A mirror pays for its field out of its aperture

Closer than they appear, by a factor with a number in it put a number on the moulded warning: a convex mirror reads distance wrong by $1 + 2d/R$, so a driver gets a wide field only by accepting a large factor, and one radius sets both. That essay ended on the natural escape. One radius sets both because the mirror has one radius, and a surface need not. A profile free to take any form might hold the field of a tight mirror and the distance reading of a flatter one over part of its aperture. How far can that go?

Not far at all, as it turns out, and the reason is a sum. But answering the question needs the problem turned round, because a mirror described by its shape hides what the mirror does. A sphere is described by a radius, and what follows for the viewer has to be worked out afterwards, angle by angle. Design the mirror by what it does, and the shape follows, and the limit appears on the page.

Any map is a mirror #

Put the eye on the mirror’s axis, with the glass a metre away. A line of sight leaves the eye at an angle $\theta$ off the axis, strikes the mirror, and goes out into the world at an angle $\phi$ off the backward axis. Choosing the mirror’s shape chooses the function $\phi(\theta)$.

The reverse is also true, in one line. The mirror’s surface at each point must be perpendicular to the bisector of the incoming and outgoing rays, by the law of reflection. Write the mirror in polar form about the eye, at a distance $R(\theta)$, and work out the tangent, and everything cancels into

$$\frac{d\ln R}{d\theta} = \tan\frac{\theta + \phi(\theta)}{2}.$$

So any smooth, increasing map $\phi(\theta)$ is a mirror. Integrate from the axis and the profile comes out. For $\phi = \theta$ it gives $R = R_0/\cos\theta$, a flat mirror. For the map $\phi = k\theta$ it integrates to a closed form, $R = R_0\cos^{-2/(1+k)}\!\left(\tfrac{1+k}{2}\theta\right)$, and the integrator agrees with that to nine digits. As a further check on a map with no closed form, a mirror built from one is traced by reflecting a ray off the integrated surface, and the outgoing direction returns the map to five digits, with nothing but the law of reflection in between.

This one line replaces the previous essay’s whole problem with a simpler one. Nothing further about curves or caustics is needed to discuss the field. The designer has a free function, and the question is what limits it.

Two factors, and the sum they obey #

A small object seen in a flat mirror looks as large as it should. Seen in a curved one, it is drawn smaller, by a factor that is what the warning on the glass is about. Two different factors are at work, because the map treats two directions differently.

Along a radius of the mirror, a small angle $d\phi$ in the world becomes $d\theta$ at the eye, so the factor is

$$M = \frac{d\theta}{d\phi}.$$

Round a ring, the geometry of a circle at angle $\phi$ on the sphere of world directions gives

$$S = \frac{\sin\theta}{\sin\phi}.$$

A flat mirror has $M = S = 1$. At the mirror’s middle every convex mirror has $M = S$, and for $\phi = k\theta$ that value is $1/k$. A size factor of a third is what the previous essay’s distance factor of three looks like, read as a size.

Now take the radial factor and integrate. The field shown by a mirror filling a half-angle $\theta_{\max}$ at the eye is

$$\phi_{\max} = \int_0^{\theta_{\max}} \frac{d\theta}{M(\theta)}.$$

That is simply the definition of $M$, added up across the glass. It holds for every mirror of revolution with the eye on its axis. And it is the whole of the answer to the question that was asked: if the radial factor must stay above some limit $m$, the field cannot exceed $\theta_{\max}/m$, however the surface is shaped.

A mirror about 21 cm across, a metre from the eye, fills 6° on each side. Flat, it shows 6°. With every part of it reading a size factor of at least a half, 12°. A third, 18°. A fifth, 30°. Nothing in the profile can buy past that, because it was never a matter of profile. An aspheric surface can redistribute a factor across the glass, and it cannot change the sum.

The upper limit is also reached, simply. The map $\phi = k\theta$ has a constant $M = 1/k$, so it fills the budget exactly. And its round factor $S = \sin\theta/\sin k\theta$ is never below $1/k$, since a sine bends over, so it meets a limit on both factors at once. That closes the design question as originally posed. Under a limit on the size factor in both directions, the widest field is the aperture over the limit, and the mirror that achieves it is $\phi = k\theta$. The earlier essay asked whether the answer was a smooth surface or a small number of zones. It is a smooth surface, it has one free number, and the limit is geometry rather than anything about the varying curvature. What the varying curvature does cost is a separate matter, and it is what the mirror with an eye goes on to measure.

