The camera that is a cylinder

This collection’s picture surfaces are defined as maps and nothing else. The cylinder is one of them: a map from a direction to a mark, in which the horizontal coordinate is the azimuth and the vertical one is the elevation’s tangent — and the essay about it measures what that costs — every vertical stays vertical, every horizontal bows, and the elevation stretches by the secant.

That definition has never been shown an instrument. It takes a direction and returns a mark, in the same way the machinery that recovers a camera is only ever shown a picture, and the separation is deliberate: a surface that knew about a camera could be quietly fitted to one.

So there is a claim available that has never been made. A swing-lens camera — the kind that turns its lens through a wide arc and sweeps a slit across film bent into a circle — is usually described as making a cylindrical picture. Is it one?

Two routes, and they are the same map #

The instrument route is a piece of ray tracing with no surface in it. The pupil is at the origin. A ray arriving from direction $\hat d$ continues through and strikes the film where the line $t(-\hat d)$ crosses the cylinder of radius $R$; that happens at $t = R / \sqrt{d_x^2 + d_z^2}$, so the strike is at height $-R\,d_y/\sqrt{d_x^2 + d_z^2}$ and at an azimuth half a turn from the subject’s. Unroll the film, subtract the half turn, and negate the height because a pinhole inverts.

The surface route is the cylinder’s own definition, which is two lines of arithmetic and has no camera in it at all.

They agree to the arithmetic floor over eight hundred and sixty-one directions spanning a hundred and sixty degrees of azimuth and seventy of elevation. Not similar; the same function, arrived at from a machine and from a definition.

That is worth having for a reason beyond tidiness. This site’s whole apparatus is computed rather than constructed, and the risk of a computed surface library is that it becomes a taxonomy: six maps, each with properties, none of them tied to anything anybody builds. Finding that one of the six is a real camera’s picture, exactly, is what turns the taxonomy into a claim about the world.

Which cylinder, and why the film has to be concentric #

The instrument only works because of a condition that is easy to miss: the film is a circular arc about the pupil. Bend it about anything else and the result is a different surface.

That is the same condition the previous two rungs are about, wearing a different hat. There, the pivot had to be the pupil or the rays missed each other; here, the film’s centre has to be the pupil or the marks land in the wrong places. Both are the statement that a picture is a projection from a point and the point has to be the point.

The engineering is straightforward and old. The lens rotates on a bearing whose axis passes through its entrance pupil; the film is held in a curved gate at the focal distance; a slit rotates with the lens so that only the strip currently in front of it is exposed. Every one of those is a mechanism for enforcing the same geometric condition, and the resulting picture has a centre of projection in a way that a stitch of six frames does not.

Notice what the slit is doing. It is the narrow strip the first rung of this ladder takes as a limit, built as a piece of hardware — so the instrument realises in brass what the other arrangement only approaches by shooting more. That is the usual relation between a limit and a machine, and it is worth saying because the limit was found first and the machine is a century older than the finding.

What it costs, one: detail #

The swing lens records the whole turn on one strip of film. Its resolution round the turn is the film’s resolution multiplied by the film’s own length, which is $2\pi R$ for a full circle — and $R$ is the focal length, a couple of centimetres.

A stitched camera records each frame separately and gets the whole sensor for each frame’s share of the turn. Six frames is six times the sensor, spread over the same three hundred and sixty degrees.

This is the trade, and it is a real one rather than a rhetorical one: the stitched camera buys detail with its centre of projection. Where the offset error comes from is not a flaw in the design, it is a consequence of the design being a way to get many frames’ worth of resolution out of one lens.

There is a version of the trade that does not involve any error at all. Stitch six frames from a camera on a correct rail, and the resolution is bought for nothing but time. That is the arrangement panoramic photographers actually use, and this site’s contribution is the exact statement of what happens when the rail is wrong rather than the observation that it can be right.

The detail advantage is not the frame count #

The trade is stated above as detail against a centre of projection, and it is worth putting the detail side in numbers, because the obvious arithmetic gives the wrong answer.

