A mirror corridor shows the room once, and the room's lines are a reference by where they are

Two mirrors nearly facing bend the corridor two ways photographed the corridor of self-images between a barber’s two mirrors and read it. Each round trip steps the image down by the sum of the mirrors’ leans and sideways by the sum of their turns, so two slopes through the images read the angle between the mirrors about both axes with no reference at all. Sharing each sum between the two mirrors needed the camera’s own attitude — its pitch against gravity for the leans, its heading against the room for the turns — and a camera believed level and square but a tenth of a degree out carried that tenth into every share.

That essay closed on a hope. Every self-image, it said, comes with a reflection of the shop around it, turned with each round trip as the person is; a straight line on the shop’s floor, running square to the mirrors, should read each mirror’s own turn round trip by round trip, with the room itself as the reference, and a line on the wall behind the person might do the same for the leans without a level.

The first half of the hope is wrong in an instructive way, and the second half is right in a way that moves the problem rather than removing it.

The corridor is a box, and the room is outside it #

The shop is the earlier essay’s: mirrors 2.5 m apart, glass 1.2 m wide and from 0.9 to 1.9 m up, the front mirror leaning back 0.2° and turned 0.3°, the back one leaning 0.05° and turned 0.2°, a person standing 1 m from the front glass with their eye 1.6 m up. The photograph is taken from the eye towards the front glass, level and square, with a phone’s 50° lens.

A tile joint on the floor running square to the mirrors is never in the photograph. A rail on a side wall at 1.2 m, square to the mirrors, is seen once, in the first reflection. A rail on the wall behind the sitter, above the back glass, and a door jamb beside the back glass, are each seen once. Only a rod held level between the two panes is seen in every reflection from the first to the twelfth.

The reason is a property of every path the corridor is made of. A thing seen after $k$ reflections is reached by a path that leaves the eye, lands on glass $k$ times, and ends at the thing. From the second reflection on, the last leg starts on one pane and ends at the thing, and the leg before it ran between the panes; for the whole path to stay on glass, the thing has to stand roughly where the panes are, inside the box their two rectangles bound. The floor is 0.9 m below the bottom of the glass and the walls are beside it, so the paths that would reach them leave the glass on their way. Only the first reflection, which starts at the eye and needs only one pane, reaches outside the box — and of the floor, even that reflection lies more than 26° below a level eye, out of the frame.

The first reflection also shows the back glass, and in it the corridor again, nested: the rail and the jamb are seen beside the back glass’s outline, and the outline’s own reflections go on down the corridor inside it. That nesting is the box drawn in perspective — two mirrors are three cameras used the same nesting to put several viewpoints of one scene into one picture, and every one of those viewpoints looks into the box and nowhere else.

So the shop is not reflected round each self-image. It is reflected once, round the first, and the corridor beyond is the box and whatever stands in it: the person, and anything they hold.

One sight of the wall behind #

The first reflection is the ordinary view in a mirror of the room behind oneself. It shows the back glass with its corridor, and the wall the back glass hangs on, and on that wall there is usually something straight: a rail, a shelf, a door frame.

The reflection of the rail is the rail turned about the front mirror’s foot by twice the mirror’s lean and twice its turn. A camera turned or pitched moves the reflection too, but about the eye, a metre from the mirror’s foot, and the two motions are not the same motion: they differ by parallax, by how the rail’s reflection falls against the self-images, which the mirrors move by rules of their own. So the four mirror angles and the camera’s three — heading, pitch and roll — can be fitted together to the photograph’s marks: each self-image a point, each line’s marks by their distance from the line its model draws. With exact marks the fit gives back every angle to a millionth of a degree, the camera’s roll included.

A line alone does not do it. With the camera’s attitude unknown, the rail by itself or the jamb by itself leaves the fit singular: three unknowns of the camera and two of the front mirror are more than one line and the corridor’s slopes can separate. The rail and the jamb together, one running across the wall and one up it, are enough.

Better than a believed level #

With every mark read to 0.4 px, a camera that knows its attitude exactly reads the front mirror’s lean and turn to 0.027° and 0.026°. One that believes it knows, and is a tenth of a degree out both ways, reads them to 0.100° and 0.107° — the tenth, carried whole into each share. With the rail and the jamb in the picture and the camera’s attitude fitted, the shares come back to 0.049° and 0.065° for the front mirror and 0.044° and 0.060° for the back. Rolled a degree, turned half a degree and pitched seven-tenths, the camera reads them exactly as well: nothing the camera does to every mark alike survives a fit that includes the camera.

The reading scales with the marks, as a least-squares reading should: at 0.2 px the front mirror’s lean and turn come back to 0.022° and 0.031°, at 1 px to 0.11° and 0.15°, at 2 px to 0.22° and 0.31°. Fitting the camera costs about a factor of two against knowing it exactly, at every fineness. A phone’s picture read to a fifth of a pixel — the corridor’s images are small round marks and the rail and jamb long edges, both of which can be read that finely — gives each mirror’s share to a few hundredths of a degree with no level and no knowledge of where the phone pointed.

The reference is where the lines are #

The obvious worry is that the rail and the jamb are only as good as the assumptions about them, and the obvious assumptions are about their angles: that the rail is level, the jamb plumb, both square to the mirrors.

The angles hardly matter. A rail whose ends are 0.2° off level moves the front mirror’s lean share by 0.010°; a jamb 0.2° out of plumb along the wall moves its turn share by 0.033°. The fitted camera takes most of either up as a roll or a turn of its own, since a slightly sloping rail looks very like a slightly rolled camera. Against that, a camera believed level and pitched 0.2° moves the lean share by 0.201°, and one believed square and turned 0.2° moves the turn share by 0.200°.

