A bowed stick is read as a wrong index

A straight stick tests the index only across the sight asked whether holding a submerged stick to a straight line stops a refraction fit from sliding. Three eyes 1.6 metres above a pool read a metre-long stick leaning thirty degrees two metres out, and fit the water’s index and its surface’s height together; with a quarter pixel of reading error the fits slide down a curved valley towards a weaker index, 0.058 low on average. The answer depended on which way the stick leaned. A wrong index carries a stick leaning away from the eyes into another straight stick, so a line constraint learns nothing from it. A stick leaning across the line of sight it bows — by 2.2 millimetres at an index of 1.25 — and there the line constraint refuses every bowed reading and removes the slide entirely.

That cure rests on a two-millimetre signal, and the essay ended by naming the obvious threat to it. A real stick is never perfectly straight: a pole sags, a branch curves, a plastic rod bends. A stick bowed by a millimetre or two in the right direction would look, to the constraint, like a straight stick seen through the wrong index, and the fit would move the index to straighten it. The question was how much bow that takes — whether it is well above what ordinary poles carry, or within a millimetre.

It is within a millimetre.

A bow is what a wrong index looks like #

The rig is the earlier essay’s throughout: a level pair of eyes 65 millimetres apart with a third raised a hundred millimetres above their middle, all 1.6 metres above still water; a metre-long stick entering the water two metres out and leaning thirty degrees from the vertical, across the line of sight; eight points read along it. The stick is now bowed into a gentle arc, its middle standing a stated distance off the chord from where it enters the water to its tip, in a stated direction round its own axis.

With every point held to one line, the fit asks one question at each assumed index: how nearly can 24 rays, bent at the surface by that index, be made to meet a single straight line? For a straight stick the answer is exactly, at water’s 1.333, and less and less nearly either side, because a wrong index bows the read stick and a line cannot follow a bow. For a stick bowed a millimetre the answer is never exactly. The rays of a bowed stick do not meet any line at any index. But they come nearest at 1.407, and that is where the fit settles, 0.074 above water and with the surface put 19 millimetres low.

The reason is the earlier essay’s finding run backwards. A wrong index bows a straight stick, in one particular direction set by how the rays cross the surface. A stick that is already bowed in that direction is straightened, as the rays see it, by an index wrong the other way. The constraint asks only whether the stick is straight; it cannot ask whether the straightness belongs to the stick or to the index. To a line constraint a bow and a wrong index are one thing.

The misfit at that best index is the uncomfortable part. Bowed a millimetre, the stick’s rays miss their best line by 0.020 pixels root mean square — a thirteenth of the quarter pixel the eyes read to. Bowed half a millimetre, 0.010; two millimetres, 0.039. Nothing in the fit’s residuals separates a bowed stick from reading error, which is the old lesson of the residual does not warn arriving from a new direction: a model that is wrong in a way the fit can absorb leaves no trace in what is left over.

How far a bow moves the index #

The effect is a matter of how much bow, and which way.

Read without error, a quarter of a millimetre of bow in the worst direction moves the line-held index 0.017; half a millimetre, 0.035; a millimetre, 0.074; two, 0.165; four, 0.43. Bowed the opposite way it moves the index down instead: 0.017, 0.032, 0.062, 0.115 and 0.20. The response is nearly linear at first and grows faster than linear one way and slower the other, because the index’s effect on the bow is itself not symmetric about water’s value. Bowed square to both of those directions, the stick barely moves the index at all — 0.004 at a millimetre.

The dashed lines mark 0.058 either side, the slide that a quarter pixel of reading error put into the same stick’s index with its points free. That slide is what the line constraint was taken up to remove. A millimetre of bow, read with no error at all, moves the index as far as the slide did — further, in the worst direction. So for a stick bowed by a millimetre the constraint does not remove the slide; it replaces it with a bias of the same size whose sign depends on how the stick happens to be bent.

Only the part of a bow along a wrong index’s bow counts #

A bow has a direction round the stick, and the measurement says that direction decides almost everything.

