A ball of water has no eye either

The mirror field settled its version of this question two phases ago. A curved mirror has no eye shows that a flat mirror is a projection from a point — reflect the eye in the plane and every line of sight passes through the reflected point, to 2.8 × 10⁻¹² millimetres — and that curving it does not make the point worse but removes it. Over twenty centimetres of a mirror ball two metres across, the lines of sight miss their own best-fitting point by 2.8 millimetres.

The refraction field has never asked the same question of a curved surface. Every interface it has traced is flat: a pane, an aquarium wall, a flat port, the surface of a pool. A picture through water has no viewpoint establishes that even a plane interface destroys the centre, and misses by ten millimetres.

So the case that is missing is the curved refracting surface, and the obvious specimen is a ball of water.

Why a ball rather than a lens #

A ball is the right specimen for three reasons and each is worth a sentence.

It is one surface repeated, so nothing about the result can be attributed to a designer’s choice of two radii, a thickness, or a glass — the arrangement has one number in it, and that number is the radius.

It is a thing readers have looked through: a paperweight, a marble, a drop of water on a leaf, the flask of water a lacemaker once put in front of a candle. Nobody expects a good picture from any of them, and the interest is in what precisely is wrong.

And it does have a paraxial focus, so the fit has somewhere to converge to rather than merely failing. That is the difference between this and the curved mirror, and it is what makes the measurement here a crossing rather than a verdict.

The ray, traced #

A ray parallel to the axis at height $h$ meets the front of the ball, refracts by Snell’s law, crosses, meets the back, refracts again, and leaves. Four steps, all of them exact, none of them approximate — the geometry is planar by construction because the incoming ray and the surface normals share a plane, so the trace is two-dimensional and is written that way rather than lifted into three and brought back.

At small $h$ the emerging rays cross the axis at a common place, and the place is the ball’s paraxial focal point, measured from the far surface:

$$f = \frac{nR}{2(n-1)} - R$$

For water, $n = 4/3$, and a fifty-millimetre radius gives 50.08 millimetres behind the glass. That is the prediction. The measurement is where the rays actually cross.

What opens up as the aperture opens #

Take nine rays out to a stated fraction of the radius and record where each crosses the axis. At two per cent of the radius they cross within 0.0136 millimetres of each other — three ten-thousandths of the radius. At ninety per cent they are spread over 34.2 millimetres, which is more than two thirds of the radius.

The rays through the outer part of the ball cross the axis much nearer the glass than the paraxial ones do, which is the ordinary sign of spherical aberration and is not the point of the essay. The point is what happens to the centre.

The fit, and its control #

Continue the emerging rays and fit them to a common point by least squares, which is the instrument a picture through water has no viewpoint uses on the flat case and a curved mirror has no eye uses on the mirror.

At seven tenths of the aperture the residual is 1.31 millimetres on a ball of fifty-millimetre radius — 2.6 per cent of the radius. At two per cent of the aperture it is 0.024 micrometres. The ratio between the two is fifty-four thousand.

That ratio is the whole answer, and it says something more careful than “a curved refracting surface has no centre”. A ball does have a centre of projection. It has one in the limit of zero aperture, where it is a thin lens with a focal length and every ray obeys it. What it does not have is a centre at any aperture where it is doing anything: gathering light means using the outer part of the surface, and the outer part is where the centre goes.

Why the control has to be the paraxial fan and not a ball of air #

The obvious control for a claim of this shape is the same computation with the effect removed — a ball of air, with the refractive index set to one. It is the wrong control, and finding out why is worth recording.

A ball of air does nothing to the rays. They pass through parallel to the axis and parallel to each other, and a bundle of parallel rays has no closest point at all: the least-squares system is singular, and the honest answer is “at infinity” rather than “at the origin, exactly”. A routine that reported a number there would be reporting a division by something near zero.

So the control has to be an arrangement where a centre genuinely exists and the same fit finds it, and the paraxial fan is that arrangement. It uses the same rays, the same solver and the same tolerance, and it differs only in how much of the ball is used — which is exactly the variable the claim is about.

Two exponents, and why they differ #

The sweep runs from a thirtieth of a per cent to two thirds, which is a factor of two thousand in the answer, and the numbers behind it obey two different power laws. Separating them says which of them a reader should design against.

