Two free frames cost one right angle

The cut along the axis is the one view of a shell that carries all three of its numbers, and a photograph squashed along the axis can pass for a different shell. A cut seen in perspective is not another shell showed that a photograph of a cut shell can have its camera fitted alongside the shell. A free opening lets the camera trade again found the limit of that: a section with an opening of its own shape, photographed straight across or along the axis, fits every camera along a line, and each camera along that line reads a different shell. Two photographs pin the shell only from two sides found the way out. A second camera, its tilt turned by ten degrees or more round the face from the first, reads the translation to a quarter of a per cent, and the two lines no longer agree on any shell but the right one.

That essay carried a stated simplification. Each picture’s axes were the face’s own, so the two cameras knew exactly how their frames were related. No real pair of photographs has this. A person with a shell and a camera moves round it and holds the camera differently each time; the pictures are rotated against each other by however the camera was held, at a scale set by how far away it was and shifted by where the shell fell in the frame. Before a joint reading can be applied to real pictures, those four numbers have to come from somewhere. So the essay’s closing question was whether registering the frames reopens the slide for pairs turned by a few degrees, how much turn a pair then needs, and whether a pair at one bearing that differs in rotation alone helps at all.

The answer is unexpected in both directions. Registration is much cheaper than it had any right to be. And what it costs, it costs at one bearing, the one that had been the best.

A frame the reading is not told #

The set-up keeps everything from the earlier reading. There are two shells, a high spire and a low one, each with an opening that is not round, an ellipse of aspect 1.25 turned by twenty degrees. The first photograph is straight across the axis, tilted ten degrees at ten rim radii. The second is tilted the same amount from the same distance with its bearing turned, here by 5, 10, 20, 45, 70, 80 or 90 degrees. Each picture has its own noise of a thousandth of the rim, and three seeds of each.

The new thing is what happens to the second picture before it is read: it is rotated by 37 degrees, scaled by 1.3, and shifted by about a fifth of the rim. The joint reading gets four more free numbers to undo that, a rotation, a scale and two shifts, on top of the fifteen it had, nine for the shell and three for each camera. To stay honest about the start, it is not given these four. A scale and a shift are guessed from each picture’s overall size and centre of mass, eight rotations round the circle are each tried with a short fit, and the best is carried forward, with the two cameras then restarted from several tilts and bearings. The joint fit has only the two sets of points to go on.

The frame comes back by itself #

The first result is how little the frame costs to find. Across every bearing on the high spire, the rotation between the two pictures is read to between 0.010 and 0.030 degrees. On the low spire it is read to between 0.023 and 0.044. That is a rotation of 37 degrees recovered to about a two-thousandth of its size, from a start that may have been forty-five degrees wrong.

There is a simple reason it is so easily found. A cut through a coiled shell is not an isotropic object. It has an axis and a sequence of openings of diminishing size along it, on two sides, and any one picture of it says where that axis lies in the picture, to about as many figures as the picture has. The frames are not being compared with each other at all. Each is being compared with the shell they share, and the shell has an orientation of its own. Registering two pictures of something with so much structure is hardly a question at all. It would be if the subject were a featureless blob or a shell’s outline alone, or a picture with its features all near a single point.

So the answer to whether registering the frames reopens the slide for a pair turned by a few degrees is simply no. With its frames free, the pair turned by ten degrees reads the translation on the high spire to 0.0054, against 0.0026 with its frames given. At twenty degrees it reads 0.0020, against 0.0018. At forty-five, 0.0013, against 0.0016. Those are the same number, to within what three seeds can say. Fitted, the two tilts come back at about ten degrees each, which is what they were.

How much turn a pair needs #

The essay’s second question has a very different answer from the one the frames-given reading implied. With frames free, the error in the translation falls as the bearings part, from 0.0152 at five degrees on the high spire to 0.0054 at ten, 0.0020 at twenty, and 0.0013 at forty-five. Then it does what no one would have predicted from the earlier essay. It rises again: 0.0023 at seventy degrees, 0.0069 at eighty, and 0.0298 at ninety. The low spire does the same at a smaller scale, from 0.0006 at forty-five up to 0.0078 at ninety.

