A tilt undone beats the calipers

A section seen from the wrong angle priced an oblique photograph of a shell section exactly. The photograph is the coiling plane compressed along one direction by the cosine of the tilt; the fit that reads a growth factor off it takes the logarithm of radius against angle, and the compression adds a term of period half a turn to precisely those data. Over half a turn of shell at twenty degrees off, the answers the fit gives as the span’s starting point moves round the shell fill a band 23.7 per cent wide. The calipers, which read a ratio of lengths along one line through the centre, are not moved at all.

That essay also found the tilt coming back out of the residual: the largest departure from the fitted line is half the logarithm of the cosine, so one number inverts to one angle. What the residual’s size does not carry is the axis the photograph was tilted across, and undoing the compression needs it. The question left was whether the axis can be read too — from where the perturbation sits round the shell, its phase angle, rather than from how large it is — on a span short enough to need correcting, and whether a corrected fit then meets the caliper reading.

Reading the tilt and its axis together #

The reading drawn here does not take the residual’s size and then look for its phase angle. It asks a single question of the whole outline: which tilt, across which axis, when undone, leaves the straightest line in the logarithm of radius against angle? Every tilt up to 44 degrees and every axis bearing is tried on a grid, the best pair is refined by halving steps, the picture is stretched back by the reciprocal of the cosine across that axis, and the straight-line fit is read off the result.

Tilt and axis are found together, so neither is assumed. Nothing about the shell beyond its being a logarithmic spiral is used: not its growth factor, not where its centre is, not which part of the shell the span covers. The shell is the one the oblique essay used, built at 3.2 per turn, and every span is seen from twenty-four views — axes 7.5 degrees apart and starting points a twenty-fourth of a turn apart — so a band is the spread over where the shell happened to be cut and how the camera happened to be turned.

An exact outline corrects exactly #

On an outline traced without error the question is settled at once. At every span from half a turn to three turns, every tilt from five degrees to thirty and every axis, the search returns the tilt it was photographed at to within the resolution of its own search and the growth factor to within the last few digits a double holds. The round trip closes.

That was not obvious in advance, and the oblique essay’s worry was specific: the amplitude of the residual is a peak-to-peak and survives a short span, while the angle at which it peaks is a position and might not. On an exact half turn it does survive, because the search is not reading a phase angle off a fitted sinusoid. It is asking which undoing makes the whole curve straight, and half a turn of a logarithmic spiral compressed across a wrong axis is not straight in any other way.

A traced outline #

An exact outline is not what a person has. A section photographed and traced by hand, or by an edge-finder, carries an error at every point, and the question that matters is what that error does to each reading. Here every traced point’s distance from the centre is multiplied by one plus a relative error — 0.2, 0.5 or one per cent of the radius, independently from point to point — over the 900 points the outline is sampled at.

At twenty degrees and half a per cent, the half turn drawn here is fitted at 2.909 as photographed. The search names a tilt of 20.02 degrees across an axis at 37.6, where the true axis was at 37.5, and undone the tracing fits 3.205. The jagged edge is the tracing’s own error, drawn at its true size; it is what the calipers will have to read three points of, and what the fit will average.

Turned to ten degrees, the photographed fit reads 3.126 and the corrected 3.206; at thirty, 2.588 and 3.205. At no tilt, the search still names one — 3.4 degrees in the tracing’s noise — and the fit moves from 3.202 to 3.192.

Three bands on one tracing #

The picture the oblique essay drew for an exact outline reappears here with a tracing error added, and with a third line. The fit as photographed has the band it had: 23.9 per cent wide on half a turn, 6.0 on one turn, 1.5 on two, 0.68 on three — and 2.6 per cent on three quarters of a turn, where it sits beside the span at which a tilt costs nothing, 0.7151 turns. The tracing error adds almost nothing to it; the tilt is the whole of its band.

The calipers’ band does not depend on the span at all, since they read three points a whole turn apart wherever the span is: between 2.2 and 4.0 per cent at every span, set entirely by the half per cent of error on each of three distances.

