A curved mirror has no eye

A flat mirror is a second camera. That is one of the tidier results on this site: reflect the eye in the plane of the mirror, and the picture in the mirror is a photograph taken from the reflected point. Every line of sight, continued behind the glass, passes through it — to 2.8e-12 mm, which is arithmetic and not geometry.

The obvious next question is what happens when the mirror bends, and the obvious answer is that the picture gets distorted. That answer is wrong in an interesting way. What goes is not the picture’s accuracy; it is the point.

What “a second camera” means, exactly #

The flat case is worth restating precisely, because the precision is what makes the curved case a loss rather than a degradation.

Light from a point in the room strikes the mirror and reaches the eye. The eye interprets it as having come from somewhere along the incoming ray, continued backwards through the glass. Do that for every point and the collection of backward rays is a bundle.

For a flat mirror that bundle is concurrent: every one of its rays passes through one point, the eye reflected in the plane of the mirror. Concurrency is exactly the definition of a projection through a centre, so the reflected picture is a projection, and it is a projection from a place a camera could actually be put.

That is why a mirror in a photograph is not a puzzle. It is a second view of the room, taken from behind the wall, and everything this site knows about pairs of views applies to it unchanged.

Measuring the loss #

The question “do these rays meet” has an answer with a number in it, and this site already had the machinery, written for a different subject.

A picture taken through water is not a projection either — continue its rays into the water and they miss each other. The routine that measures that fits the point minimising the summed squared distance to a bundle of lines, and reports how far the lines actually pass from it. A bundle that meets gives zero; one that does not gives the size of the region the rays pass through.

The same routine is used here, deliberately, so that the millimetres in this field and the millimetres in the refraction field are the same millimetres. Two pictures fail to have a viewpoint for entirely different physical reasons — one because the medium bends the light, the other because the surface does — and putting the numbers in one column is a thing a site that measures both should be able to do.

Over a 20 cm patch of mirror, with the eye 1.4 m away:

Three things about that table are worth saying separately.

The flat row is the control. It is not merely small; it is the arithmetic floor of the same computation, so every other row is a real number rather than a residual of the method. And the point the flat bundle converges on is checked against the reflected eye rather than against the fit — a fit that returned a plausible point would not pass that.

The miss vanishes with the curvature. A ball twenty metres across is nearly a flat mirror and misses by less than a fifth of a millimetre. That monotone approach to zero is what says the number is about the shape.

And it grows without bound. A shop mirror ball half a metre across misses by five centimetres over a patch a hand can cover. That is not a subtle effect; it is larger than most of the objects being reflected.

Why “distorted” is the wrong word #

There is a temptation to file all this under distortion, and it is worth resisting because distortion is a much weaker statement.

A fisheye picture is heavily distorted and is still a projection from a point — and a mirror ball’s own rule turns out to be one of the named fisheye rules exactly, which is the sharpest way to see that the two properties come apart. Every ray of it passes through the eye; what varies is the rule mapping direction to position on the picture. Knowing the rule, a fisheye picture can be re-projected onto any other surface exactly, and the result is a picture a different camera would have taken.

A curved mirror’s picture cannot be. There is no rule from direction to position, because there is no single place the directions are measured from. Two points of the mirror show the world from two different virtual positions, and the positions vary continuously across the surface.

The practical consequence is sharp. A photograph containing a curved mirror cannot be rectified into a second view of the room. What it can be asked for is a ratio rather than a size, which is the one recovery that survives the loss of the centre. No amount of processing produces the image a camera behind the wall would have taken, because there is no such camera. The information is there — it is a picture of the room — but it is not a picture from anywhere.

The table has a knee, and it is where the focal length arrives #

The seven rows are usually read as “the miss grows as the ball tightens”, which is true and leaves the most informative thing about them unsaid. The growth is not a single power law.

