A cut seen in perspective is not another shell

A photograph squashed along the axis is another shell read photographed cuts of coiled shells with Raup’s numbers — the expansion of the whorl W, its distance from the axis D and its translation along the axis T, read from the round sections every whorl leaves on a cut through the axis. A camera held a few degrees off the cut face’s normal squashes the whole picture by the cosine of the tilt, and that essay found two kinds of squash. Straight across the axis or straight along it, the squashed picture is exactly the picture of another shell — same W and D, T divided or multiplied by the squash, the round openings turned into upright ellipses — and nothing in it can say which. At any bearing between, the squash turns every opening the same way on both sides of the axis, which a real opening never does, and it is caught.

Every camera there was infinitely far away. That essay ended on the case a worker with a camera actually meets: a lens close enough to the cut that the whorls nearer it are larger in the picture than the whorls further off. The squash then grows across the picture rather than being one factor everywhere, and a whorl’s size changes from one side of the picture to the other in a way a shell’s own expansion along its spiral does not. It asked whether perspective, unlike the squash, can be told from a shell even straight along the axis.

It can, and once it is told it can be undone.

A pinhole at a stated distance #

The cuts are the earlier ones: a high spire (W = 2, D = 0.2, T = 2) and a low one (W = 2.4, D = 0.42, T = 0.5), seven round sections over three turns, forty-eight points on each, moved by noise of a thousandth of the rim. Each is photographed by a pinhole camera tilted 10 or 20 degrees across the axis, along it or at 45 degrees between, from infinitely far and from 40, 20, 10, 5 and 3 rim radii — a cut two centimetres across photographed from forty centimetres is forty rim radii away, and from six centimetres, three. Twenty pictures are taken at each camera.

Each picture is read three ways. As a shell: round-or-elliptical openings, the photograph taken square, the reading an opening that is not round built. As a squash: round openings seen through one squash with a factor and a bearing, the reading that caught the oblique squash. And as a camera: round openings seen through a pinhole, fitted beside the shell’s seven numbers with three of its own — the camera’s tilt, the bearing it is tilted across and the reciprocal of its distance. A camera square on has no perspective whatever its distance, as a real one has none, so the three are stated that way. Each reading reports how far it misses the points, over the noise.

A cut from ten radii #

Photographed along the axis from ten rim radii at twenty degrees, the cut looks like a cut. The whorls are slightly too large at one end and too small at the other, which is also what a spire looks like, since its whorls grow by W a turn. Read through one squash, the first picture gives W = 2.0802, D = 0.2003 and T = 1.8728: the expansion four per cent too large and the translation six per cent too small. It misses its points by 3.6 times the noise. Read through a fitted camera, the same picture gives W = 2.0007, D = 0.2001 and T = 1.9975, the camera at 10.0 rim radii and 19.8 degrees, and misses by the noise.

Turned by the dial to three radii, the whorls at the near end are visibly swollen and the squash reads W = 2.35 and T = 1.57. Turned out to forty, the picture and its readings are hard to tell from a distant one — which is the question.

Even along the axis, a squash does not fit #

The squash reading’s miss over the noise is the test, and the noise sets its scale. Over every picture taken from infinitely far — both spires, both tilts, all three bearings, 240 pictures — the squash reading misses its points by at most 1.03 times the noise. That is what a correct reading of a noisy picture leaves. Taken in perspective it misses by more, at every bearing.

Across the axis it misses most: at a ten-degree tilt the high spire’s median miss is 1.23 at forty radii, 1.73 at twenty, 2.99 at ten and 9.29 at three. At 45 degrees the oblique squash it was built for takes up some of the perspective, and it misses by 1.10, 1.35, 2.09 and 6.74. Straight along the axis, the blind bearing of the earlier essay, it misses by 1.08, 1.30, 1.99 and 6.70. The low spire, whose sections stand closer together along the axis, misses less at every distance along the axis — 1.01, 1.04, 1.15 and 2.25 — but by three radii, plainly more than the noise.