What cannot be held #

The map $\phi = k\theta$ holds one factor constant. The other it leaves to run. Compare three mirrors that agree exactly at the middle, each with a factor of a third on the axis.

Over a half-angle of 10° at the eye, $\phi = 3\theta$ stays at exactly 0.333 along each radius, while its round factor rises to 0.347. The conformal mirror, with the map $\tan(\phi/2) = 3\tan(\theta/2)$, holds the two factors equal to each other at every point and lets both rise to 0.354, so a small square stays square, everywhere, at a size that varies. The sphere, which most convex mirrors roughly are, holds neither: its radial factor falls to 0.312 and its round factor rises to 0.341, so a small square is stretched round a ring by nearly a tenth at the edge.

Could a mirror hold both, with one constant value everywhere? That needs $d\theta/d\phi = \sin\theta/\sin\phi = 1/k$ at once, which integrates to $\sin k\theta = k\sin\theta$. Only $k = 1$ satisfies it, at all $\theta$. Therefore:

A mirror that reads true in both directions at one constant size is a flat mirror, with no other shape possible, however free the profile. This is the same kind of statement as the cartographer’s old result that a map of a sphere cannot keep all its distances: a curved mapping has to give up something in the small, and all it can choose is what. At 15° off the axis, $k = 1.5$ is only 1.4 per cent out of square, $k = 2$ is 3.5 per cent, $k = 3$ is 9.8 per cent and $k = 5$ is 34 per cent. In a small mirror at normal strength this is a mild thing. At high strength it starts to show, since a mirror with a wide field has both a low factor and a high ratio.

The conformal choice deserves one more look, because the figure shows its factor rising with the field rather than falling. It uses up its budget the opposite way from a mirror built for a wide edge: its largest factor is at the edge, and at 3-to-1 the field it shows is 29.4°, against 30 for the $\phi = 3\theta$ mirror.

Four things a map can hold, and the one it cannot #

Those mirrors are members of a short list, and it helps to see the list whole, because each entry is one small line of calculus.

Hold the radial factor: $d\theta/d\phi = 1/k$, so $\phi = k\theta$. The rings of world direction lie evenly spaced across the glass, and a reader can measure angles off the axis with a ruler held to the mirror.

Hold the round factor: $\sin\theta/\sin\phi = s$, so $\sin\phi = \sin\theta/s$. Sizes measured round each ring are constant, and the field stops abruptly at the point where that sine reaches one, which limits the design’s use.

Hold the two factors equal: $d\theta/\sin\theta = d\phi/\sin\phi$, which integrates to $\tan(\phi/2) = c\tan(\theta/2)$. That is the conformal mirror.

Hold their product: $\sin\theta\,d\theta \propto \sin\phi\,d\phi$, which integrates to $1 - \cos\phi = K(1 - \cos\theta)$. Every patch of world then takes the same area of glass, which is what a mirror ball does for an eye far away. With $K = 9$, for a factor of a third on the axis, the same 10° of glass shows 30.3°. Along a radius the factor falls to 0.323 at the edge and round a ring it rises to 0.344, and their product stays at a ninth.

These are the same four options a map projection has, for the same reason: a sphere of directions is being drawn on a patch of something else, and the choice between keeping angle and keeping area or keeping a radial scale is forced. Any of the four can also be blended into another, so the design space is really a single free function. But what the free function cannot do is what the list shows: choose all of them at once. One line of arithmetic gets each entry, and the same line shows that no entry but the flat mirror satisfies the others. The cylindrical mirror has a distinct entry of its own, since it treats one axis as flat, and it pays the price of a picture that has two scales by construction. And conformal is not undistorted has already warned against the name of the third entry: it keeps small shapes, not sizes.

Reading true in the middle #

The design most drivers have actually met is a different compromise. Some wing mirrors have a flat or nearly flat inner part, reading true, and a marked outer band reading much smaller. The obvious appeal is that a driver gets a true reading where it is usually wanted and a wide field too. The sum says exactly what that costs.

