Two photographs pin the shell only from two sides

A free opening lets the camera trade again photographed axial cuts of coiled shells whose openings were not round but ellipses a quarter taller than wide and turned twenty degrees, as a snail’s opening is, and read each picture with the shell’s nine numbers — its placing, Raup’s W, D and T, the opening’s aspect and turn — and the camera’s three: its tilt, the bearing of the tilt and the reciprocal of its distance — the reading a cut seen in perspective built for round openings. At a bearing between the axis and the line across it, every number came back. Straight across or straight along the axis, none of the camera’s did. A pinhole tilted $\tau$ at $L$ rim radii is exactly a pinhole tilted $\tau'$ at $L'$ followed by a squash, whenever $\tan\tau'/L' = \tan\tau/L$, and a squash of a turned-opening cut is the cut of another shell. Every camera on that line fitted a noiseless picture to rounding, each with a shell of its own.

That essay ended on the obvious remedy. A person with a cut shell and a camera does not take one photograph; they take several, and two photographs of the same cut share the shell’s nine numbers while each has its own camera. If each picture alone slides along its own line, two together have to agree on one shell, and the two lines need not allow the same one. The question was whether two degenerate pictures pin the shell, whether a pair at one bearing from two distances or two tilts does it, and how much the cameras must differ before the joint reading stops sliding.

Fifteen numbers for two pictures #

The joint reading is the one-picture reading with the shell held in common. Each picture’s points are un-projected through its own camera onto the face of the cut, and every point’s distance from its section’s ellipse is measured there; the fit adjusts the shell’s nine numbers and both cameras’ three, fifteen in all, to make the sum of squared distances over both pictures smallest. The pictures here share the face’s frame: neither camera is rotated in the picture plane, so the two pictures need no registering. That is a simplification a real pair would not enjoy, and it favours the joint reading. The shell’s numbers are Raup’s three and the opening’s two that an opening that is not round added.

The first photograph is always the degenerate one: a camera tilted ten degrees straight across the axis at ten rim radii. The second is taken either at the same bearing — farther away at twenty radii, or steeper at twenty degrees from five — or with its tilt turned by five to ninety degrees from the first’s. The shells are the two spires the earlier essays used, a high one at $W = 2$, $D = 0.2$, $T = 2$ and a low one at $W = 2.4$, $D = 0.42$, $T = 0.5$, both with the turned opening.

Two pictures at one bearing, held #

The cleanest test is noiseless. Take two exact photographs, hold the first camera’s tilt at a value other than the truth, and let the other fourteen numbers fit. If the best fit still matches both pictures to rounding, the held tilt costs nothing and the pair slides; if it cannot, the pair pins the tilt.

Both cameras straight across the axis — the first at ten degrees from ten radii, the second at twenty degrees from five — the best joint reading misses the two pictures by about $10^{-13}$ of a noise at every held tilt from two degrees to eighteen. That is the rounding of the arithmetic, not a fit. Both cameras straight along the axis behave the same. Holding the tilt anywhere on that range costs nothing at all: the second picture has not removed the slide.

One across and one along is different. Held at ten degrees, the truth, the pair fits to rounding; held two degrees away, at eight or twelve, the best reading misses by half to six tenths of the noise, and by 1.3 and 2.2 noises at the two ends of the range. Every step away from ten degrees costs more than the one before. The pair at two bearings pins the tilt.

The closed form of the slide #

Why the pair at one bearing slides follows from the single-picture result. Sliding one camera along its line from $\tau$ to $\tau'$ squashes its picture across the bearing by $\cos\tau'/\cos\tau$, and the squash is absorbed by moving to another shell. Two pictures share the shell, so they must share the squash. Each camera can make any squash its line allows, so the pair slides together along the family

$$\frac{\cos\tau_1'}{\cos\tau_1} = \frac{\cos\tau_2'}{\cos\tau_2}, \qquad \frac{\tan\tau_i'}{L_i'} = \frac{\tan\tau_i}{L_i}.$$

Fitted, the second camera follows that form exactly. Held at two degrees the first camera pulls the second to 17.52 degrees at 4.34 radii; at eighteen, to 24.84 degrees at 6.36 radii; every fitted tilt is within $2\times10^{-14}$ of a degree of the form, and every distance within a hundredth of a radius. The family runs out at one end where the less tilted camera, sliding towards square on, comes to a single rim radius from the cut: with $\tan\tau'/L'$ held, that happens at about a degree of tilt, before square on is reached. At the other end it runs on until the tilts approach a right angle. Over the range a person would photograph from, the pair is no better pinned than one picture was.