How many pictures show it #

Counted picture by picture, against the largest miss any distant photograph shows: the high spire photographed along the axis at a tilt of ten degrees is missed beyond that on 15 of 20 pictures from forty radii and on all 20 from twenty in; at twenty degrees, on all 20 from forty radii. The low spire along its axis, on 5 and 6 of 20 from forty radii at the two tilts and on 12 and 20 from twenty. Across the axis and obliquely, 19 or 20 of 20 from forty radii in on the high spire, and 15 to 20 on the low.

So the answer the earlier essay could not give — that along the axis the photograph is exactly another shell — does not survive a near camera. A squash along the axis is a shell; a perspective along the axis is not, and with noise of a thousandth of the rim the difference shows on a high spire from forty rim radii and on a low one from twenty. What makes it show is the one thing a shell cannot do: change a whorl’s size across the picture independently of the whorl’s place on its own spiral.

What it does before it is seen #

Between infinitely far and the distance at which it is seen there is a range in which a perspective photograph is read as a shell and the numbers are wrong. Along the axis the squash reading puts W too large and T too small. On the high spire at twenty degrees, W is 0.95 per cent too large at forty radii, 1.93 at twenty, 4.03 at ten and 17.3 at three; T is 1.6, 3.2, 6.3 and 21 per cent too small. On the low spire, W 0.21, 0.42, 0.87 and 3.1 per cent too large, and T 0.23, 0.46, 0.92 and 3.0 too small.

The size of W’s error has a simple shape. On the high spire it is 1.1 times the sine of the tilt over the distance in rim radii, at both tilts, from forty radii to ten, and grows faster closer in. That is the fraction by which a pinhole magnifies the near side of the picture against the far, the same quantity a seed head photographed in perspective failed its correction at, and the shell’s reading takes it into the expansion per turn, since along the axis the near and far ends of the picture hold whorls of different ages. Where most pictures are not yet seen — the low spire from forty radii — the errors are a tenth to a quarter of a per cent: what a perspective hides, it hides because it is small.

Twice the tilt, twice the error #

The error’s shape can be checked against the other tilt. Along the axis on the high spire from twenty radii, W is 0.97 per cent too large at ten degrees and 1.93 at twenty — a ratio of 1.99, against 1.97 for the ratio of the two sines. T is 1.61 and 3.15 per cent too small, a ratio of 1.96. From forty radii the ratios are 1.97 in W and 1.95 in T. Doubling the tilt doubles the magnification across the picture and doubles what the shell reading has to absorb, so the rule of thumb is sound in both directions: a camera tilted half as far can stand half as far away for the same error, and the error falls as the camera backs off in proportion to the distance.

The low spire’s errors are smaller by a factor of four to five at the same camera — W 0.42 per cent too large from twenty radii at twenty degrees against 1.93 on the high spire. Its seven sections stand a quarter as far apart along the axis, so the magnification that differs between its nearest and furthest sections differs by less. That is also why its miss over the noise is smaller and why its perspective is seen at half the distance.

Across the axis, D takes the error #

Across the axis the shell’s numbers take the perspective differently. A camera tilted across the axis magnifies one side of the cut against the other, and the two sides are the sections half a turn apart; the squash reading has W too small and D too large — the number an assumed centre costs most, and the one an outline alone has to trade against W. On the high spire at ten degrees, W is 0.13, 0.27, 0.53 and 1.6 per cent too small and D 0.33, 0.68, 1.4 and 5.0 per cent too large from forty radii to three; T barely moves. The miss is larger across the axis than along it at every distance, so a perspective across the axis is seen sooner, and its error in D is, in proportion, the largest of the three numbers.

A camera beside the shell #

Three numbers more undo all of it. Fitted beside the shell’s seven, the camera’s tilt, bearing and reciprocal distance leave every picture missed by the noise alone — a median of 1.00 times it at every shell, tilt, bearing and distance. Over the twenty pictures of every cell, the camera reading’s worst error in W, D or T is between 0.13 and 0.37 per cent, about what the noise leaves on a distant picture read the right way, while the squash reading’s grows from a tenth of a per cent at forty radii to 21 per cent at three.

The camera itself is read back too. At every finite distance its median tilt is within a quarter of a degree of the truth and its median distance within five per cent: 38.2 to 40.9 radii at forty, 2.9 to 3.0 at three. From infinitely far it reads hundreds of radii, as near to infinity as the points can say, and the tilt still to a fifth of a degree, because the round openings it is held to fix the squash the tilt makes.