The corrected fit is narrower than both at every span: 0.86 per cent on half a turn, 0.39 on three quarters, 0.21 on one, 0.10 on three.

The fit catches up, then passes #

That is the reversal. The oblique essay found the calipers exactly immune to the tilt and the fit badly exposed to it, and recommended the calipers on any section whose residual named a tilt. On an exact outline that was the end of it. On a traced one it is half of the comparison, because the calipers’ immunity is bought by reading only three points, and three points carry the tracing’s error without any averaging.

Once the tilt is undone, the fit has nothing left against it but the tracing error, and it reads nine hundred points. At half a per cent of tracing error the calipers’ band is 2.6 times the corrected fit’s on half a turn, 7 times on three quarters, 13 times on one turn and 40 times on three. At ten degrees and at thirty the ratios are of the same size — 3 to 42 — and the corrected fit is narrower at every span and every tilt from ten degrees up.

Where the tracing now sits among the errors #

The other errors priced on a section give the corrected band a scale. A centre a quarter of the innermost radius out moves the growth factor by 4.56 per cent; stepping dividers along an outline with too few steps inflates the answer, toward a golden spiral; a tilt left uncorrected on half a turn opens a band of nearly 24. Against those, the corrected fit’s 0.86 per cent on half a turn and 0.21 on one turn are small — the tracing error, averaged over the outline, has become the smallest thing wrong with the reading.

That ordering is worth having in view because it turns the correction into a question of priorities. On a photographed section whose centre is uncertain, how far a centre must move to change the story is the first thing to settle, and a tilt undone on top of a misplaced centre gives an exact answer about the wrong curve. The correction here assumes the centre and makes the camera the last error standing; it does not rank above the centre, and it was measured with the centre where it should be.

Against the tracing’s error #

On the shortest span, where the corrected fit has least to average over, its band and the calipers’ both grow in proportion to the tracing error — 0.29 and 1.30 per cent at two tenths of a per cent of error, 0.86 and 2.23 at half, 1.45 and 6.41 at one per cent — and the fit as photographed stays near 24 per cent throughout, because its band was never about the tracing. The corrected fit’s advantage holds at every error drawn, and it does not shrink as the error grows.

The reason is counting, not cleverness. The fit’s slope is an average over every point of the span; the calipers’ ratio is two sums of three distances. An error on each point shrinks in the first as the root of the number of points and not at all in the second. Undoing the tilt is what lets the fit use that advantage at all.

How well the tilt comes back #

On the shortest span and the largest tracing error drawn, the tilt the search names stays within a degree of the truth at every tilt drawn: 19.62 to 20.62 at twenty, 29.93 to 30.10 at thirty with half a per cent of error and 29.67 to 30.25 at a full per cent on one turn. The axis is read to under a degree at twenty and thirty degrees on every span, to under three degrees at ten.

At five degrees the axis wanders — up to twelve degrees from the truth on half a turn, seven from one turn up. That is the worry about the angular position come true in the place where it costs nothing. A five-degree tilt compresses the plane by four parts in a thousand, and an axis misread by twelve degrees moves the compression a fraction of that; the corrected band at five degrees is 0.69 per cent on half a turn, as narrow as at twenty. The axis is hard to read exactly when it does not matter where it is.

A section that needed no correcting #

A search that can only add a tilt will find one in a square-on tracing, because some small compression always straightens a noisy curve a little. At half a per cent of error it names up to 3.5 degrees on half a turn and up to 2.6 to 3.2 on longer spans; at a full per cent, up to 4.6.

The cost falls on the shortest span. On half a turn the corrected band is 0.63 per cent against 0.36 uncorrected, nearly twice as wide, because two extra numbers read from half a turn of noisy outline take some of the precision the growth factor would have had. From three quarters of a turn the two bands are the same to within a quarter. So the correction is cheap insurance on a span of a turn or more, and on half a turn it is worth applying only when the residual has already said the section is tilted.

A shell whose expansion slows #

The oblique essay’s own warning was that a residual is one number asked to carry one cause, and a section can depart from a straight line in log radius for reasons that are not a camera. The sharpest test of the correction is a section that departs without being tilted: a shell photographed square on whose expansion slows along the span, its log radius bending away from a line as $-\varepsilon\theta^{2}$, at two strengths.

