The mean over the draws is a few thousandths at every count, −0.004 for one stick and 0.0005 for eight: no bias is shared by every draw. That is the first half of the question answered in the hoped-for direction. A bow is a bias for the stick that has it, but bows in random directions are noise for a set of sticks, and noise averages.

More slowly than the square root #

The second half is how fast.

Eight equal, independent pulls would leave a set’s error at one stick’s over the square root of eight, a factor of 2.8: from 0.037 to 0.013. The eight crooked sticks reach 0.016, a factor of 2.4. Two reach 0.029 against the 0.026 independent pulls would give; four reach 0.024 against 0.019. The set behaves like a set of fewer, equal sticks — about five and a half sticks’ worth from eight — because its sticks do not pull equally and the fit does not listen to them equally.

Sticks leaning whichever way they were pushed in start worse: one such stick leaves the index ±0.053 out, half again as far as one leaning across the sight, and eight leave it ±0.017, about where the eight leaning across end. The next section is why.

A stick pulls hardest just off the line of sight #

How hard a bow pulls depends on how the stick stands, and not in the way the earlier essays might suggest.

A stick standing 2.4 metres out and leaning straight away from the eyes is pulled 0.0015 by a millimetre of bow in its worst direction: a wrong index does not bow such a stick either, so the line constraint has nothing to mistake the bow for, and it barely moves the index. Leaning fifteen degrees off that line, the same bow pulls the index 0.100 — sixty-six times as far. Leaning across the sight it pulls 0.052, and leaning straight towards the eyes, 0.0025 again. A pull is a ratio: the bow a millimetre adds, over the bow a unit of index adds. Across the sight, the index’s own bow is large and a millimetre is worth little of it. Leaning nearly along the sight, the index barely bows the stick, while a bow in the stick still shows in full; a small bow is then a large change of index. Only exactly along the sight, away or towards, do both vanish together.

Distance matters as well. Leaning across the sight, a stick 1.6 metres out is pulled 0.099 by its millimetre, at 2.4 metres 0.052, at 3.2 metres 0.040; and at every lean the nearest stick pulls hardest, up to 0.146 for one 1.6 metres out leaning forty-five degrees off the sight. The same millimetre is a larger angle at the eyes from nearer, and the rays’ disagreement the line constraint minimises is measured in angles. So in a set spread from 1.6 to 3.2 metres, the nearest sticks pull hardest — and they are also the ones the fit learns the index best from, so their pulls carry the most weight. That is the reason the set cancels slowly: a few sticks dominate it. It is not one stick dominating, which was the question’s worse case; it is the near half of the set.

The random-lean sets start worse for the same reason. A stick that happens to lean fifteen to forty-five degrees off the line of sight is the hardest-pulled stick there is, and a set of one at random leans has a fair chance of being one of them.

At a quarter pixel, eight crooked sticks are one tape #

The reading error does not stand still while the bows average, and the comparison that matters is the one with error in it.

With every observation read to a quarter pixel, one straight stick held to a line reads the index to ±0.068, and two, four and eight straight sticks to ±0.041, ±0.034 and ±0.026: more sticks are more readings, and they also cross the fit’s valley at more angles. Bowed a millimetre each at random, the same sticks read it to ±0.087, ±0.056, ±0.048 and ±0.037. The bows cost the set about a third more error at every count — they never become negligible, because the reading error shrinks with the count as fast as the bows’ pull does.

One straight stick with its length taped to a millimetre, its points left free, reads the index to ±0.037. Eight crooked sticks held to lines read it to ±0.037. The earlier essay’s guess — that matching a tape would take a large number, since the tape’s length was never in the bow’s reach — was half right. The tape is out of the bow’s reach. The number is eight.

A tape does not notice, and adds nothing to eight #

The last figure sets the instruments side by side.

A tape on one bowed stick reads the index to ±0.041, against ±0.037 on a straight one: a millimetre’s bow shortens a metre’s chord by a few thousandths of a millimetre, far below the tape’s own millimetre, and what little the bow costs comes from the free points following the bowed shape a little less well. The same bowed stick held to a line reads ±0.087, more than twice as badly. For one crooked stick, a tape is the instrument.

For eight, it no longer is. Eight bowed sticks held to lines, ±0.037; the same eight with one of them taped, ±0.037. Taping one stick of eight adds nothing the eight did not already know, because the tape and the set of lines are now measuring the index about equally well and the tape is one reading among nine. Eight straight sticks held to lines, ±0.026, are better than either: the best of all is still a set of straight sticks, which is the remedy the earlier essays started from and could not always have.

A set of eight is worth about five and a half #

The slow cancellation can be put as a count. If eight sticks pulled equally and independently, the set’s error would be one stick’s over the square root of eight. It is one stick’s over 2.4, which is what five and a half equal sticks would give. The two and a half sticks that went missing are the far ones, which pull less and are listened to less; they still add readings, which is why the set keeps improving, but they add less to the cancellation than a near stick would.

