A free opening lets the camera trade again

A cut seen in perspective is not another shell photographed axial cuts of coiled shells with a pinhole camera close enough to see the near whorls larger than the far ones, and read them with Raup’s three numbers and three for the camera: its tilt, the bearing of the tilt and the reciprocal of its distance. The camera came back to within a quarter of a degree and five per cent, and W, D and T to within a third of a per cent, at every bearing and every distance down to three rim radii. Every opening on those shells was a circle.

An opening that is not round had found that a snail’s opening is closer to an ellipse, and that reading an elliptical opening with circles puts the whole misfit into D and T. So a reading meant for real shells has to let the opening have a shape — an aspect, how much taller it is than wide, and a turn, how far its long axis leans from the shell’s axis. And a photograph squashed along the axis had found that a squash straight across the axis or along it is exactly the picture of another shell with elliptical openings, while a squash at any bearing between turns the openings in a way no real opening is turned.

So the question the camera left was whether it survives the opening getting its freedom. Twelve numbers — the shell’s nine with the opening’s two, and the camera’s three — fitted to a turned, elliptical opening seen in perspective: can the turn and the camera’s bearing both be recovered, or is this where a camera and a shell trade places again?

Three hundred and sixty photographs #

The shells are the two the earlier essays read, cut through the axis as the cut along the axis reads them: a high spire, W = 2, D = 0.2 and T = 2, and a low one, W = 2.4, D = 0.42 and T = 0.5. Their openings are a quarter taller than they are wide, and either upright or turned twenty degrees. Each cut is drawn at 48 points a section with noise of a thousandth of the rim, and photographed by a pinhole tilted 10 or 20 degrees across the axis, along it and at 45 degrees, from 20, 10 and 5 rim radii — five photographs a cell, 360 in all.

Each is read twice: as a shell with elliptical openings photographed square, nine numbers, and with the camera fitted beside the shell and its openings, twelve.

Tilted across the axis, the fit returns W at 2.0001 and D at 0.1974 — and T at 2.119, six per cent high, an aspect of 1.313, a turn of 15.8 degrees and a camera tilted 4.2 degrees one rim radius away. It misses the points by 0.79 times the noise. Turn the dial to 45 degrees and everything comes back: T at 2.0008, the aspect at 1.251, the turn at 19.9 degrees and the camera at 20.0 degrees and 5.0 radii. Along the axis it fails again, the other way: T at 1.892, the aspect at 1.200, the turn at 26.0 degrees.

At a bearing between, the camera is recovered #

The 45-degree cells settle the first half of the question. At every shell, opening, tilt and distance the fitted camera’s tilt is within three tenths of a degree of the true one, the opening’s turn within seven tenths of a degree, and its aspect within one and a half per cent. On the high spire T comes back to within a tenth of a per cent and D within fifteen hundredths. The low spire reads less well from twenty radii, where the perspective is weakest — T within 1.4 per cent at worst — and its camera’s distance is the loosest number in the fit.

So the turn and the camera’s bearing can both be recovered. A camera tilted at a bearing between the axis and the line across it squashes the openings on the two sides of the axis in two different directions, and no single opening shape, turned however it likes, can make both sides at once. That was the earlier essay’s argument for a distant camera, and it carries over to a near one.

The same camera on round openings #

The comparison that makes the rest of this worth reading is the same camera on the same spire with its openings held round. Photographed twenty degrees off from ten rim radii, the round-opening camera came back at 19.98 degrees and 9.94 radii across the axis, 20.04 and 9.97 at 45 degrees, 20.02 and 10.00 along it, and its worst error in W, D or T was a third of a per cent at every bearing. The two readings differ in two numbers, the opening’s aspect and turn, and in nothing about the camera.

So whatever goes wrong across and along the axis below is not a weakness of the camera model. It is something the opening’s two numbers can do that a circle cannot, and the place to look for it is in what a tilted camera does to a picture that a shell can do too.

Across and along, the picture fits everywhere on a line #

The other two bearings are where it fails, and the failure is exact. A pinhole tilted τ at L rim radii takes a point at distance u along the tilt’s bearing to u·cos τ / (1 + u·sin τ / (L·R)). Pull the cosine out and the camera becomes two things in sequence: a squash by cos τ across the bearing, and a pure perspective whose strength is tan τ / L. So two cameras with the same tan τ / L differ only by a squash, and straight across or along the axis a squash is another shell.

