Closer than they appear, by a factor with a number in it

A curved mirror has no eye is a statement about rays. A flat mirror sends every line of sight through one point — the eye reflected in the plane — and a curved one does not: over twenty centimetres of a mirror two metres across, the lines miss their own best-fitting point by 2.8 millimetres.

That is a statement about what a curved mirror is not. The moulded warning on every convex mirror in the world is a statement about what it does, and the two have never been put together. This essay makes the second a number.

What “apparent distance” has to mean #

A single eye receives a single ray from each point of the mirror, and a single ray carries a direction and nothing else. So “how far away does it appear” is not a question a fixed eye can answer, and asking it of a curved mirror without saying what would answer it is how the subject stays vague.

Two things settle it, and they are the two an observer actually has. Two stations — two eyes, or one eye at two moments of a turning head — give two apparent directions, and where those cross is the apparent place. A known size gives a distance directly, because an object of known width subtending a known angle is at a known distance.

Both are traced here rather than taken from a formula. The reflection point is solved for each station: the point of the glass at which the law of reflection carries a ray from the object to that eye, found by a root search on the surface rather than by assuming the ray goes to the middle of the mirror. The two apparent directions are then exact, and so is the place they cross.

The flat mirror, to fix what a right answer looks like #

A flat mirror gives the same answer both ways, and it gives the true one.

Its virtual image is the object reflected in the plane, so the apparent distance is the length of the light’s own path — eye to glass to object — rather than the straight-line distance to the object. Measured on a run of objects from two metres to sixty-four, the two-station intersection lands on that path length to six decimal places at every distance, and the apparent distance rises without bound.

That is worth saying plainly because it is the property a convex mirror loses, and it is not obvious that it is a property at all. A flat mirror is an honest instrument for distance: what it reports is a real length in the room, just not the one a driver wants.

The ceiling #

A convex mirror’s apparent distance does not rise without bound. It rises to a limit and stops.

The reason is one line. The image of a point infinitely far away sits half a radius behind the surface — that is what the focal length of a mirror is — so as the object recedes, its image runs toward a fixed place a few centimetres behind the glass and goes no further. The apparent distance from the observer therefore approaches the distance from the eye to the glass, plus half a radius, and nothing beyond that distance can be told from anything else.

Measured: a mirror of one metre radius, held eighty centimetres from the eye, puts an object at sixty-four metres at an apparent 1.296 metres. The predicted ceiling is 0.80 plus 0.50, which is 1.30. A flat mirror in the same place reports 65.8 metres for the same object.

So the distance information is not compressed by a convex mirror. Past twenty metres or so it is gone, and what a driver reads is a place a few centimetres behind the glass which every distant object shares.

Where the warning’s number comes from #

The size reading is what a driver actually uses, and it does not collapse. An object of known width still subtends an angle, the angle still falls with distance, and the distance is still recoverable from it — wrongly, by a factor.

Traced on a 1.5-metre object from four metres to two thousand, the factor falls from 3.39 to 2.60 and settles there. The limit has a closed form and the tracing confirms it to a part in a thousand:

$$\frac{\text{distance read}}{\text{distance true}} \;\longrightarrow\; 1 + \frac{2d}{R},$$

with $d$ the observer’s distance from the glass and $R$ the mirror’s radius. For the mirror above, $1 + 1.6/1.0 = 2.60$. For a two-metre radius, 1.80.

Two things follow that the warning does not say, and both are useful.

The factor is bigger for a mirror further from the eye. A passenger-side wing mirror stands about twice as far from the driver as the driver’s own, so the same curvature there reads twice as far wrong. That, rather than the curvature alone, is why the legend is moulded into the passenger’s mirror and not the driver’s, and it is a fact about where the mirror is mounted rather than about how it is ground.

And the factor does not fall with distance once it has settled. A reader might expect the error to matter most for close objects — it is largest there, 3.39 at four metres — but the close case is the one the driver has other information about. At the distances where a mirror is the only evidence, the factor is constant, so the whole reading is a fixed multiple and the mis-estimate is proportional.

