The mirror with an eye is the one that is sharp

A mirror pays for its field out of its aperture found that a mirror of revolution can be designed by its map, that the field is a sum along the glass of one over the radial size factor, and that this sum is the whole of the limit. It also noted what the arithmetic of the sum could not touch. A surface whose curvature changes across the glass, and that differs along a radius and round a ring, brings a distant point to two different focuses. An eye can accommodate to one depth at a time. The essay that first asked what a mirror would have to be to read true wondered whether this blur, rather than the geometry, would be what sets the limit.

So there are two things to measure: how large the blur is, and what it depends on. The second turned out to have a much simpler answer than the shape of the question suggested, and the simple answer led somewhere nobody asked to go.

Two focuses, traced #

Take a mirror from the previous essay, a metre from the eye, and pick a point on the glass at some angle off the axis. Light from a distant object arrives from the direction the mirror sends that line of sight, as a bundle of parallel rays. Reflect each ray off its own piece of the glass, and ask where the reflected rays seem to come from.

There are two neighbouring rays worth tracing. One strikes the glass a little further along the same radius; the other strikes the patch of glass turned slightly about the axis. Continue each reflected ray back behind the mirror until it comes nearest the central ray. Those two places are the two virtual images of the distant object, one for detail running across a radius and one for detail running round a ring. They are the same place only by a special accident of the surface. The depth behind the glass tells the eye where to focus, and the eye’s demand in dioptres is one over the total distance, $1/(R + s)$, with $R$ the distance from the eye to the glass and $s$ the depth behind it.

On a sphere near its middle, this has a known answer, which serves as the test: both depths should be half the radius. With a radius of two metres, the trace returns $1.0000$ m and $1.0008$ m at a small angle off the axis.

For the mirror with the map $\phi = 3\theta$, the radial depth stays near half a metre over the whole mirror, from $0.500$ m near the axis to $0.516$ m at 10°, and the round depth rises to $0.549$ m. That is the ceiling on apparent distance that the warning’s number found, now read separately in each direction and at each part of the glass. And it is a surprise that the second curve in the figure is a single line.

The gap in focus is the gap in size #

The factor and the focus look like different kinds of thing. The size factor is about how large a distant object looks. The focus is about where a bundle of rays converges. Yet a few rows of numbers suggest a relation, since the radial factor of $1/3$ and a depth of $1/2$ go together, and so do a factor of $1/2$ and a depth of $1$, and a factor of $1/5$ and a depth of $1/4$.

They obey $M = s/(R + s)$. Rearranged, the eye’s focus demand in each direction is simply

$$D = \frac{1}{R + s} = \frac{1 - M}{R}.$$

So a direction’s focus demand and its size factor are the same number, up to the distance. It is not a coincidence, and it has a simple explanation. Tilting a parallel bundle by a small angle $d\phi$ moves its image behind the glass sideways by $s\,d\phi$, and the eye a further $R$ away sees that move as a change in angle of $s\,d\phi/(R + s)$. That quantity is what the size factor means.

Therefore the difference between the two focus demands is

$$\Delta D = \frac{|M - S|}{R}.$$

The blur is not a new property of the mirror. It is the same property that the previous essay drew as the two size factors drifting apart.

Those two sides are computed quite separately. One reflects rays off neighbouring bits of glass, and the other takes two derivatives of the map. Across three kinds of mirror and two strengths, they agree with a worst difference of $1.2 \times 10^{-4}$ dioptres.

This answers the question that was set. The limit on field was already set by the sum, and blur does not impose a new one. There is one fact about each point of the mirror, the pair of factors. How much field it spends is one reading of that pair. How much it blurs is another. And the first reading can be held fixed while the second is moved, as these mirrors do, because they read different parts of the pair: field depends on the radial factor, blur on the gap between the two.

How much it is #

The sizes are small, which is itself a useful thing to find.