Turning the same one-millimetre bow round the stick in thirty-degree steps traces a sine: 0.074 high at its peak, 0.062 low at its trough, and nothing in between, where the bow lies square to the direction a wrong index bows the read stick. A bow in that square direction is a kind of crookedness no index can produce or remove, and the fit, which is looking for index-shaped crookedness, ignores it. Every other bow is read through its component along that one direction.

So the threat is not “a stick that is not straight”. It is a stick that is not straight in a particular plane — roughly, here, a bow with its middle both raised and pushed sideways, the shape a sag and a twist together would give a pole held at an angle. A stick whose bow lies square to that plane passes the constraint harmlessly. A reader cannot usually know which way a pole bows under water, which is the practical point: the bias has either sign and a size set by a direction nobody measured.

With reading error, the constraint’s bias overtakes its cure #

The figures so far read the stick without error. The case that matters is the one the constraint was for, a quarter pixel of reading error.

Forty rigs at each bow, read to a quarter pixel, give the same picture with noise on it. Straight, the line-held fits come back 0.004 high with a spread of 0.060, the earlier essay’s cure. Bowed half a millimetre the worst way, 0.042 high; a millimetre, 0.084; two millimetres, 0.188, with spreads growing to 0.109. Bowed the other way, 0.031, 0.062 and 0.116 low, with spreads shrinking to 0.038. The constraint’s own bias passes the 0.058 slide it removes at about 0.69 millimetres of bow one way and 0.94 the other.

That is the answer to the question as the earlier essay posed it. The sag at which the constraint’s bias exceeds the slide it was removing is under a millimetre over a metre of stick. Precision-ground rod is straighter than that; ordinary bar stock and pipe may not be, and a wooden pole, a branch or a plastic rod certainly is not. So the earlier essay’s conditional resolves the cautious way: hold a stick to a line only if it is known to be straight to a few tenths of a millimetre over its length, and otherwise do not.

A tape does not care #

The earlier essay’s other cure was a tape: the stick’s length, read off a tape to ±5 millimetres and added to the fit as one more reading. A wrong index shortens or lengthens the read stick by about a hundred millimetres per tenth of index, which is a far louder signal than a bow, and the tape crosses the valley at its first step.

A bow does not touch it. The stick’s read length is the span of its points along their best line, and a millimetre of bow shortens a metre’s chord by a few thousandths of a millimetre — far below the tape’s ±5. With the stick bowed a millimetre the worst way, the taped fit comes back 0.006 high with a spread of 0.039, exactly what it gave with the stick straight. The tape was always the stronger cure, by a factor of about two in the spread; it is also the one a bent stick cannot fool, because a bow changes the stick’s shape and leaves its length.

That is the practical form of the result. A scale bar is worth its ends, not its tape found, for cameras in air, that a known length is used through the two points it joins and is indifferent to anything between them. Here that indifference is a virtue: whatever a stick does between its ends, its length is still its length.

Leaning away, nothing to fool #

There is one stick the bow cannot hurt, and it is the one the line constraint could not help.

Leaning straight away from the eyes, a stick bowed a millimetre in any of eight directions moves the line-held index by at most 0.004, against 0.074 for the stick leaning across. The earlier essay found that a wrong index keeps a stick leaning away straight to a few hundredths of a millimetre, so the constraint had nothing to work with there. By the same token, a real bow cannot be mistaken for a wrong index there, because no wrong index draws one. The constraint is fooled exactly as far as it is informative.

That symmetry is the general lesson, and it is worth stating plainly because it applies to every shape constraint, not only straightness. A constraint helps a fit in proportion to how much the wrong answers distort the shape it demands, and it is fooled in proportion to the same thing: a departure from the shape is read as evidence about whatever the shape is sensitive to. Where the shape is sensitive to nothing, it neither helps nor misleads. Where it is sensitive to the index, a real departure from it is evidence about the index, and wrong evidence.

What the bow was compared with #

The comparison that decides it is between two millimetres and a millimetre. A wrong index of 1.25 — 0.083 below water — bows the stick leaning across by 2.2 millimetres as its eight points are rebuilt, and that is the whole of the signal the constraint uses. A real bow of a millimetre in the stick is read as an index 0.074 out. The two numbers are close because they are the same quantity measured from opposite ends: how much crookedness a unit of index puts into this stick, as these three eyes rebuild it. The constraint can therefore be no more precise about the index than the stick is known to be straight, divided by that rate — and with a rate of roughly a millimetre per 0.07 of index, a stick has to be straight to a few hundredths of a millimetre for the constraint to read the index to a few thousandths.