What the curvature is bought with #

None of this is a mistake in the mirror’s design. The curvature is there for the field of view, and the exchange rate can be measured on one pair of axes.

A nine-centimetre mirror eighty centimetres from the eye shows 12.1 degrees of the world when it is flat. Curve it to a radius of two metres and it shows 21.8; to 1.4 metres, 26.0; to 0.7 metres, 40. The distance factor moves the other way over the same range — 1.0 flat, 1.80 at two metres, 2.14 at 1.4, 3.29 at 0.7.

Both quantities are set by the same number, which is what makes this a trade rather than a design problem with a solution. There is no radius at which a small mirror is both wide and honest, and the only free choice left is where to mount it: moving the mirror closer to the eye reduces the distance error, since the factor depends on $d$, and leaves the field of view alone.

The number is two numbers #

Everything above quotes one apparent distance, and the previous finding says there cannot be one.

If a curved mirror had an eye behind it, every pair of stations would give the same answer. It has not, so they do not: an observer whose two stations are separated across the plane of incidence intersects a different pair of rays from one whose stations are separated along it, and the two apparent distances differ.

Measured on an object twenty-four metres away in a mirror of one metre radius, the gap is 0.02 per cent at two degrees off the mirror’s axis, 0.50 per cent at ten, and 0.72 at twelve. It grows as the square of the obliquity, fitting that law to within five per cent over the whole range, and it vanishes on the axis.

So the single figure quoted in the sections above is exact on the mirror’s axis and an approximation everywhere else, and the size of the approximation is now stated rather than assumed. For a wing mirror at ordinary obliquities it is well under one per cent, which is far smaller than the factor of 2.6 it sits inside — which is the honest way to report it: the astigmatism is real, it is the same absence of a viewpoint measured in a viewer’s units, and it is not what makes the mirror misleading.

The same instrument with the radius turned down #

A mirror ball is a wing mirror taken to its limit, and the two numbers behave exactly as the law says they should.

A ball of fifteen centimetres radius, held eighty centimetres away, shows 150 degrees of the world — most of a hemisphere — and its ceiling is 0.875 metres, which is eighty centimetres plus seven and a half. So everything beyond a few metres appears seven and a half centimetres behind the ball’s surface, and the size factor is 11.7. A thirty-centimetre ball shows 78 degrees, its ceiling is 0.950, and its factor is 6.3.

That is why a mirror ball is so good at the job it is actually used for and so useless at the one it is not. A mirror ball is an equal-area fisheye establishes what its map of directions is, exactly, and a map of directions is all a lighting probe or a shop’s security mirror wants. What it cannot do is tell a viewer how far away anything is, and the ceiling says why in one number: the whole room arrives in a shell seven centimetres thick.

The cone that reads the floor is the other extreme of the same family — a surface curved in one direction and straight in the other, which has a ceiling along one axis and none along the other. The two apparent distances it delivers are not nearly equal; they are a finite number and infinity.

To use the factor, the radius has to come from somewhere #

The correction is a factor of one plus twice $d$ over $R$, and both of those are lengths that have to be known. That is not a difficulty for a driver, who can measure both with a tape. It is a difficulty for anybody reading a photograph.

A picture of a convex mirror does not give its radius. A mirror ball does not know its size is the general statement — a ball twice as large twice as far away draws the identical outline and the identical reflected field — and the argument applies unchanged to a wing mirror: the map of directions a convex mirror produces depends on the ratio of the mirror’s size to its distance and not on either alone. So a photograph containing a mirror leaves the pair $(d, R)$ free along a whole curve, and the factor along that curve is not constant.

That is the same missing length the one thing a single view cannot give records for every other recovery in the subject, arriving here in a place that looks like it should be exempt. A convex mirror is a measuring instrument with no scale on it, and one length — the radius, the mounting distance, or the width of anything visible in the glass — restores the whole reading.

Fitting the radius from the reflection itself is possible and it is not free. A fitted radius is wrong before it is uncertain measures what happens when a shape is fitted to a mirror’s own picture: the answer has a bias before it has a spread, because the model is being asked to absorb a departure it does not contain. A wing mirror fitted as a sphere when it is aspheric is exactly that case.