The places matter a great deal. A rail 5 mm higher than believed moves the front mirror’s lean share by 0.14°; a jamb 5 mm further along the wall moves its turn share by 0.16°. The reason is the parallax the reading rests on. A mirror’s lean is told from the camera’s pitch by where the rail’s reflection falls against the self-images, and where it falls depends on how high the rail really is: 5 mm at 3.5 m is 0.08° of angle, and the fit has nowhere to put it but in the mirror. The reference the wall supplies is a position, and a position is measured with a tape, not a level. To beat a camera believed level to a tenth of a degree the rail’s height above the mirror’s foot has to be known to about 3 mm.

That is a fair exchange rather than a free one. A tape read to 3 mm over two metres is easier to come by than a camera level to a tenth of a degree, and a barber’s shop has both its floor and its rail where a tape can reach them. But the hope the earlier essay ended on — a reference that comes with the picture and costs nothing — is not what the room provides. The room provides a reference that costs a measurement of where its lines are.

A camera can turn but cannot move #

Why an angle costs so little and a place so much has a short answer, and it is the same answer a turn of the head is not a step sideways gave for a stereo pair. Everything a camera can do to a photograph without moving is a rotation about the eye, and a rotation about the eye moves every mark in the picture by a rule that depends only on the mark’s direction, never on its distance. A rail whose ends are a little off level is a rail turned about its own middle, and seen from 3.5 m that is almost exactly what the camera would draw if it were rolled a little: the fit, free to roll the camera, rolls it and leaves the mirrors alone.

A rail that is higher than believed is not a turned rail. It is a moved one, and a move is drawn differently at different distances: the rail’s reflection, 3.5 m away along its folded path, shifts by one angle, while the self-images, at 5, 10, 15 m and beyond along theirs, would have to shift by others. No rotation of the camera produces that pattern. A mirror’s lean does produce something like it — the lean turns the reflection about the mirror’s foot, a metre in front of the eye and well below it, so it too moves near things and far things differently — and the fit, with nowhere else to put a misplaced rail, puts it in the mirror.

So the reading divides a photograph’s errors into two kinds by the one test that can tell them apart: whether they could have been made by a camera standing where the eye is. Those that could are absorbed by the fitted camera and cost nothing. Those that could not — a line in the wrong place, an eye in the wrong place — are taken for the mirrors. A level checks a direction, and directions are cheap here; a tape checks a place, and places are what the corridor reads.

What a rod in the box could do #

The one thing that is seen again and again is anything in the box, and a person in a barber’s chair is often holding something. A rod held level between the panes, running square to them at 1.3 m, is seen in every one of the first twelve reflections. Its reflections are turned by each round trip as the person’s images are, and unlike the person it is straight, so each sight of it is a line with a direction as well as a place.

That is the measurement the earlier essay’s hope was really reaching for — a reference repeated in every round trip — and it moves the question from the room to the rod. A rod held level by a spirit level is a gravity reference carried into the corridor; a rod laid along a known line in the room, a floor joint or a counter edge, carries the room’s direction in. What it costs depends on how the rod is held, and that is the next question rather than this one.

What the reading leans on #

The eye’s place is known. As in the earlier essay, each share depends on where across the gap the eye stands, and that is taken as measured. It is another position, and like the rail’s height it enters the shares through parallax.

The mirrors are flat and their pivots known. A lean is a turn about the mirror’s foot and a turn a swing about its middle; a mirror hung from a cleat at the top leans about the top instead, and the parallax that separates its lean from the camera’s pitch changes with it.

The lines are straight and seen in one piece. The rail and the jamb are long edges; a shelf with things on it or a door frame half hidden by a coat gives a shorter, noisier line, and the reading scales with what is seen of it.

Every mark is told from its neighbours. The corridor’s images crowd together after a dozen round trips, and the rail’s reflection crosses the back glass’s own outline; a reader has to know which edge is the rail.

What the room can and cannot tell a mirror #

A mirror is a second camera made the general point the earlier essay leant on: a row of virtual eyes can be read for their arrangement among themselves without any reference, and for their place in the world only with one. Two mirrors make one turn is why the arrangement is a pair of slopes, and a corridor of mirrors counts their angle and two mirrors show fewer images than they make why it ends where it does. What this essay adds is where the world is, in a photograph of two mirrors: once, round the first reflection, and nowhere else. A mirror that is not parallel to the wall measured what a single mirror’s turn does to the room it reflects; the first reflection here is that single mirror, and it is the only one in the corridor with the room in it.

Measured down from the waterline is the contrast worth keeping. A pond is a mirror gravity holds level, and its reflection needs no reference at all. A pair of wall mirrors is held by nothing, and the photograph of their corridor has to be told about the world from outside — by a camera’s level, by a tape to a rail, or by something held in the box between them.

Still open: whether a rod held level in the corridor reads both mirrors without a level on the camera #

A rod held level between the panes is seen in every round trip, and its reflections turn with each round trip as the self-images do. Its direction in each reflection is a reading of the corridor’s rotation that carries its own reference — level, by the spirit level on it — and its position is held by the person’s hands, which nothing measures.

The measurement that settles what that is worth puts a rod of stated length in the box, held level to a stated precision and at an unknown place, traces its reflections in each round trip, fits the mirrors and the camera with the rod’s place as three more unknowns and its direction known only to be level, and asks whether the leans’ shares come back without the camera’s level or the rail’s height; whether the turns’ shares need the rod laid along a known line of the room as well; and how the reading improves with the number of round trips in which the rod is seen, since the rod, unlike the wall, is seen in all of them.