What the shell does on the slide #

The shell slides as the single-picture essay found it does: W stays exact and T, the aspect, the turn and D move together. Both across the axis, T runs from 2.028 with the first camera held at two degrees to 1.935 at eighteen, the aspect from 1.264 to 1.219 and the turn from 18.8 to 23.3 degrees. Both along it, the slide runs the other way, T from 1.972 to 2.068. Each of those is a shell that two photographs, at two different tilts and distances, see exactly.

The crossed pair’s curve in that figure is not a slide: it is the shell the fit returns when the first camera is forced to a wrong tilt, and it fits only at ten degrees, the truth. A forced wrong reading moves T by about as much as the free slide does — but it leaves a misfit the pictures can see.

Why two directions pin it #

A squash across the axis turns a shell into another shell, and so does a squash along it — a photograph squashed along the axis is another shell found the second for round openings — which is the whole of the single-picture trouble. What no shell absorbs is two squashes at once, in two directions by two independent amounts — or one in a direction between the axis and the line across it, which is why a single picture at a bearing of 45 degrees was already enough. Two pictures at one bearing can always agree on one squash; two pictures at two bearings are asked for two, and the shell can supply only the one that both share, which is none.

That also says which pairs help. A second camera at the first’s bearing, at any tilt and any distance, adds a picture that slides on the same family; a second camera whose tilt runs across a different direction adds a picture whose slide is a different family, and the two families cross at one point.

Noisy pairs at one bearing run to the end of the family #

Real photographs carry noise, here a thousandth of the rim at every point of both, on its own seed for each picture. Read together from several starts, a pair at one bearing does what the noiseless family says it can: the first camera slides, and the noise decides where it stops. On every seed, on both shells, with the second camera either farther away or steeper, the fit runs the first camera to the end of the family — 1.0 degree of tilt, one rim radius from the cut — and stops there only because a nearer camera is not one the reading allows.

The shell it returns there is wrong by the amount the family’s end implies. The high spire’s T comes back 0.030 off its true 2, one and a half per cent; the low spire’s 0.0077 off its 0.5, the same share; the opening’s aspect is 0.016 off and its turn 1.3 degrees. W stays within a sixteenth of a per cent, as the single-picture essay found it always does. And every one of those wrong shells fits both pictures to 0.92 of the noise, slightly closer than the turned pairs’ readings of the true shell do — which is why the fit went there. Neither a second distance nor a second tilt at the same bearing gives the reading anything to disagree with.

No better than one picture #

The single-picture essay read this same first photograph on its own, five seeds. Its noisy fits slid down the same line, three of the five to the end of it at one rim radius and the other two stopping at 2.0 and 3.9 degrees, and returned the high spire’s T between 2.026 and 2.030 and the low spire’s between 0.506 and 0.508. Read with a steeper second picture at the same bearing, all five seeds of both shells run to the end, and T comes back between 2.0286 and 2.0298 on the high spire and 0.5072 and 0.5077 on the low. The second camera sits where the closed form puts it for a first camera at the end of the family: 17.4 to 17.5 degrees at 4.3 rim radii.

So the second picture has not made the reading less wrong. It has made it more consistent: the spread of T over five seeds has narrowed by about a third, around a value one and a half per cent off. A repeatable wrong answer is worse than a scattered one in one respect — nothing in it suggests anything is wrong — and the residual is not the test is the standing reminder that a good fit to the points is not evidence for the shell.

Turned by a few degrees, the pair reads the shell #

Turn the second camera’s tilt away from the first’s and the slide stops. At five degrees of turn the worst T over five seeds is 0.0052 off on the high spire and 0.0057 on the low; at ten, 0.0026 and 0.0034; at twenty, 0.0018 and 0.0017; at ninety, 0.0013 and 0.0005. On the high spire that is a quarter of a per cent at five degrees and well under a tenth at ninety; on the low spire, whose T is a quarter the size, just over one per cent at five and a tenth at ninety. Every reading fits both pictures to 0.93 or 0.94 of the noise, the value the true shell gives, because these readings are the true shell to within the noise.