At 10° off the axis, $\phi = 3\theta$ has a gap of 0.0135 D and the sphere 0.0274 D. An eye tolerates a defocus of about a quarter of a dioptre with hardly any loss, so the two smooth mirrors are ten or twenty times below that limit, and it would take a much stronger mirror at a much wider angle to reach it. So in the case the earlier essay imagined, of a varying curvature holding the field and the distance reading together, the limit was certainly not astigmatism. The varying-curvature mirror $\phi = 3\theta$ has half the sphere’s gap, because it keeps the two factors closer together than the sphere does.

The two-zone mirror is different. Past the join of a flat zone with a factor of a fifth, the radial factor has fallen abruptly to 0.2, and the round factor has not, because it depends on the whole of the field accumulated so far. Just past the join at 3°, the gap is 0.44 D. That is well above the eye’s tolerance, and over ten times anything else in the figure. It falls away going out, to 0.08 D at 10°, as the round factor catches up. A driver’s eye focused on the flat zone and glancing across the seam meets a blur it cannot focus away, so one reason for the marked line is more optical than it looks.

In daylight, with a 3 mm pupil, the smear is the pupil’s width times the gap, in radians: 0.14 arc minutes for $\phi = 3\theta$ at 10° and 0.28 for the sphere, far below the one arc minute a good eye resolves. For the zoned mirror it reaches 3.1 arc minutes just past the join. A larger pupil makes it worse in proportion, so a seam that is mildly visible by day can be noticeably soft at night, and that is what the slider shows. The direction of the smear also has a character of its own. Detail running round a ring and detail running along a radius are sharp at different focuses, so a person watching a small object cross the seam may see its horizontal and vertical edges go soft at different moments.

A mirror with no gap at all #

One of the five mirrors has a gap too small for the measurement to resolve. For the map $\tan(\phi/2) = c\tan(\theta/2)$, which holds the two factors equal by construction, the formula predicts zero gap everywhere, and the trace reports $7 \times 10^{-5}$ D at the edge, which is its own numerical noise.

So holding the two size factors equal, which looked like a concern with how small squares are drawn, also removes the astigmatism. An eye looking into that mirror can focus on any distant object, in both directions at once, anywhere on the glass. It is a remarkable property for a surface to have at all, and it invites the question of what the surface is.

The profile came out of an integration, so there is no reason to expect it to be anything in particular. But any conic with a focus at the eye has the form $R = \ell/(1 + e\cos\theta)$, so $1/R$ is linear in $\cos\theta$, and this is easy to test. It holds, to the rounding of the arithmetic: $2 \times 10^{-16}$ across the whole mirror.

The conformal mirror is the hyperboloid with the eye at one focus, with an eccentricity of $(c+1)/(c-1)$, which is 2 at a factor of three. Four values of $c$ from 1.5 to 5 were tested, and each is a hyperbola to the same degree, with the eccentricity predicted.

It is the mirror with an eye #

A hyperboloid’s defining property is old. It sends every ray aimed at one focus toward the other. The one shape that focuses lists it for exactly that reason, as the mirror used in a Cassegrain telescope. The caustic is the mirror’s own ruler describes the concentration of rays any other mirror has instead. In the present arrangement, with the eye at one focus, it means that every line of sight, continued behind the glass, passes through the other focus.

So this mirror has a single centre of projection. A curved mirror has no eye said the opposite of curved mirrors in general, and measured it for the sphere: over twenty centimetres of a mirror two metres across, the lines miss their own best-fitting point by 2.8 mm. That was the sphere’s result. Measured here, as the distance along the axis over which the virtual crossing wanders across a whole mirror, it is a matter of millimetres for every shape, except this one.

$\phi = 3\theta$ wanders over 7.1 mm, the sphere over 14.7 mm, $\phi = 5\theta$ over 14.0 mm. The conformal mirrors wander over nothing. Omnidirectional camera designers have known the fact from the other direction for a long time: Baker and Nayar’s analysis of single-viewpoint catadioptric sensors, in 1999, showed that the mirrors with a single effective viewpoint are the conic sections with the camera placed at the right spot. The route here came at it from a different question entirely, and it arrives with the addition of three properties that turn out to be one thing:

• a single centre of projection, in the lines of sight;
• equal size factors, so that the map from the eye’s angles to the world’s is conformal;
• no astigmatism for a distant object, for an eye on the axis.

