A photograph squashed along the axis is another shell

The cut along the axis showed that one saw-cut through a shell’s axis returns Raup’s three numbers — the expansion of the whorl W, its distance from the axis D, and its translation along the axis T — from the circles every whorl leaves on the cut face. An opening that is not round fitted the same cut with ellipses, and read the opening’s aspect and turn beside W, D and T, all five nearly as precisely as circles read three.

Both assumed the cut was measured as it is. In practice a cut face is photographed, and a camera held a few degrees off the face’s normal squashes the whole picture by the cosine of that angle, across whichever bearing it was tilted along. A round opening becomes an ellipse of aspect near one, and the ellipse reading will read it as one. That essay ended by asking whether a single squash fitted beside the shell’s own numbers can tell a squashed photograph of round openings from a shell whose openings really are ellipses, photographed square.

Two readings of one picture #

Any picture can be read two ways with nine numbers each. As an opening: the shell’s axis, scale, W, D and T, and an elliptical opening’s aspect and turn, with the photograph taken square — the reading the last essay made. As a photograph: the same axis, scale, W, D and T with round openings, seen through one squash with a factor and a bearing.

Each reading leaves a residual — how far the points miss the sections it redraws — and on a picture whose only error is noise the right reading’s residual is the noise. The test is which reading misses by less. Every picture here is a cut of a high spire (W = 2, D = 0.2, T = 2) or a low one (W = 2.4, D = 0.42, T = 0.5), seven sections over three turns, forty-eight points on each, moved by noise of a thousandth of the rim; twenty pictures at each camera tilt and bearing.

Straight across the axis, a photograph is a shell #

Before anything is fitted, there is a case the geometry settles. Suppose the camera is tilted so that the picture is squashed straight across the axis — the shell upright in the frame and the camera swung to one side. Every distance from the axis shrinks by the same factor c, and nothing else changes.

D is the ratio of two distances from the axis, the opening’s inner edge over its outer, and keeps its value. The scale of the shell takes the factor. T is a height over a radius and becomes T/c. And every round opening becomes an ellipse whose height is 1/c times its width, standing straight along the axis. That is the description of a real shell: same W, same D, a translation of T/c, openings of aspect 1/c. Tilted the other way, so the picture is squashed along the axis, every height shrinks instead: the same W and D, a translation of c·T, openings of aspect c.

Read without noise, the opening reading returns exactly that. At a camera tilt of ten degrees the picture squashed across the axis reads aspect 1.0154 and T = 2.0309 with D and W unmoved, missing by nothing; squashed along it, aspect 0.9848 and T = 1.9696. There is nothing left over for any test to find.

At other bearings the squash shows #

Squash the picture across any other bearing and every round opening becomes an ellipse turned by the same angle in the picture, on both sides of the axis. A real opening cannot look like that. The two sides of an axial cut are mirror images of each other, so an opening turned by ψ against the axis on one side is turned by −ψ on the other. The opening reading has one turn to give all seven sections, applied mirrored, and it cannot match a picture in which all seven lean the same way.

Drawn at a fifteen-degree tilt and a bearing of forty-five degrees, the opening reading’s ellipses miss the points by 3.34 times the noise, and it reports an opening turned −37.5° with an aspect of 1.019 — a large turn of a nearly round opening, which is what a squash at forty-five degrees looks like to it. The photograph reading misses by 0.92 times the noise, the noise itself, and reads the squash as 0.9655, the cosine of fifteen degrees to three figures, with W, D and T within a fifth of a per cent. Moved by the dial to a bearing of nought, both readings miss by the noise.

How far off, and at which bearings #

The opening reading’s miss rises from the noise at a bearing of nought to its largest at forty-five degrees and falls back to the noise at ninety. At a ten-degree tilt it misses by 1.73 times the noise at fifteen degrees, 2.65 at thirty and 3.01 at forty-five; at twenty degrees, by 5.60, 9.73 and 11.39. The photograph reading misses by the noise at every bearing and tilt, 0.97 to 1.00.