Take the successive ratios as the radius halves: 2.11, 2.18, 3.37, 3.22, 5.87. Read as exponents those are 1.08, 1.13, 1.75, 1.69 and 2.55 — so the miss goes as roughly $1/R$ for the large balls and steepens toward $1/R^{2}$ and beyond for the small ones. There is a knee, and it sits between the five-metre ball and the two-metre one.

That is exactly where it should be, and the reason is one line of elementary optics. A convex mirror of radius $R$ puts the virtual image of a point at distance $d$ at

$$v = \frac{Rd}{2d + R}$$

behind the surface. For $R \gg d$ that tends to $d$ — the image sits as far behind the glass as the object is in front, which is what a flat mirror does. For $R \ll d$ it tends to $R/2$, the focal length, and the image stops caring where the object is at all. The crossover is at $R = 2d$, which for the eye at 1.4 m is 2.8 m — between the five-metre ball and the two-metre one, which is where the exponent changes.

So the table is two regimes with a transition, and knowing which regime a mirror is in says which description of it is true.

Above $R = 2d$ the mirror is a flat mirror with a defect. The virtual image tracks the eye’s own distance, the bundle is nearly concurrent, and the miss is a small correction growing linearly in the curvature. A shaving mirror gently bowed, a large sheet of glass sagging under its own weight, a shop window: all of these are in this regime, and treating them as flat mirrors with an error bar is honest.

Below it the mirror is an optical element. The virtual image sits near the focal point wherever the object is, the bundle’s spread is dominated by the astigmatic separation the next section derives, and the miss grows faster than the curvature does. A mirror ball, a spoon, a doorknob: these are not nearly-flat mirrors at all, and the reason the caustic is the right object for them and a fitted point is not.

Two further readings are worth having, because the knee explains both.

Why the “swimming” test works so well. A slightly curved mirror is above the knee, so its miss is linear in the curvature — first order — while the visible distortion of shapes is second order. A first-order effect against a second-order one is why walking sideways detects a bow that looking cannot.

And why a ball of water behaves differently. That rung finds a refracting sphere’s miss growing as the cube of the aperture rather than as a power of the radius, because there the variable swept is how much of the surface is used and here it is how tightly the surface bends. Two sweeps, two exponents, and the same underlying statement — which is also the scroll’s, where the miss is the spread of the eye’s own track, and the paraboloid’s, where one shape escapes the whole difficulty by construction. Four objects, four laws, one solver, and it is worth reading a miss’s exponent as carefully as its size: the size says how bad the arrangement is and the exponent says what kind of thing it is.

Why the bundle spreads: two focal surfaces, not one #

The reason a curved mirror has no centre is worth one level of detail, because it also explains why the miss is what it is rather than something else.

Take a narrow pencil of rays leaving the eye and striking the mirror obliquely. Reflected, that pencil does not converge to a point even in the limit of a vanishingly narrow pencil. It converges to a short line segment in one plane and to a different short line segment in the perpendicular plane, at a different distance. That is astigmatism, and it is present for every off-axis pencil on every curved mirror.

So even before the aperture is opened up, the local behaviour of the bundle is two focal surfaces rather than one focal point: a tangential one, from rays in the plane containing the axis, and a sagittal one, from rays across it. On a flat mirror the two coincide, everywhere, and the common surface is a single point — which is the flat result all over again.

Two consequences that show up in the numbers. The best-fitting point is not on either focal surface, because it is a compromise between two families of rays that go to different places. And the miss does not shrink when a smaller patch of mirror is sampled as fast as a naive argument suggests, because part of it is the separation between the two focal surfaces rather than the spread within one.

The one curved mirror that does have an eye #

There is a single configuration in which a curved mirror’s lines of sight are concurrent, and it turned up here as the symptom of a sign error rather than as a discovery — which is the honest way to report it.

An early version of the machinery had the sag of the spherical cap taken the wrong way, so that a mirror labelled convex was in fact concave. The tell was a curvature at which the lines of sight met exactly. No convex mirror does that; a concave one does, when the eye sits at its centre of curvature.