A straight stick in water is a kink and a curve found that one eye’s image of a straight stick is already curved below the surface, by much more than a millimetre; it is the three-eye reconstruction, not the image, that comes back straight at the true index. A bow of a millimetre in the stick is invisible in any one image beside that curve, and becomes visible only in the reconstruction — which is precisely where it is mistaken for a wrong index.

What a reader holding a crooked stick can do #

A reader who has only a pole of unknown straightness, and no tape, still has three honest options, and the measurements here rank them.

The first is not to use the line at all. With its points free, the stick leaning across slides 0.058 low at a quarter pixel with a spread of 0.130, but that slide does not depend on the stick’s shape: a bowed stick read with free points is read exactly as a straight one is, because every point is placed by its own three rays and nothing asks them to agree on a line. A known slide is better than an unknown bias.

The second is to measure the stick’s shape before it goes into the water. The same three eyes that read it submerged can read it in air, where no surface bends the rays and the three rays to each point meet without any index to fit. Holding the submerged stick to that measured shape, rather than to a line, is the same constraint with the stick’s own crookedness taken out of it. Its worth is then set by how well the eyes read the shape in air — and a point under water has two depths is a reminder that depth along the line of sight is what any pair of eyes reads worst, in air as well as in water. A bow lying along the line of sight is the hardest to measure and, for a stick leaning across, the part that matters.

The third is the earlier essay’s tape, which the evidence here makes the clear first choice: it is the stronger cure for the slide even when the stick is straight, and a bow cannot reach it.

There is a fourth thing worth knowing, which is where the dangerous bow lies. The wrong index’s own bow appears where the rays to the stick’s upper and lower ends cross the surface in different directions and bend in different planes of incidence; the read stick is pulled sideways and up together. A pole that sags under its own weight bows downward in the vertical plane through it, which for a stick leaning across the sight is partly along that dangerous direction and partly square to it. A reader cannot remove the sag, but can at least know it is the kind of crookedness the constraint will read as index — the kind the stick a stereo pair puts back and a third eye sees the water the pair cannot never had to face, because they never asked a stick to be straight.

What was assumed #

The bow is a parabola. A gentle arc, deepest at the middle. A stick bent near one end — a branch that kinks, a pole that has taken a knock — puts its departure where the parabola does not, and the index the constraint returns will differ; the direction-dependence and the absence of a warning will not.

The bow is in the stick, not the water. A ripple, a current that bends a flexible stick and a surface that is not quite flat all distort the read stick too, and each would be read by the constraint through its component along a wrong index’s bow, as a bow is. None was measured here.

The rig is the earlier essay’s. One stick, two metres out, leaning thirty degrees. On other rigs the wrong index’s bow is larger or smaller — from a few hundredths of a millimetre to a couple of millimetres, the earlier essay found — and the bow a stick may carry before it fools the constraint scales with it. A rig that looks more steeply into the water bows its sticks more under a wrong index, and tolerates more crookedness in them.

The tape is right. The tape’s indifference to bow is a strength only if the tape’s own number is right; a water surface’s height trades against its index found the valley a wrong number would move the fit along, and a wrong tape moves it there as surely as a bowed stick moves a line-held fit.

Still open: whether two sticks bowed independently average the bow away #

A single bowed stick biases the line-held index with a sign set by its bow’s direction. A rig that looks at several sticks, each bowed by a millimetre or so in its own random direction, gets a bias from each, of either sign, and a fit that holds every stick to a line combines them.

The measurement that settles what that is worth places two, four or eight sticks across the water at stated distances and leans, each bowed by a stated amount in a random direction round its axis, reads them to a quarter pixel and fits the index with every stick held to a line. It asks whether the biases average away as one over the square root of the count, so that eight crooked poles do what one straight one did; or whether the sticks at different distances and leans respond to their bows so differently that one bad stick dominates; and how many crooked sticks it takes to match a single tape — which, the evidence here says, will be a large number, because the tape’s length was never in the bow’s reach.