Held anywhere on that line — two degrees and two rim radii, twenty degrees and twenty-one — a camera fits a noiseless picture of a cut photographed across or along the axis to 2 × 10⁻¹⁶ of the rim, which is the arithmetic’s own rounding. At 45 degrees the same held cameras miss the noiseless picture by one to eight thousandths of the rim, more than the noise any photograph here carries, and the fit cannot slide.

So the fit cannot tell the true camera from any other on a line, and it is a line in two numbers: what it reads of the camera is tan τ / L, the strength of the perspective, and the bearing. The tilt and the distance apart are not in the picture.

What the shell becomes on the line #

With the opening upright, the trade has a closed form. W and D come back exact on every held camera. T and the aspect come back scaled by cos τ′ / cos τ, with τ′ the held tilt and τ the true one: 1.48 per cent high at two degrees and 4.58 per cent low at twenty, for a camera that was really at ten. Along the axis the ratio inverts. The squash the camera has handed to the shell is the squash a distant camera would have made, and the shell carries it in exactly the way the squashed photograph found.

With the opening turned, the squash has more to work on. A squash across the axis changes a turned ellipse’s turn as well as its aspect, so the turn comes back wrong, by −1.17 degrees at two and +4.64 at twenty, and D takes part of it too, −0.39 to +1.61 per cent. T stays within three tenths of a point of the cosine ratio and is no longer on it.

Where a noisy fit settles #

On a noiseless picture every point on the line is a perfect fit. On a noisy one every point is a fit to the noise, and where the search stops is set by where it started and which way the noise tips a nearly flat valley.

Of the 240 photographs taken across or along the axis, most fits run down the line to its near end — a tilt of a third of a degree to four degrees, most of them with the camera at the closest distance the fit allows, one rim radius. There T comes back about 1.5 per cent high for a 10-degree camera across the axis and 6.4 per cent high for a 20-degree one; along the axis, 1.4 and 6.0 per cent low. Four run the other way, all on the low spire with upright openings photographed along its axis, to tilts of 34 to 40 degrees, and return T between 18.5 and 24.8 per cent too large.

Every one of the 240 sits on its cosine line to within four tenths of a percentage point, and every one is fitted within the noise — 0.79 to 0.89 times it. The fit is not failing to converge. It is converging to a point on a line of equally good answers, and reporting that point as the camera.

The bearings, side by side #

Read at a fixed tilt of twenty degrees across the three distances, the bearing decides everything and the distance nothing. Across the axis the high spire’s T comes back 5.9 to 6.2 per cent high at every distance; along it, 5.4 to 5.7 per cent low; at 45 degrees, within a tenth of a per cent. The low spire is the same at both failing bearings and recovers less exactly at 45 degrees.

A closer camera does not help. The perspective is stronger from five rim radii than from twenty, and the fit reads its strength better — tan τ / L comes back within four per cent on the high spire at every cell — but the strength is the one thing about the camera the trade leaves alone.

The low spire’s camera is the loosest number in the whole fit, even where it is recovered. Photographed at 45 degrees from twenty rim radii, its fitted camera reads the perspective’s strength 19 per cent too high and puts itself at 17.1 radii rather than 20: its whorls are wide and its spire short, so the cut spans less depth from the lens than the high spire’s does and the perspective has less to show. Its W, D and T are still recovered there, within 1.4 per cent, because a camera misplaced along a weak perspective costs the shell little.

The turn and the axis distance #

A turned opening photographed across or along the axis comes back turned by the wrong amount: at a tilt of twenty degrees, by about four degrees too little across the axis and six and a half too much along it, on both spires. D, which upright openings keep to within fifteen hundredths of a per cent, moves by 1.4 to 2.2 per cent on the high spire and by under half that on the low. An upright opening stays upright, because a squash across or along the axis keeps a symmetric ellipse symmetric.

What the axis distance costs had found D the number that a tilted camera cost nothing. That was a squash on a round opening. On a turned one photographed straight across or along the axis, the camera reaches D through the turn.