Called by the rule that a picture is a photograph when the opening reading misses by more than a tenth again as much as the photograph reading, a high spire photographed ten degrees off is called a photograph on all twenty pictures at every bearing from fifteen to seventy-five degrees, and on none at nought or ninety. At five degrees the window narrows to thirty, forty-five and sixty. A low spire, whose whorls are fewer and closer together in the cut, is called a photograph on all twenty from ten degrees up at every oblique bearing, and at five degrees on at most one picture of twenty at any bearing: a five-degree tilt of a low spire’s photograph is invisible however it was taken.

What the squash costs when it is not seen #

The bearings at which a squash cannot be seen are the ones at which it costs the most. The opening reading’s T is off by +1.5 per cent at a ten-degree tilt squashed across the axis and −1.5 along it; at fifteen degrees, +3.5 and −3.4; at twenty, +6.4 and −6.0. At forty-five degrees, where the squash is caught, T is right to a fifth of a per cent and the squash shows instead as a turn of the opening and a small bias in D, 1.3 per cent at ten degrees.

Where the squash is caught it is not free either, if the picture is read as a shell anyway. At forty-five degrees the opening reading’s D is 1.3 per cent high at a ten-degree tilt, 3.3 at fifteen and 6.7 at twenty on the high spire, and W 0.4, 1.0 and 1.7 per cent high; on the low spire D moves by about two fifths as much. The bias in D is the opening reading spending its one turn on a squash it cannot represent, and the residual that catches the squash is the warning that D has gone with it.

The photograph reading, when it is the right one, is right everywhere. On round openings it returns T within a hundredth of a per cent and D within three hundredths at every tilt and every bearing, the blind ones included — because at the blind bearings it is not wrong, it is merely not the only reading that fits. A worker who knows the openings are round loses nothing to the camera. A worker who does not know cannot tell, at those bearings, which reading to believe.

W is the most robust of the three throughout. It is a ratio of section sizes along the spiral, and a squash that shrinks every section by the same factor in the same direction leaves it alone; across or along the axis W is exact, and at oblique bearings it moves by under half a per cent at a ten-degree tilt and under two per cent at twenty.

A turn that is not there #

What the opening reading reports at oblique bearings is a turned opening. At a ten-degree tilt it reports −5° at fifteen degrees, −14° at thirty, about −40° at forty-five and +16° at sixty — turns large enough to look like a real feature of the shell, from openings that are perfectly round. The size of the reported turn barely depends on the tilt; it is set by the bearing, since the squash’s direction is what the reading is trying to explain. Only the miss reports how strong the squash is.

So a photographed cut read as a shell does not just read slightly wrong numbers. Near the diagonal bearings it invents a turned opening, and along the axis it invents an elliptical one. The residual flags the first and not the second, which is the pattern a residual that is not the test found in a spiral fit: the instrument is quiet on exactly the input it handles worst.

Real ellipses, photographed square #

The question also runs the other way: is a real elliptical opening, photographed square, ever mistaken for a squash? A straight one always is, in the sense that nothing separates them — it is the same picture as a round opening squashed along or across the axis, and the two readings miss by the same noise on every one of the twenty pictures, at every aspect from 0.94 to 1.06 and on both shells. Read as a photograph it would give T off by +6.4 per cent at an aspect of 0.94 and −5.7 per cent at 1.06. The photograph reading explains a tall opening of aspect 1.03 as a squash of 0.971 straight across the axis, and a squat one of 0.97 as a squash of 0.970 straight along it — each the exact mirror of the shell the opening reading would have made of a round opening behind the same squash.

A turned opening is never mistaken. Turned ten degrees against the axis, a high spire’s openings are read as openings on all twenty pictures at every aspect, and a low spire’s on 16 to 20 of twenty: the photograph reading cannot mirror its squash across the axis, and misses by 1.26 to 1.85 times the noise on the high spire and 1.12 to 1.43 on the low.