Whether moving the far sticks nearer would help was not measured, and it is not obvious: a near stick is weighted more and also pulls harder, and the two work against each other. The arithmetic of a set of crooked sticks is the arithmetic of any weighted average of independent errors: the effective count is the square of the sum of the weights over the sum of their squares, and a set whose weights are uneven has fewer effective members than members.

The random-lean set makes the same point from the other side. Its sticks’ pulls range from almost nothing, for a stick that happens to lean along the sight, to the largest there is, for one fifteen degrees off it; and since the near, hard-pulled sticks dominate, a set at random leans is at the mercy of whichever near stick leans worst. At eight sticks the luck evens out, and the random set ends within two thousandths of the across set.

Where the set’s knowledge of the index comes from #

None of this would matter if the fit had another way to the index. It does not, and the earlier essays are the reason. A third eye sees the water the pair cannot found that two eyes cannot tell a submerged stick from a dry one at another place, and that the third eye’s disagreement with the pair is the whole of the evidence for the water; the stick a stereo pair puts back found what the pair alone reconstructs instead. The disagreement is small, and the line constraint is what turns it into an index by saying what shape the stick must have. A crooked stick says the wrong shape, and that is the only channel through which a bow reaches the index.

A miss-weighted fit trusts the surface weighted each ray by how much it could be trusted, and a set of crooked sticks invites the same idea one level up: weight each stick by how little its bow can pull. Of the sticks that inform the index at all, one leaning across the sight and far out has the smallest pull, and a reader who knew which sticks those were could lean on them; whether that buys more than it costs in information was not measured. The figures here weight every reading alike, which is what a reader without that knowledge does.

What a reader with crooked poles should do #

The practical rule has three parts, and each comes from one of the figures.

Use sticks that lean across the sight, not along it. A stick leaning straight away from the eyes is immune to its own bow but tells the fit nothing about the index either; a stick leaning fifteen to forty-five degrees off that line tells it little about the index and is the most sensitive to its bow. The safe lean is across the sight, where the index’s own bow is largest and a bow in the stick is worth least of it. A straight stick tests the index only across the sight found the same lean was the only one where straightness was evidence at all.

Spread the sticks in distance, and do not crowd them near. A near stick pulls hardest and is weighted most, so a set crowded close to the eyes behaves like a smaller set.

Count eight, or tape one. One crooked stick is better taped than held to a line, by two to one; eight crooked sticks held to lines are as good as one taped; and a set of straight sticks beats both. Which to do depends on what the shore offers: a tape and one stick, or a jetty of crooked posts.

The bias, and the noise it becomes #

There is a general point under the arithmetic, and it is the one the residual does not warn made about reading error: a model error is invisible in the fit that commits it, because the fit absorbs it. One bowed stick is a model error — the model says straight, the stick is not — and the line-held fit absorbs it by moving the index. Several sticks bowed independently turn the same model error into something the fit can partly see: their bows disagree about which way to move the index, and the residual of their combined fit rises by what they disagree about. That is the step from a bias to noise, and it is what makes a set of crooked sticks measurable where one is not.

A water surface’s height trades against its index found the valley the fit slides down; the line constraint closed it for straight sticks, and bows reopen it a little for crooked ones, in a random direction each time. Eight sticks narrow the reopened valley back to the width one tape leaves it.

What the measurement assumes #

The bows are independent and random in direction. Sticks cut from one bundle, dried in one sun, may all bow the same way — towards the side that faced the light. Bows that share a direction do not cancel, and a set of them is one bowed stick with more readings; the mean over draws here was a few thousandths precisely because the directions were drawn independently.

Every stick is bowed by the same millimetre. Real crooked poles vary, and the set’s error would be set by its most crooked near members.

The sticks lean thirty degrees. A stick leaning steeper or shallower changes both the index’s bow and the bow’s visibility, and so the pull; the azimuth sweep here holds the lean fixed.

The water is still and flat. What a ray does at a surface is the reminder that every number here rests on one flat surface; ripples bend each ray differently and are a reading error of their own, larger near the surface where the sticks enter.

Still open: whether a crooked stick can be told by its residual and dropped #

Eight crooked sticks bowed at random disagree with each other, and the disagreement shows in the combined fit’s residual. Each stick’s own share of that residual is larger when it disagrees more — when its bow pulls the index further from where the others put it. A reader could then fit the set, find the stick whose residual is largest, drop it and fit again, as a surveyor drops a reading that fails a test.

The measurement that settles what that is worth fits eight crooked sticks at a quarter pixel, ranks them by their share of the residual, and asks whether the stick ranked worst is the one whose bow pulls hardest more often than chance; whether dropping it and refitting improves the index or only removes a reading, since the hardest-pulling sticks are also the nearest and most informative; and whether a stick bowed three millimetres among seven bowed one — one genuinely bad pole in a jetty of fair ones — is found and dropped reliably, which is the case a reader would actually face.