A hand that drifts takes back what the fit won

A tilt undone beats the calipers was a clean result. A shell’s section photographed off its plane is squashed along one axis, and that squash spreads the growth factor a fit reads into a band that depends on the view. So search over every tilt and axis for the one whose undoing leaves the straightest logarithmic spiral, undo it, and fit again. On a whole turn at twenty degrees off, with each traced point in error by half a per cent, the band falls from 6.1 per cent to under a third of one. The same outline read by calipers across three diameters gives a band about nine times wider, because calipers throw away everything but three points on a diameter and a fit averages them all.

That word averages is where the trouble lies. Every traced point in that essay erred independently of its neighbours, so a turn’s worth of them made hundreds of separate estimates of the same thing. A real tracing is made by a hand or a line-following routine. The pen drifts out for a few degrees of arc and comes back, and the errors in neighbouring points are the same error. A long arc has fewer independent errors than it has points, and one wander of about half a turn has the very shape that the correction is looking for.

So that essay ended by asking three things. At what length of drift does the search start naming tilts that are not there? Does the corrected fit then still beat the calipers? And does the residual test, which told a slowing shell from a tilted one, still tell a drift from a tilt? All three are quickly measured, on sections with stated drift.

An error with a length #

The tracing error is a relative error on each point’s distance from the centre, as before, with a stated overall size. Only now it is correlated along the outline, by smoothing an independent sequence with a gaussian of a stated length: a fiftieth of a turn, a twentieth, a tenth, a quarter, half, or a whole turn, and rescaled so that its overall size stays the same. So every setting has the same root-mean-square error and they differ only in how that error is arranged. That is a stricter test than the usual error budget for a nautilus, which priced each source by its size and never by its shape.

The dial makes the distinction concrete. With no length, the error is hash, a new value at every point. At a fiftieth of a turn it has some texture, and still looks like noise. At a tenth, it is a series of slow wanders, about ten in a turn, and a person looking at that would call it a hand. At half a turn, there are only a couple of very broad excursions. Each section is read at twenty-four views, the same axes and starting points as before, at spans of half a turn, one and two, with the outline either square on or twenty degrees off.

Tilts that are not there #

A search over a positive quantity will always return something, as the residual that was not the test warned about fits generally, and on a section with no tilt the question is how much. With an independent error of 2 per cent, the search names about four degrees at every span, which is a modest figure given that it can only add tilt. With a drift, that climbs a long way. On one turn traced, the mean tilt named is 10.8 degrees with a drift of a fiftieth of a turn, 12.0 at a twentieth and 12.5 at a tenth. On two turns it reaches 10.7 at a twentieth. A section with no tilt at all gets called a dozen degrees off.

The essay that asked this question expected the worst of it near half a turn, on the ground that a wander of that length has the shape of a tilt: a squash across an axis changes the radius with twice the angle, so it has a period of half a turn. The measurement does not agree. A drift of half a turn names 3.5 degrees on one turn, under what an independent error does. The peak lies far shorter, between a twentieth and a tenth of a turn.

The reason takes a moment. A tilt-shaped wander fits a tilt, but a drift half a turn long has only about two such wanders in a whole turn, and is mostly a smooth, long excursion in which there is very little left for a search to do. What gives a search a bite at a short drift is the extra freedom of two free numbers against a residual made of only a handful of independent lumps. With a dozen lumps a turn, a tilt at some bearing can usually be found that fits several of them. A search works best against an error that is rough enough to be fitted, though not so rough that its fit means nothing, which turns out to be a length of about a tenth of a turn. There is a warning in that about where a real hand’s error sits: a drift of a few tens of degrees is the kind of thing a hand does.

What the correction loses #

The band of readings tells the rest. On one turn at twenty degrees off with an independent error of 2 per cent, a fit as photographed spreads over 6.4 per cent, a corrected fit over 0.72 per cent, and calipers over 14.7 per cent. That is the earlier essay’s result, and the corrected fit is twenty times better than calipers.

Give the same error a length of a fiftieth of a turn and the corrected fit’s band is 6.8 per cent, nine times wider than before, while the calipers have barely moved at 13.5. At a twentieth of a turn it is 10.6 per cent, with the calipers at 9.9. At a tenth, 11.2 against 11.7. From there to a whole turn the corrected fit, the calipers and a good part of the fit as photographed are all around 10 to 15 per cent.

A twentieth of a turn is eighteen degrees. So a pen that wanders with about that length is enough to erase the entire advantage of fitting over measuring, on one turn. It should be stressed what the calipers are doing: the same pen made the same tracing, and the three points they pick up are on its wanders too. Their band does not get any better as the drift lengthens. It simply does not get worse, since three points never were an average of anything and they had no independence to lose.

What a longer span buys back #

The way to restore an average is to have more independent lumps to average, and that needs a longer span. On two turns, the same comparison gives a ratio of corrected fit to calipers of 0.04 with an independent error, 0.17 with a drift of a fiftieth of a turn, 0.41 at a twentieth, 0.32 at a tenth, 0.48 at a quarter and 0.58 at half. So on two turns the fit stays better than calipers, by about two to one, right across the range. On one turn, the corresponding row is 0.05, 0.50, 1.07, 0.95, 1.14 and 1.06: nothing to choose between them from a twentieth of a turn onward.

That is a very sharp dependence on span. Nothing has been changed about the shell or the error, and doubling the span takes the correction from no use to a real improvement. The conclusion from the earlier essay, that a tilt correction beats the calipers, survives with a condition added: it holds for sections of about two turns or more if the tracing is a real hand’s. Short fragments are exactly what is usually available, so that is a significant limit. Three entries and one span found the span setting nearly every error budget, and this one is no exception.