The two shells differ in how fast they settle. The high spire’s T is nearly as good at twenty degrees of turn as at ninety. The low spire keeps improving out to forty-five, and its first camera’s tilt, which the high spire recovers to within a degree at every turn, comes back between 4.1 and 7.4 degrees against the true ten at a turn of five, and within a degree and a half only from twenty. A likely reason, not measured here, is that a low spire’s sections are flatter and nearer one another, so a squash across them and a squash along them are nearer to being the same squash, and two cameras need more angle between them to tell them apart.

Every number, worst case #

The table settles the opening as well. At one bearing the aspect is 0.016 to 0.017 off and the turn 1.2 to 1.4 degrees on both shells — the shape of the opening trades along the family with T. Turned by ten degrees or more, the high spire’s aspect is within 0.0016 and its turn within a tenth of a degree; the low spire’s within 0.0063 and 0.4 degrees at ten, 0.0023 and 0.09 degrees at forty-five. D is within a thousandth everywhere, and within four ten-thousandths once the pair is turned.

Two photographs or a tape measure #

The single-picture essay found that one number from the bench closes the line: the camera’s distance, or its tilt, measured rather than fitted. The bench measurement it means is the kind three points on a diameter priced for the shell itself. That number fixes where on the line the camera sits and so the squash, and the shell follows. A second photograph does the same job without a measurement, but only if it is taken from a different side. Taken from the same side, at any distance and any tilt, it adds fifteen numbers’ worth of fitting and nothing the first picture did not already allow.

Of the shell’s numbers, D is the one no camera disturbs: what the axis distance costs found a tilted camera costs it nothing at all, and here it stays within a thousandth whether the second picture helps or not. T and the opening’s shape are what the second picture is for — and T is the number what a spire buys prices, since translation along the axis is what makes a spire. The practical rule is short. Two photographs of a cut shell taken with the camera tilted the same way — both leaning across the axis, or both along it — read no better than one. Two taken with the tilt turned by ten degrees or more between them, about the turn of a wrist, read the shell’s numbers to a quarter of a per cent and the camera’s to a degree, and no bench measurement is needed. A single photograph tilted at forty-five degrees to the axis does as well, which the earlier essay found; two from different sides are the version a person is more likely to have taken by accident.

What the pair assumes #

Both pictures share the face’s frame: neither is rotated in its own plane and their scales are fixed, so the joint reading need not register them. A real pair would bring a rotation and a scale for each picture and would have to fit them, which adds freedom the joint reading here does not have and could reopen part of the slide. The cameras are pinholes, the face is flat, the noise is independent at each point, and every reading starts from a dozen places and keeps the best. Two shells and one opening are read. Nothing here measures a real shell.

What would show two pictures at one bearing pin it after all #

A pair at one bearing whose best joint reading cannot fit both noiseless pictures when the first camera is held away from the truth, or whose second camera leaves the closed-form family. A crossed pair that fits to rounding with its first camera held two degrees away. A pair turned by ten degrees or more whose T misses by a hundredth. Each is checked on the readings above.

Still open: two pictures that must be registered #

Every pair here was given its two pictures in one frame. The realistic pair is two photographs whose frames are unknown to each other — each rotated and scaled in its own plane as the camera happened to be held — so the joint reading needs a rotation and a scale for each picture as well as the camera. The measurement is the same fifteen-number reading with those four numbers added: whether registering the frames reopens the slide for pairs turned by a few degrees, how much turn the pair then needs to pin the shell, and whether a pair at one bearing that differs in rotation alone — the camera turned in the hand rather than moved — helps at all.

Measured, two free frames cost one right angle: the rotation between two pictures comes back to a twentieth of a degree from the cut itself and registering them costs almost nothing, except at a right angle. There a free scale takes one camera’s squash whole, the fit slides exactly, and the pair that was best reads the translation twenty-three times worse than a pair at forty-five degrees, until the scale is measured.