Five degrees is already larger than the noise #

The tilts here are not large. A camera held by hand over a bench is easily five degrees off the face’s normal, and a squash of cos 5° = 0.9962 is under half a per cent. Across the axis it moves T by +0.4 per cent and along it by −0.4. Set beside what the cut along the axis found the cut itself can deliver — T to 0.031 per cent on a high spire drawn to a thousandth of its rim — that is thirteen times the reading’s own noise error, from a tilt nobody would notice, with the points missing their sections by exactly the noise.

So the photograph is not a small correction to a precise instrument. On the bearings it cannot see, it sets the instrument’s precision. Whatever the outline of the shell and Raup’s three numbers were read to, a photographed cut is read to the cosine of the camera’s tilt in T, and to that only if the tilt was along or across the axis rather than between.

Ten degrees, which a photograph taken at arm’s length without a stand reaches readily, costs 1.5 per cent; at fifteen, 3.5; and the photograph adds them without a mark.

Why the common photograph is the blind one #

A cut shell is usually photographed with its axis upright in the frame, and a camera that is off the face’s normal is usually off to one side or above — tilted about the frame’s vertical or its horizontal. Those squash the picture straight across the axis or straight along it, the two bearings at which the photograph is exactly another shell. The squash that is caught on every picture comes from a camera tilted about a diagonal of the frame, which nobody does on purpose.

That is also the reason the last essay’s warning was the right one. A nearly round opening moved D by fourteen of its own noise errors while the points missed their circles by only a quarter more than the noise; a squash of the same size along the axis moves T by the same kind of amount with the points missing by nothing at all.

What a person photographing a cut can do #

The geometry suggests the remedy without this essay having tested it. A squash is fixed to the camera, while the shell’s axis can be turned in the frame. A cut photographed with its axis at forty-five degrees to the frame’s edges puts any accidental tilt about the frame’s vertical or horizontal at an oblique bearing to the axis, where it is caught. A circle drawn or placed on the cut face beside the shell — a coin, a ring of known shape — measures the squash directly, since a circle is the one thing a squash cannot hide in.

A head photographed from the side met the same problem on a seed head and found that the head’s own second moments could measure its squash. A shell section has no such internal round: every opening’s aspect is a property of the shell or of the camera, and only their arrangement on the two sides of the axis can say which.

A squash is not a lens #

That a camera’s error is a squash. A lens close to the face sees it in perspective, the near side larger than the far, so the factor changes across the picture; and a face that is not flat — a cut that wandered — is not one plane to be squashed. Neither is modelled here.

Nor that Raup’s model with elliptical openings is the shell. The fourth number divides the third found that an opening’s height and the spire’s translation enter the shell’s shape together; this essay finds that a camera can trade between them too, along exactly the bearing where the trade is invisible.

Pictures that would contradict this #

A photograph squashed straight across or along the axis whose opening reading leaves a residual above the noise, or does not read aspect 1/c and T/c, or c and c·T. A high spire photographed ten degrees off at a bearing between fifteen and seventy-five degrees read as a photograph on fewer than nine pictures in ten. A turned elliptical opening photographed square read as a photograph.

Two blind bearings, and every other one seen #

A photograph of an axial cut squashed across or along the axis is exactly the picture of another shell — same W and D, openings of aspect 1/c or c, T off by the same factor, six per cent at a twenty-degree tilt — and no fit can tell. At any bearing between, the squash turns every opening the same way on both sides of the axis, which a mirrored real opening cannot do, and a ten-degree tilt is caught on every picture. A straight elliptical opening photographed square is indistinguishable from such a squash; a turned one never is. The blind bearings are the ones a camera usually gets wrong.

Still open: a camera close enough to see in perspective #

Every photograph here is a squash, the same factor everywhere in the picture. A camera close to a cut sees the near whorls larger than the far ones, which is a squash that grows across the picture — and the large outer whorl and the small inner ones of a cut do not sit symmetrically about its centre. The next measurement photographs cuts in perspective from a stated distance and asks whether the perspective, unlike the squash, can be told from a shell along the axis as well, since a whorl’s size changes across the picture in a way a shell’s expansion along its own spiral does not.