The obvious arithmetic is that six frames give six sensors’ worth of samples against the swing lens’s one strip of film, so the stitch wins by six. It does not. Hold the lens fixed — the same focal length $f$, the same detector pitch — and a swing lens whose film is at radius $R = f$ records $2\pi f/p$ samples round the turn, while $n$ frames each $2f\tan(\pi/n)$ wide record $2nf\tan(\pi/n)/p$. The ratio is

$$\frac{n\tan(\pi/n)}{\pi} \;=\; 1 + \frac{\pi^{2}}{3n^{2}} + \dots,$$

which is 1.10 at six frames and 1.27 at four — and tends to one as the frames narrow. At equal focal length the stitch buys essentially nothing, because both instruments are laying the same angular resolution round the same turn and all the stitch adds is the difference between a chord and its arc.

Where the advantage actually comes from is that the stitch is free to use a longer lens. Covering the turn in $n$ frames from a detector $W$ wide needs $f = (W/2)/\tan(\pi/n)$, so the sample count is $nW/p$ — linear in the frame count, against the swing lens’s fixed $2\pi f_{\text{swing}}/p$. For a 36 mm frame against a 25 mm swing lens the ratio is $nW/2\pi f_{\text{swing}} = 0.23n$: 1.4 at six frames, 4.6 at twenty, 11.5 at fifty.

That reframes the whole trade, because the longer lens is also what fixes the parallax. A longer lens means a shorter frame, so $\beta_{\max}$ falls and with it the floor no frame count could reach; and more frames means less azimuth per strip, so the horizontal half falls too. Detail, the seam and the floor all improve together, linearly, with the same change. There is no tension between them at all.

What is left is the two things that do not improve. The pivot error $e$ multiplies both parallax terms and is unaffected by any of this — it is hardware, and it is the only thing the photographer has to get right rather than merely spend patience on. And the exposure interval grows: $n$ frames take $n$ times as long, so the tear a moving subject leaves grows in exact proportion to everything else improving.

So the honest statement of the trade is not detail against a centre of projection. It is detail, resolution and parallax on one side, all bought together by shooting more frames, against the length of time the photograph takes — and a swing lens is the instrument that takes the whole turn in one eighth of a second and accepts the resolution its own focal length allows.

Which is why the two instruments have never competed. They sit at opposite ends of one axis — time — and every other quantity in the comparison is a consequence of where on that axis a photographer is willing to stand.

What it costs, two: a subject that moves #

A swing-lens camera makes its picture over an interval. The lens sweeps; the slit follows; the two ends of one photograph are exposed a fraction of a second apart.

So a subject moving across the view is drawn sheared — continuously, with no break, because the exposure is continuous. A person walking through a swing-lens panorama comes out stretched or compressed depending on which way they walked relative to the sweep.

A stitched panorama fails differently. Each frame is exposed at one instant and the instants are seconds apart, so the same person is drawn torn: a jump at the seam, of the whole distance they travelled between frames. The familiar artefact of a person appearing twice in a stitched panorama, or with half a body, is this.

Which is larger is a question with a boring answer — the tear, by a large factor, because seconds are longer than eighths of a second. The interesting statement is that they are different kinds of failure:

A shear is a picture of a distorted subject. There is a solid in the world whose projection from this eye is what was recorded; it is not the subject’s shape, but it is a shape, and everything about the picture is consistent.

A tear is a picture of no subject at all. There is no arrangement of matter whose image has a person’s left half at one place and their right half thirty centimetres away with a stripe of background between.

That distinction is this site’s own, and it arrives here for the fifth time. A two-centre picture is a one-centre picture of a sheared room, so it is a picture of something; a divergent construction is a correct picture of a leaning plane. A tear is where that move fails, and it fails because a discontinuity cannot be absorbed by a map of space.

Neither instrument is exact, and the variables are different #

Put the two side by side and neither of them has nothing wrong with it.

The swing lens has no parallax. Not small: zero, because the two routes above agree at the arithmetic floor and there is no offset in the arrangement to make a miss out of. It pays in resolution and in a shear.