What the trade costs a place in Raup’s space #

Raup’s three numbers matter most for where they put a shell against the line on which its whorls come apart, and that line depends on T as well as on W and D: a taller spire lets its whorls part at a smaller distance from the axis. On the high spire, far from the line, a T six per cent off moves nothing that matters. The low spire is the sharper test. Its contact line sits at an axis distance of 0.319 for its true T of 0.5; read with the quarter-too-large T of the worst fit, it sits at 0.267, and the spire’s own D of 0.42 stays on the free side either way.

So the trade leaves a shell’s coarse place in the space alone and spoils its spire, which is the number that distinguishes a turret from a disc. A survey of turreted shells photographed upright in the frame would read the spires and not the coiling wrong, and nothing in its residuals would say so.

The square reading sees something is wrong #

The nine-number reading that assumes the photograph square is not fooled the same way. It misses nearly every perspective photograph by more than the noise, by up to 22 times it, and only the low spire photographed along its axis comes close to fitting; its T errors across and along the axis run from one to eleven per cent. So a person who reads a perspective photograph with a square model is told the model is wrong; a person who fits the camera too is told nothing, and is better off for T only by a few points.

That is the uncomfortable half of the result. Fitting the camera is the right thing to do at 45 degrees and in every case the camera has three numbers it can find; across and along the axis it turns a reading that knew it was wrong into one that does not.

The opening’s freedom reopens it #

With round openings the same camera recovered every picture at every bearing. Nothing about the perspective changed between that essay and this; what changed is that the opening may now be an ellipse. Round, it cannot absorb a squash, so the squash inside the camera has nowhere to go and the camera’s tilt and distance are pinned apart. Free, it absorbs the squash exactly at the two bearings where the squash is symmetric about the axis, and the camera slides.

The turn does not change that. A turned opening is recovered at 45 degrees and not across or along the axis, where an upright one is not either. What the question imagined — that an oblique opening in perspective might be the special case — is the wrong way round: the opening’s turn is a casualty of the trade, not its cause.

One number from the bench closes it #

The line has one free number, so one number measured off the bench rather than off the picture fixes the point on it. The camera’s distance from the cut face is the obvious one: a tape measure gives it to a per cent, and with L known the fitted tan τ / L gives τ, the squash is the camera’s and not the shell’s, and the shell on the line is the true one. A spirit level on a camera held at a known tilt does the same.

Short of that, a photograph with the cut turned in its own plane — the shell’s axis laid on the diagonal of the frame — moves the tilt’s bearing off the axis whatever the bearing was. A person cannot easily see which way a hand-held camera is tilted; they can choose which way the shell lies. Upright in the frame, with the camera a little off to one side, is the common photograph and exactly the one that cannot be read.

Pinholes, flat faces and two openings #

A pinhole camera with no lens distortion, a cut face that is flat and passes through the axis, and openings that are true ellipses with one aspect and one turn throughout the shell. Two aspects of opening — upright and twenty degrees — and three bearings are read; the bearings between 0 and 45 degrees were not, and the squashed photograph found a distant camera’s squash caught from fifteen degrees, which is where the camera’s own escape from the line would be expected to begin. Five photographs a cell measure where fits settle, not how often.

A held camera that cannot fit would end it #

A noiseless photograph straight across or along the axis that a camera held elsewhere on the line cannot fit to rounding. A noisy fit whose T is off its cosine line by more than the noise allows. A photograph at 45 degrees whose camera is not recovered. A round opening — held round — that slides along the line. Each is checked against every photograph read above.

Still open: two photographs of one cut #

Every reading here is of one photograph. A person with a cut shell and a camera takes several, and two photographs of the same cut share the shell’s nine numbers while each has its own camera. If both were taken straight across the axis with different tilts, each alone slides along its own line; together they must agree on one shell, and the two lines need not allow the same one.

The measurement is a pair of photographs read jointly, fifteen numbers for two cameras and one shell: whether two degenerate pictures pin the shell when their tilts or distances differ, whether a pair at the same tilt from two distances does it, and how much the two cameras must differ before the joint fit stops sliding — which would make two photographs, rather than a tape measure, the instrument.

Measured, two photographs pin the shell only from two sides: two pictures at one bearing, at any tilts and distances, slide together along the closed form $\cos\tau_1'/\cos\tau_1 = \cos\tau_2'/\cos\tau_2$, fitting both to rounding, and noisy pairs run to the end of it with T one and a half per cent off — no better than one picture. A second camera whose tilt is turned by ten degrees or more reads T to a quarter of a per cent and the camera to a degree.