The pair at right angles had been the natural choice, and on frames-given pictures it had read the high spire’s translation to 0.0013. It is the bad case now, twenty-three times worse than forty-five degrees on the high spire, and a good deal worse than a pair at five degrees.

That is a remarkable result for how little has changed. Two photographs at right angles are the ones anyone would take, because intuition says that the most different views give the most information. With a scale nobody knows, they give the least.

What goes wrong at a right angle #

The tilts say what has happened. For bearings up to about seventy degrees, the fit reads both cameras at close to their true ten degrees, 9.8 and 10.0 at forty-five. At eighty it is starting to go, at 9.2 and 10.7. At ninety it reads 1.0 and 14.1, on every seed, on both shells, while fitting both pictures at the level of their noise. That is a wrong pair of cameras that fits as well as the right one.

The arithmetic behind that is short. A camera tilted by $\tau$ at some bearing squashes the picture by $\cos\tau$ across one axis, and leaves the axis perpendicular to it alone, to first order. Two cameras at right angles squash along perpendicular axes. And a squash of $c$ along one axis and a squash of $c$ along the other, applied together, are a change of scale by $c$ and nothing else. So when a joint reading is free to change one picture’s scale, it can give that picture a squash of any amount along the other axis, and absorb the rest in the scale. A pair at right angles has one free number that it did not have before: an exact trade among the first camera’s tilt, the second’s tilt and the scale, at which

$$\cos\tau_2' = \frac{\cos\tau_1 \cos\tau_2}{\cos\tau_1'}.$$

With both true tilts at ten degrees, setting the first to nothing gives a second whose cosine is $\cos^2 10°$, or 14.1 degrees. That is what the fit found. It runs to one end of the slide because noise has no preference among a family of exactly equal fits, and the end has the most room.

An exact trade, with no noise #

A slide of this kind can be checked directly, by holding the first tilt at various values, with no noise, and letting the fit find the rest. At ninety degrees, held at 3, 5, 7, 9, 10, 11 or 13 degrees, the fit misses by about $1.5 \times 10^{-16}$ of a rim radius, which is the last bit a double holds, and the second tilt runs from 13.8 through 13.2, 12.3, 10.9, 10.0, 8.9 down to 5.5, each as the formula says. The trade is exact. It ends where the formula does: held at 15 degrees, the required second cosine would be above one and the miss is $3.3 \times 10^{-4}$.

At forty-five degrees there is no such thing. Held at 9 degrees it misses by $1.7 \times 10^{-4}$, at 7 degrees by $4.5 \times 10^{-4}$, at 1 degree by $9.9 \times 10^{-4}$, and only at 10 degrees does it fall to rounding. That is a pinned reading. The two pictures disagree with any wrong camera by something that scales with how wrong it is, and noise a thousandth of the rim is enough to find that.

So this is a real degeneracy, in the same family as the slide the earlier essay found for pairs at one bearing, whose form was $\cos\tau_1'/\cos\tau_1 = \cos\tau_2'/\cos\tau_2$. Both are about squashes that a free number can absorb, and neither has anything to do with the shell. They are properties of photographs. A pair at one bearing shares one squash and slides with it. A pair at right angles, with a free scale, shares it the other way.

What the scale gives away #

The scale is where to see it on any one pair. At 45 degrees it reads 1.3 to 0.06 per cent on both shells, and a good part of the range does about as well. At ninety it is out by 1.52 per cent on the high spire and 1.58 on the low. That is one minus the cosine of ten degrees, 1.52 per cent exactly, on the high spire. The scale has quietly taken one camera’s squash.