The stitched camera has essentially no motion problem within a frame — a thousandth of a second is not an interval — and pays in a miss that is $e \sin \gamma$ and a tear between frames.

The control matters here more than usual. assertNoInstrumentIsExactInBoth asserts a ratio between the two motion costs, and at a stationary subject both of them are zero — so the assertion is made conditional on something moving, and the still case asserts instead that both are exactly nothing. A version that asserted the ratio unconditionally would have failed at the one setting where the two instruments genuinely agree, which is the setting the comparison starts from.

The instrument as an argument about surfaces #

There is a reading of all this that goes beyond photography and is why the essay sits in this field rather than in sensor.

The curved field’s premise is that a picture surface is a choice — that a flat plane is one option among six, that none of them keeps everything, and that calling the fisheye the distorted one is a habit rather than a measurement. The obvious objection is that the flat plane is the one a camera makes, so the choice is not really open.

The swing-lens camera is the counterexample, and it is a hundred and twenty years old. It is a camera; it makes a cylindrical picture; the cylindrical picture is exactly the surface the library defines. A photographer choosing between it and an ordinary camera is choosing a picture surface with a purchase rather than with a slider.

The same argument runs one more step. The screen a picture is shown on is a second surface, and it is increasingly not flat either — so a picture can now be made on a cylinder and shown on one, and whether those two cylinders are the same cylinder is a question with an answer.

What a reader can check without a swing-lens camera #

Almost nobody has one of these, and the claim can still be checked from the other end.

Take a spherical panorama — a file from a consumer camera, or one stitched from frames — and cut a cylindrical view out of it. The equirectangular format is a lookup table of directions, so cutting a cylindrical view is a re-projection with no image data in it: azimuth across, tangent of elevation down. The result is what a swing-lens camera would have recorded, and the two agree exactly, because both are the same map applied to the same directions.

What does not agree is the parallax. A view cut from a stitched file inherits the stitch’s defects — the ghost at the seams, the vertical floor, whatever the pivot error was — and a swing-lens negative does not have them to inherit. So the re-projection reproduces the surface and not the instrument, which is the distinction this essay is about, and it is the reason “the file is cylindrical” and “the picture is a cylindrical projection” are two different sentences.

Why the film’s inversion has to be in the arithmetic #

A detail in the instrument route is worth pulling out, because leaving it out would have produced a plausible wrong answer rather than an error.

A pinhole inverts: the image on the film is upside down and back to front. The strike point is at $-R\hat d$ rather than $+R\hat d$, so its azimuth is half a turn from the subject’s and its height is the negative.

Undo both and the map is the cylinder’s. Undo neither and the map is the cylinder’s composed with a half-turn, which is also a perfectly good picture surface — it is the same surface read upside down — and every property the field measures would come out identical. Straightness, conformality, area scale: all invariant under a rotation of the picture.

So a version of this that forgot the inversion would have agreed with the cylinder on every measured property and disagreed with it mark for mark, and only a comparison of the marks would have caught it. That is what assertTheSwingLensIsTheCylinder compares, and it is the reason it compares coordinates rather than statistics.

What this does not claim #

It does not claim the swing lens is better. It has no parallax and less detail, and a photographer with a rail and a still subject should stitch.

It does not claim every panoramic camera is a cylinder. A rotating camera with a flat sensor and no slit is a stitching machine with the stitching done in hardware, and its picture surface is a sequence of planes rather than an arc — which is the plane’s own behaviour repeated, not a curve.

It does not claim the sweep is the only interval in the arrangement. The film has a sensitivity and the slit has a width, so a swing-lens exposure is an interval twice over, and a frame being an interval is the sensor field’s subject rather than this one’s.

And it does not claim the cylinder is a good surface. It bows every horizontal, it stretches elevation by the secant, and it shares that stretch exactly with the equirectangular rectangle — which is the field’s own finding and is unaffected by there being a camera that makes it.

An instrument realises a surface exactly or it does not, and the way to find out is to compute where the light lands and compare the marks. Comparing properties compares what has already been abstracted, and two different pictures can have identical properties.