The same arithmetic gives a useful way of thinking about an error of a stated length. A drift of length $\ell$ on a span of $s$ turns leaves about $s/\ell$ independent values, and the corrected fit’s band should scale as the square root of $\ell/s$ if nothing else is going on. Going from an independent error to a drift of a twentieth of a turn on two turns multiplies the band by about seven, and that is the same order as a shortened effective sample would give, with a whole span’s worth of points reduced to about forty lumps. A shaken stem’s window is the corresponding story for counts, where more organs buy back independent error. Here, more points on the same arc buy nothing. Only more arc does.

The residual still works #

Against all that, the third question has the better answer. Undoing a tilt that is there removes nearly all the residual left in a spiral. Undoing a tilt invented to fit a drift removes very little of it, since the rest of the drift is still there. So the residual left over, as a share of what it was before, ought to separate the two, and it does.

With an error of half a per cent on one turn, a real twenty-degree tilt leaves between 23 per cent of the residual with an independent error and 1 per cent with a very long one. A section with no tilt and a drift leaves 92, 78, 81, 95, 98 and 98 per cent across the range of lengths. On two turns the separation is wider still: 22 to 4 per cent for the tilt, and 92 to 99 for the drift. So on a turn or more a person can look at how much the correction helped. If the residual falls by most of its size, a real tilt was there. If it falls by a fifth or less, then the tilt it named was not.

That rule of thumb works where the error is half a per cent, the size a careful tracing of a section has generally been credited with here, and a larger error muddies it. At 2 per cent on one turn, with a real tilt the residual falls only to about two thirds with an independent error, since the tracing’s own error is then about the size of the tilt’s effect. At a tenth of a turn it falls to 43 per cent. A section with no tilt leaves 75 per cent at the same setting. That is still a separation, though not one that could be read off a single section. And it is the only reading in this essay that is not hurt by a longer drift, since a longer drift has less residual that a tilt can claim.

On half a turn it does not #

That reassurance stops at half a turn. There, a drift of a tenth of a turn on a section with no tilt loses 76 per cent of its residual to the correction, a quarter of a turn loses 78 per cent, and half a turn 69 per cent, even at half a per cent of error. A real tilt loses 97 to 100 per cent over the same range. Those remain different, by a factor of ten in what is left, but no single section could tell the difference, since a person cannot know what would have been left. A residual falling by three quarters looks like a success.

The cause is straightforward. On half a turn a tilt is a very flexible curve, since most of a squash’s behaviour lies in a half-turn period, and that arc holds only one period of it. With two free numbers it can fit a slow wander very closely. So on a fragment of half a turn, which is a common thing to have in hand, a person can have a named tilt, a fall in the residual of three quarters, and a corrected reading that is no better than an uncorrected one, with nothing in the numbers to warn them. A drift of a fifth of a turn or so on a half-turn section has the same fingerprint as a real tilt, as far as any one section can show.

What the correction is worth now #

Taking the three questions together, the earlier recommendation stands in a narrower form. On two turns or more, a correction is worth doing with any reasonable tracing: it beats calipers by about two to one under a real hand’s drift, and its residual says whether the tilt was real. On one turn, it is no better than calipers once the drift is a twentieth of a turn long, though its residual still warns of false tilts. On half a turn it should not be trusted. It invents a tilt and then shows a misleading improvement in the residual, and the corrected band is as wide as the uncorrected one.

It also suggests something practical about tracing. Since what ruins the average is correlation along the arc, a hand tracing an outline twice, independently, gives two sets of wanders, and their average has half the correlated part in it. A line-following routine probably has a different arc from a person, and its errors would be worth measuring before anyone trusted a fit. The shortest such drifts come from the pixel grain and the longest from lighting and wall thickness. Nothing here has measured any of them, on anyone’s tracing.

What this does not establish #

The drift is a stationary gaussian process with one length, and a real hand’s error has a mix of lengths and a tendency to lean one way, for which the search might do something else. Its overall size has been taken as a free number. The error is relative to the radius, so that a shell’s outer whorl gets larger absolute wanders, and a pixel-limited trace would have nearly the opposite. The calipers here read the same tracing, so they are correlated with the fit, which is fair for comparing two readings of one tracing and unfair if calipers would really be applied to the specimen itself rather than to a tracing of it. A caliper on a real shell has its own errors, which earlier essays have priced separately.

What would overturn it #

A corrected fit on a turn of section, traced by hand, whose band over many views stayed well under the calipers’. A real hand’s error turning out to have a length well below a fiftieth of a turn, which would put this whole question out of scope. A section with no tilt on a turn or more whose residual fell by most of its size under the correction, with a drift of any length. On half a turn, a way of reading residuals that did separate a drift from a tilt, which would withdraw the warning.

Still open: two tracings of one outline #

The cheapest remedy nobody has tested is to trace twice. If a tracing’s drift is a property of the hand on that pass and not of the outline, the difference between two passes is pure error, and its size and length can be read straight off, with no model of the shell and no knowledge of any tilt. The next measurement takes two tracings of one section with stated drifts, a shared part from the specimen and a part from each pass, and asks how far averaging them restores the fit’s advantage, whether the difference between them gives a reliable estimate of the drift’s length and size, and whether a corrected fit can then be trusted on a span as short as a turn by a reader who has measured how the pen moves.