That gives a practical repair, and it is the cheapest in this essay. A second picture’s scale need not be a free number, if the two pictures were taken with one scale bar in shot or at a known distance. Measure the scale, and the slide is gone: it needs the scale to take up the slack, and it can’t if the scale is held. Measured, on the same pictures with the scale held at its true value and the other three frame numbers still free, the slide closes. On the high spire the pair at ninety degrees goes from a translation error of 0.0298 to 0.0018, sixteen times better, and it reads its tilts at 9.8 and 10.2. At eighty degrees it goes from 0.0069 to 0.0019, at seventy from 0.0023 to 0.0017. At forty-five and below nothing changes at all: 0.0013 both ways at forty-five, and 0.0054 and 0.0053 at ten degrees. The low spire reads 0.0005 at ninety with the scale held, against 0.0078 with it free. So the scale is what is costing the right angle, and only the scale. A person with a ruler in both pictures can take any bearing they like. A person with no ruler should take their second picture at about forty-five degrees.

The width of what goes wrong #

The slide at exactly ninety degrees is a point in the range of bearings, and the essay would mislead if it suggested that the trouble lived only there. Near a right angle the trade is nearly exact rather than exact. At eighty degrees the translation is already out by 0.0069, five times worse than at forty-five, and the tilts are moving apart at 9.2 and 10.7. At seventy, 0.0023, nearly twice as bad. So a reader with no ruler should stay well clear of a right angle, and should treat seventy degrees as about the limit. That leaves a wide window from about twenty to about seventy degrees in which registration costs nothing that three seeds can detect.

Nor does a pair need to be held with any care inside that window. Across it the bearing can be off by ten or twenty degrees without changing the reading noticeably: 0.0020, 0.0013 and 0.0023 on the high spire at twenty, forty-five and seventy degrees. Compared with what a second bearing bought in the first place, that is generous. It also means that the second picture need not have been taken for this purpose. Two photographs of a sectioned shell taken from different places, for any reason, are likely to fall in that window, and an old pair could be read.

A camera turned in the hand #

The third question is the shortest. A pair at one bearing that differs in rotation alone, the camera turned in the hand and not moved, adds nothing, as might have been expected. On the high spire it reads the translation at 0.0298, the same as the ninety-degree pair and for a closely related reason, and the tilts come back at about one degree each. Rotating a camera about its own axis does not change what it sees of the shell, only how the frame lies against it. And a reading that has been told the frame is free has been told to ignore exactly that. It is two copies of one picture with separate noise, and it slides as one picture does.

That is a useful thing to know because it is an easy mistake to make. Two pictures at different orientations in a set of photographs may look like two views and not be. What matters is where the camera stood relative to the shell, and nothing about how it was held.

What this does not establish #

The pictures are carried through a similarity, which is what holding a camera differently does. Real lenses have distortion and real pictures may not be flat to the film, and a registration by a similarity would then miss whatever those contribute. The points are correctly labelled by section and opening in both pictures. Matching points between two real photographs is its own problem, and the frames being so easily found here depends on the openings being known. Only ten-degree tilts at ten rim radii have been read at the second camera, so the numbers at ninety degrees are for one tilt. And the slide’s formula is first-order in perspective: at very short camera distances a perspective term enters that could weakly break the trade, and these fits are too noisy to say if it has. It would be a natural small test to hold the distance at two or three rim radii.

Measurements that would overturn it #

A pair at right angles, with a free scale, that named both tilts correctly on real pictures would be remarkable. A pair at forty-five degrees sliding like the pair at ninety would mean the account is wrong about where the degeneracy lies, as would a fit at ninety degrees whose residual rose steadily with the held tilt with no noise. A free frame that could not be recovered from a cut with a known opening would remove the simple account of why registration is cheap.

Still open: a registration with no labelled openings #

Every point here carries the opening it belongs to, on both pictures, and that is a large part of why registration is free. A real pair of photographs gives two sets of unlabelled outlines, and matching them is the first thing a reader would have to do. The next measurement drops the labels and matches them by the fit itself, assigning each point to the nearest fitted section and iterating, and asks how far apart the two views can be before the matching itself goes wrong, whether a slide in the matching can take the place of the slide in the cameras, and whether two views of an unlabelled outline can do without the scale bar this essay ended by asking for.