The geometry is immediate once seen. If the eye is at the centre of curvature, every ray from it strikes the surface along the normal, so every ray reflects straight back along itself. The lines of sight are then concurrent to 5.0e-12 mm — and the point they are concurrent at is the eye.

So the exception exists and it is empty. A picture that is a projection from where the viewer already is contains no second viewpoint at all; it is a view of the room from the room. And one step off that curvature, the concurrency is gone.

That is worth keeping as the shape of the whole field. A curved mirror is a projection from a point exactly when the projection is worthless, and every configuration that would be interesting is one where the centre does not exist.

Where the eye went instead #

If the rays do not meet at a point, they still do something, and what they do has a name.

A bundle of lines that is not concurrent is generally tangent to a surface — an envelope — and for a rotationally symmetric mirror the meridional rays are tangent to a curve. That curve is the caustic, and it is what a curved mirror has instead of an eye. The bright cusped shape in the bottom of a coffee cup is one, seen directly, and its length is a measurement of the mirror rather than a nuisance.

The fitted point in the figures above is not on that curve, or anywhere in particular. It is a least-squares compromise, and the useful thing about it is precisely that it is a compromise: every routine that recovers a viewpoint returns something, and the number that matters is not what it returned but how far the rays are from it.

What a reader can check in a room #

Two of these statements can be checked without any equipment, and checking them is more convincing than the arithmetic.

Look at a doorframe in a flat mirror and walk sideways. The reflection behaves exactly like a view through a window into a second room: near things shift more than far ones, by the amounts a real second viewpoint would give. That is parallax from a genuine centre, and it is why a mirror never feels like a picture.

Now look at the same doorframe in the bowl of a spoon and walk sideways. The reflection does not behave like a view from anywhere: different parts of it move by amounts that are not consistent with any single displaced eye, and the whole thing swims. That swimming is the bundle’s failure to be concurrent, seen directly, and it is the reason a curved mirror never reads as a window.

The second observation is the useful one, because it is the fastest available test of whether a reflecting surface is flat. A mirror that is very slightly curved — a large sheet bowed a millimetre or two over its width — passes every visual inspection and fails this one, since the swimming is a first-order effect in the curvature while the visible distortion is second order.

What survives #

Losing the centre of projection removes a great deal and it is worth being exact about what remains, because a curved mirror is still useful and still analysable.

The reflection law survives, pointwise. At every place on the mirror the angle in equals the angle out about the local normal, and that is enough to trace any ray exactly. Nothing here is approximate.

Invertibility survives, over a region. Given the mirror’s shape and the eye’s position, each place in the picture names one direction in the world; the map is one-to-one where the surface is smooth and convex, so a photograph of a mirror ball can be turned into a map of directions.

And a picture surface survives — in the sense the curved field uses the phrase. A mirror ball takes directions to positions in the picture, which is exactly what a picture surface does. What it does not do is take them all from one point, and the two properties turn out to be separable in a way this site has not needed before.

That last observation is the useful one and it has a rung of its own. A mirror ball’s rule, read as a picture surface, turns out to be one of the four named fisheye rules — exactly, in the limit of a distant camera, and by an argument two lines long.

The three ways a picture can fail to have a viewpoint #

This site now has three, and they are worth setting beside each other because they look alike and are not.

The light bends. A picture taken through water or glass has rays that kink at the interface, and continued into the far medium they miss. The failure is in the medium, and the amount depends on how obliquely the interface is crossed.

The eye moves. A scroll, a pushbroom, a rolling shutter: each column or row of the picture is taken from a different place, so there is no one centre by construction. The failure is in time, and the miss is the spread of the eye’s track.

And the surface curves. A curved mirror sends each part of the incoming bundle off in a direction that depends on where it struck, so the virtual sources are spread over a region. The failure is in the geometry of the reflector, and the miss grows with its curvature.

All three are measured by the same solver and reported in the same units, which is the point of having had it. And all three end at the same place: a picture that is a picture of something and a projection of nothing.