The first two imply each other through a small piece of geometry. The map between the pencil of rays through one focus and the pencil through the other is a projection between the spheres of directions at those points, in the form $\tan(\phi/2) = c\tan(\theta/2)$. That is a similarity on the stereographic images of those spheres, and so it keeps angles. The third follows from the second, by the identity above. Being a centre of projection, in other words, is the same as drawing each small patch of world in the right shape. It is hard to see why this should be so until the two are put on the same page.

Now the earlier essay’s question has an unexpected answer. It asked what a mirror would have to be to read true, and showed that nothing but a flat mirror reads true at one constant size. This mirror reads with one consistent shape, and its size is what varies, from a third at the middle to 0.354 at 10°. And it has the one property a curved mirror was said to lack. So if the goal were an honest wide-angle reflection, in the sense of a picture that is locally undistorted, sharp and from a single place, then the answer would be a hyperboloid, with the eye at its focus.

What holds the mirror back #

No driver’s mirror is a hyperboloid with the driver at its focus, for two reasons that do not depend on the arithmetic above. The driver is not on the axis, and an off-axis section of a hyperboloid is only right for an eye at the focus: moving the head a few centimetres moves the eye off that point, and every one of these three properties weakens, with a loss that this measurement has not made. And as the previous essay showed, a slight change in shape makes little difference in the small factors and strengths most mirrors use. The sphere’s own gap, 0.027 D at 10°, was well below any tolerance. Most of what a hyperboloid would offer a driver is already within the noise of what a driver would notice.

A camera has no such difficulty. It holds still, and it can be placed exactly. That is where these mirrors are used, with cameras mounted on the axis, and an imaged panorama read as if from the mirror’s second focus. A mirror ball does not know its size measured the opposite case: a sphere that has no centre, whose picture carries only a ratio. There, a recovery has to treat every pixel as a ray with a place of its own, and a mirror ball’s equal areas are all the map it offers in return. Choosing a hyperboloid instead, and placing the camera correctly, replaces that with a single ordinary camera model, which is the reason for the choice.

Sharp in the limit: a far object, a thin bundle, an eye on the axis #

The object is at infinity. For a nearer object the two focuses move, and the identity above is a statement about the limit: a finite object’s bundle has its own curvature, and the two terms no longer have this one form. The hyperboloid’s freedom from astigmatism applies strictly to an object at the second focus or at infinity in this sense. How quickly it degrades for a near object has not been measured.

The factors are taken at a point on the axis and the bundle is thin. A real pupil samples a patch of glass, and higher-order terms appear in the blur for a large one. A wide-angle mirror close to the eye would show them first.

And none of these results are about a mirror off its axis, in the arrangement of real wing mirrors. They are about what is possible in the simplest geometry. They set expectations rather than a design.

Still open: how far the eye may move #

A hyperboloid’s three properties all depend on the eye being at the focus, and nobody’s head is exactly there. For a camera that is a matter of alignment. For a person, it is a matter of an eye that moves over a few centimetres at least. Move the eye along the axis and the mirror is no longer conformal about it; move it off the axis and the symmetry that made the factors simple has gone too.

The measurement that settles what the eye’s position costs moves it by stated distances along and across the axis, traces the same two neighbouring rays at each point of the glass, and reports the three quantities separately: how far the lines of sight wander, how much the two size factors differ, and the gap in focus. It asks whether those three stay tied together off the focus or come apart; whether a hyperboloid off its focus is still better than a sphere for the same eye position; and how large a head movement it takes to lose the advantage entirely.

A mirror’s blur, the shape of its picture and whether it has a centre of projection are one fact about it. The surface that gets all three right has been named since antiquity, and it only asks that the eye stand in the right place.