What the camera costs a distant picture #

A reading with three more numbers can always fit noise a little better, and a careful reader of the earlier camera on a seed head will remember that it read one head in ten fewer than the simpler correction on an affine photograph, the cost of numbers the picture had nothing to fix. On the cuts the cost is too small to see. On photographs taken from infinitely far, the median of each picture’s worst error in W, D or T is 0.087 per cent by the squash reading and 0.099 by the camera on the high spire, and 0.065 to 0.068 by either on the low; both miss the points by the noise. The camera’s three numbers are fixed by the sections’ shapes, which a seed head’s even scatter of organs does not supply, so a distant picture leaves them nothing to absorb.

That settles the order a worker should read a cut in. Fitting the camera first loses nothing on a distant photograph and recovers everything on a near one; fitting the squash first is right only when the camera is known to be far.

Why the camera can be fitted and the dome’s could not #

A dome is read as a nearer camera found a camera fitted from a seed head’s organs fooled by the head’s own shape: a dome and a near camera bend the picture the same way to first order, and the head the camera un-projected was close enough to round to satisfy its fit and far enough to misread. The cut is different, because the shell’s shape is not something the camera can be traded against. Raup’s three numbers place every section exactly, and a perspective moves sections in a way no choice of the three numbers reproduces — which is why the squash reading misses it, and why a reading that includes the camera can find it. A model of the object that is complete enough to be falsified by the picture is what makes the camera identifiable.

What a person photographing a cut can do #

The advice is cheap. Fit the camera: it costs three numbers, it never does worse than the squash reading, and on a distant photograph it returns a distance of hundreds of radii, which is the honest statement that there was no perspective to find. If the camera cannot be fitted, keep it far away: at forty rim radii the squash reading’s errors are under one per cent along the axis and a third of a per cent across it on a high spire, and smaller on a low one. And a reading whose miss is more than the noise is a reading that is wrong somewhere, which the earlier essay’s oblique squash and this one’s perspective both show.

Pinholes and flat faces #

Every camera here is a pinhole and every cut face a plane. A real lens adds radial distortion, a taper of its own centred on the lens rather than on the cut, which a pinhole camera fitted beside the shell would partly absorb and partly leave as a miss; nothing here measures how much. A sawn face that is not flat is a surface the photograph projects, and its relief would move the sections the way a dome moved a seed head’s organs. And every opening is round: the elliptical openings a shell may really have were not photographed in perspective, so whether a turned elliptical opening can still be told from a camera once the camera is fitted is not settled here.

Pictures that would contradict this #

A photograph from infinitely far that the squash reading misses by more than 1.03 times the noise. A high spire photographed along its axis from forty radii in, missed beyond any distant photograph on fewer than fifteen of twenty pictures, or a low one from twenty radii in on fewer than twelve. A squash reading of the high spire along its axis whose error in W is not between 1.0 and 1.2 times the sine of the tilt over the distance from forty radii to ten. A fitted camera leaving an error over 0.38 per cent in W, D or T, a tilt off by a quarter of a degree or a distance off by five per cent. Each is checked against the measured cuts whenever they are read.

A shell that grows is not a lens that magnifies #

A squash along the axis is exactly another shell; a perspective is not, at any bearing. With noise of a thousandth of the rim, the squash reading misses a perspective photograph beyond anything noise alone does, on a high spire from forty rim radii and on a low one from twenty, even straight along the axis. Before that the error is small and has a shape — W too large by about the sine of the tilt over the distance. A camera fitted beside the shell takes it all back, to a third of a per cent, at every bearing and distance down to three rim radii.

Still open: a turned opening seen through a lens #

The fitted camera recovers a shell whose openings are round. A shell whose openings are ellipses turned against the axis was told from a squash in the earlier essay by the turn, which a squash cannot make. A camera fitted beside an elliptical opening has eleven numbers to trade, and a perspective across the axis turns the near sections’ outlines one way and the far sections’ another. The next measurement photographs elliptical, turned openings in perspective and fits the camera beside the opening’s aspect and turn, asking whether the turn and the camera’s bearing can both be recovered, or whether a perspective across an oblique opening is the one picture where a camera and a shell trade places again.