The search names a tilt on every one of them, between 5 and 10 degrees, and on half a turn the undoing straightens the curve well — the residual falls to 28 per cent of what it was. A slowing shell is mistaken for a tilted one.

Mistaken, and harmless, and visible #

The mistake costs the growth factor nothing. On every span and at both strengths, the corrected fit returns the uncorrected fit’s growth factor to within a hundredth of a per cent: a stretch across a fitted axis changes the curve’s shape near its ends and leaves its average slope where it was. Whatever the fit says about a slowing shell, it says the same with the phantom tilt undone.

And from a turn of shell up, the mistake shows. A real tilt’s residual falls, once undone, to the tracing error — between 22 and 26 per cent of what it was at half a per cent of error, at every span. A slowing shell’s falls to 80 per cent on three quarters of a turn, 97 per cent on one turn and to all of it on longer spans: the correction names a tilt and straightens nothing. The test the corrected residual gives is the one the residual is not the test asked a residual for and did not get from its size alone — whether the departure has the shape a tilt has. Only on half a turn, where the two are both mostly removed, does the test fail to separate them.

What a person with a photograph should do #

Fit the outline as traced and read its residual, as before. If the residual is above the tracing’s own error, read the tilt and axis together, undo them, and fit again. Then compare the residual after with the residual before: a fall to about the tracing error is a tilt corrected, and the corrected growth factor is the best number the photograph holds — better than the calipers on the same tracing. A residual that barely falls is not a tilt, and the growth factor is the uncorrected one either way.

On half a turn of shell, apply the correction only when the residual says there is something to correct, since on a square-on section it doubles the spread for nothing. And on any span, the calipers remain what they were: a reading a tilt cannot touch, worth taking alongside as a second route to the same number, and a check rather than the answer.

What this round trip assumes #

The camera is orthographic, as the oblique essay assumed; perspective from a camera close to the specimen is not drawn, and it does not scale a line through the centre uniformly, so it would trouble the calipers and might not be undone by a single compression either. The tracing error is independent from point to point. A traced outline that wobbles smoothly — a hand that drifts over a quarter turn — would look more like a tilt than a jitter does, and the correction would then absorb part of it; that was not drawn.

The shell is a logarithmic spiral built at 3.2 per turn, the centre is where it should be, and the search reads tilts under 44 degrees. A section whose law changed partway is the slowing case’s relative, and the residual test above is the only protection against reading it as a camera.

Readings that would undo it #

An exact tilted outline whose tilt comes back more than a hundredth of a degree out, or whose growth factor comes back more than a millionth out, at any span, tilt or axis. At half a per cent of tracing error and any tilt from ten degrees up, a span at which the corrected fit’s band is not narrower than the calipers’. At twenty degrees, an axis read more than a degree out on any span. A square-on tracing named a tilt of five degrees or more, or narrowed by the correction to less than four fifths of its band. A slowing shell named less than four degrees of tilt, or whose growth factor the correction moves by a hundredth of a per cent, or whose residual on a turn or more the correction cuts by more than a tenth. Each is checked every time the measurement runs.

Still open: a hand that drifts #

Every traced point here errs independently of its neighbours, and against that kind of error the fit’s averaging is what beats the calipers. A tracing made by hand errs in runs: the pen drifts outward for a few degrees of arc and comes back, which is an error with a length, and an error whose length is near half a turn has the shape the correction is looking for. The next measurement traces the same tilted sections with an error correlated along the outline over a stated arc — a twentieth of a turn, a quarter, half — and asks at what arc the search starts naming tilts that are not there, whether the corrected fit then still beats the calipers, and whether the residual test still tells a drift from a tilt.

Measured, a hand that drifts takes back what the fit won: the tilts a search invents on a section with none peak at a drift of a twentieth to a tenth of a turn and not near half. On one turn the corrected fit is no better than the calipers from a twentieth of a turn on. On two it stays about twice as good, and the residual still tells a drift from a tilt on a turn or more.