The budget for a mirror filling 10° and showing 30° is 30° in total. A flat zone uses one degree of field per degree of aperture, the rate of a flat mirror. So if a fifth of the half-angle reads true, the remaining eight degrees of aperture have to supply 28° of field, a factor of 0.286. A half leaves 0.200. Eight-tenths leaves 0.091, which is an image roughly a tenth of true size, compressed into a thin band, where it is little better than a warning that something is there. The trade is exact. There is no version of a true centre that does not lower the factor at the edge, and the lower the factor, the less a driver can read from it.

That has a very different consequence from the single map. A mirror with $\phi = 3\theta$ reads the same everywhere, with one consistent factor, and a driver can learn it. A two-zone mirror has a true centre and a seam, across which the scale jumps by whatever the zones were chosen to give. As the earlier essay noted, the seam is visible because the scales disagree, and the marked line is there for that reason. The zone design buys a reading that is correct where the driver is most used to reading, at the price of a reading that is much worse where the driver must make a decision about something in the blind spot. Whether that is the right trade is a question about driving and has no answer here. The arithmetic only fixes the terms: every degree of true reading in the centre of a mirror costs a stated amount of the factor at its edge.

How little the shape differs #

All of these mirrors look like the same mirror. The difference between a sphere and $\phi = 3\theta$ on a mirror 35 cm across at a metre is small enough that nothing about the glass, held in the hand, would tell them apart.

At the mirror’s edge, 18 cm out, the two surfaces are 0.27 mm apart. So a designer specifying one has a tolerance well below what the eye can see in the glass and an effect, the 6 per cent difference in the radial factor at the edge, that is quite visible in what the glass shows. This is a familiar pattern in reflecting surfaces. The one shape that focuses separated a paraboloid from a sphere by a similar sliver, and the two have sharply different ways of forming an image. A reflecting surface amplifies its own slope by two, so small changes of shape are large changes of direction, and that is why it can be designed by its map at all.

A curved mirror has no eye measured how little a surface has to depart from flat before its lines of sight stop meeting, and a similarly small departure is doing the work here. The practical side of this is that the choice among these mirrors costs nothing in material or size. A sphere is cheap because of how glass is ground and pressed, and for a small mirror at low strength its two factors differ by a few per cent, which nobody notices. That is a reason to use a sphere, and it holds a long way. It is just no longer a reason to believe a sphere is what a convex mirror must be.

A driver’s eye is neither on the axis nor a point #

The eye has been on the axis. A driver is not on the axis of a wing mirror: the mirror is seen at a large angle, its useful field lies to one side, and the shape is off-axis. The sum generalises, since it is only the definition of a factor, but there is then no single axis of revolution to integrate along, and no reason to want a surface of revolution. A free-form mirror could trade across the two dimensions as well as along them, and this essay’s limit in that case becomes a bound on an area of world against an area of glass.

The eye has also been a point. A head moves, so a driver’s eye is in a small volume and sees a stated point of the world at slightly different places on the glass, and the factor changes with that. A consistent factor is more forgiving of this than a variable one, which is a further reason to prefer the smooth maps to the zoned one, though the size of that reason has not been measured.

And the distance reading has been converted to a size factor as if the object were far enough away. For near objects the earlier essay found the factor falls, for reasons a small-angle size factor does not see.

Still open: what the varying curvature costs in focus #

The earlier essay’s question had a second half, which is untouched. It asked whether the limit was set by the geometry or by the astigmatism that a varying curvature introduces. The geometry has now been dealt with: the limit is a sum, and it is reached. But reaching it takes a surface whose curvature along a radius differs from its curvature round a ring, and a surface like that focuses a distant point at two different depths.

The measurement that settles what that costs traces parallel light off neighbouring pieces of glass for each of these mirrors, finds where the bundle comes to its two focuses behind the surface, and reports the difference as what an eye a metre away would have to do about it, in dioptres. It asks whether a mirror’s blur at a stated place follows from something already drawn here, and whether the blur at a two-zone mirror’s join is comparable to a smooth mirror’s, or much worse.

A surface can choose where to spend its field, and how evenly, but not how much. The aperture and the smallest factor it will accept have already set the total.