The shape matters less than the curvature #

A wing mirror is not a sphere. Most are aspheric, and some are two surfaces joined — a spherical inner portion and a tighter outer band that shows the blind spot.

Nothing above depends on the sphere except the exact half-radius. Any convex surface has a focal length, the image of a distant point sits at it, and the ceiling and the factor follow with that focal length in place of $R/2$. The one shape that focuses is the standing account of which shapes have an exact focus and which have only an approximate one, and the answer there — a paraboloid for parallel rays, nothing for anything else — applies here as the reason the ceiling is approached rather than reached exactly.

The two-surface mirror is the interesting case, because it delivers two factors in one piece of glass. A driver reading across the seam reads two different scales, and the seam is visible precisely because the two scales disagree. That is the same discontinuity a joined picture surface produces, and it is the reason the outer band is usually marked.

The same ceiling turns up in water #

The mechanism is not about mirrors, and one check on that is worth making because it says which of the ingredients is load-bearing.

Nothing in the argument used reflection. It used a curved surface with a focal length, and refraction at a curved surface has one too. A ball of water has no eye either makes the ray-miss measurement for the refracting case and finds the same absence of a viewpoint; the distance reading behaves the same way for the same reason, and a glass sphere used as a lens delivers distant objects to a fixed place just as a mirror ball does.

What is load-bearing is the curvature’s sign. A concave mirror has a focal length too, and it does not compress distances into a shell — it magnifies, inverts beyond the focus, and delivers an image in front of the glass rather than behind it. The ceiling is a property of a surface that bends rays apart, and the warning moulded into a wing mirror could not be moulded into a shaving mirror because the error there runs the other way.

What this does not settle #

The observer is assumed to read the geometry. A driver does not intersect rays; they use a learned relation between the size of a car in the mirror and its distance, learned in that mirror. A driver who has learned the mirror is not making the error above at all, and the warning is for the driver who has not. Nothing here measures how quickly that learning happens or how well it transfers between vehicles.

And nothing here is about motion. The strongest cue a mirror carries is not size but rate of change — how quickly the image grows — and a convex mirror compresses that too, by a factor that is the derivative of the compression rather than the compression itself. That is a different quantity and it is not measured here.

The ceiling is quoted for one mounting. Eighty centimetres is a plausible distance from a driver’s eye to a driver’s own wing mirror and nothing more; the ceiling and the factor both move with it, and a mirror inside the cabin, a truck’s mirror on a long arm, and a shop’s corner mirror across a room are three quite different instruments made of the same glass.

Still open: what a mirror would have to be to read true #

The trade above has no good end because one radius sets both quantities. It sets both only because the mirror has one radius, and a surface need not.

A surface whose curvature varies with the angle at which it is being read could in principle hold the field of view of a tight mirror and the distance reading of a flatter one over part of its aperture. The question with a measurement in it asks how far that can go: given a rotationally symmetric surface free to have any profile, and a stated aperture and mounting distance, what is the widest field obtainable subject to the size-reading factor staying below a stated bound across the whole field — and whether the answer is a smooth surface at all or a small number of zones, which is what the two-surface mirror already is. The bound cannot be one, since a mirror that reads true everywhere is flat; the interesting number is how close to one it can be held while the field stays wide, and whether the limit is set by the geometry or by the astigmatism the varying curvature introduces.

The short version #

A convex mirror’s apparent distance has a ceiling. The image of anything far away sits half a radius behind the glass, so a mirror of one metre radius held eighty centimetres from the eye reports 1.296 metres for an object at sixty-four, where a flat mirror in the same place reports 65.8 and keeps rising.

What survives is the size reading, and it is wrong by a factor that settles at one plus twice the eye’s distance over the radius — 2.60 for that mirror, 1.80 for a two-metre one, and larger again for a mirror mounted further away, which is the reason the passenger’s side carries the warning. The curvature buys field of view at exactly that rate: 12.1 degrees flat, 40 at a radius of 0.7 metres.