The spread of the axis crossings grows as the square of the aperture. That is the longitudinal aberration, and third-order theory puts it at $C h^{2}$. Fitting $C$ from the seven-tenths reading — 18.57 mm at $h/R = 0.7$ — gives $C = 37.9$ mm, which predicts 15.2 microns at two per cent against a measured 13.6, and 30.7 mm at nine tenths against a measured 34.2. Eleven per cent either way across a factor of 1,225 in the answer, the residual being the $h^{4}$ term that third-order theory leaves out.

The miss of the fitted centre grows as the cube. That is the transverse aberration, and the distinction is the standard one: a ray crossing the axis a distance $\delta z$ early does so at an angle proportional to $h$, so its perpendicular miss is $h\,\delta z \propto h^{3}$. The measurements agree — 0.024 microns at two per cent and 1.31 mm at seven tenths is a ratio of 54,600 against $35^{3} = 42,900$, the excess again being the higher-order term.

Three consequences, and the third is the one that reframes the essay’s headline.

The caustic gets relatively thinner as the ball opens. Its length goes as $h^{2}$ and its width as $h^{3}$, so the ratio goes as $h$: a wide-open ball has a long thin caustic and a stopped-down one a short stubby one. That is why the bright arc from a glass of water is a line rather than a patch — the arrangement is far along the aperture axis, where the envelope is at its most elongated.

Designing for a centre is cheap and losing one is expensive. A cube means a tenfold smaller aperture buys a thousandfold better centre, so a stop does an enormous amount of work; and it means a doubling of the aperture costs a factor of eight, so there is no comfortable middle. The transition the essay describes — a centre at two per cent, none at seventy — is not a gradual erosion but a third-power collapse.

And the answer to “does a ball have a centre” is a tolerance, not a yes or a no. Set an acceptable miss $\epsilon$ and the aperture that delivers it is $h/R = (\epsilon/1.31\,\text{mm})^{1/3} \times 0.7$. At a tenth of a millimetre that is 30% of the radius; at a micron, 6%; at the 24 nanometres the paraxial fan returns, 2%. Every one of those is a real ball with a real centre over a real region, which is exactly the reading the eye is a place, not a point arrives at from the sensor’s side, now with the exponent attached.

The caustic is where the centre went #

A bundle of rays with no common point still has structure, and the structure has a name the mirror field has already built machinery for. The emerging rays are tangent to a surface — their envelope — and that surface is the caustic.

Where the focus went builds the caustic of a mirror and the caustic is the mirror’s own ruler shows that its shape encodes the mirror’s. The refracting case is the same construction on a different family of rays: a cusped surface, its tip at the paraxial focus, opening back toward the glass.

The reason to name it here is that it explains what a reader sees looking through a paperweight. The image is not blurred in the way an out-of-focus photograph is blurred — it is smeared along a curve, brighter where the caustic folds and dimmer away from it, and the bright arc a glass of water throws on a table is the same surface seen where it meets the table.

What this does to a picture #

Everything above is about a bundle of parallel rays, which is the arrangement an optician tests a lens with. A picture is made of bundles from many directions, and the consequence for a picture is worth stating separately.

A projection through a centre has one property that everything on this site depends on: a straight line in the world images as a straight line, because the line and the centre span a plane and the plane cuts the picture in a line. Take the centre away and that argument has nothing to stand on.

So a picture through a ball has bent lines, and it has bent them by an amount that varies across the field rather than by a fixed rule — which is the difference between this and a lens distortion. A lens destroys the invariant measures what a real photographic lens does to the cross-ratio and finds it small and correctable, because a designed lens is a designed approximation to a projection. A ball is not an approximation to anything.

The one arrangement where a ball is a good camera #

There is a real use for the arrangement, and it is worth naming because it shows what has to be given up to get the centre back.

Stop the ball down. Put an opaque screen with a small hole in it against the glass, so that only the paraxial part of the surface is used, and the ball becomes a lens with a focal length and a genuine centre — at the price of the light the rest of the aperture would have gathered. That is what the crossing measured above is: the aperture at which the centre stops being usable, and it is at a few per cent of the radius rather than at some large fraction of it.

This is also, roughly, the situation of an eye. A biological eye is a strongly curved refracting surface with a stop in front of it, and its pupil is a small fraction of its cornea’s radius — the arrangement above, with the fraction chosen by something other than geometry. What that costs when the pupil opens is the aperture field’s subject, and the eye is a place, not a point is where this collection prices it.

The two failures are not the same failure #

It is tempting to file the curved mirror and the curved refractor together as “curved surfaces have no centre” and move on. The measurements say otherwise, and the difference is instructive about what a centre of projection actually requires.

A spherical mirror sends a ray parallel to the axis through a point at half the radius, and it does so for paraxial rays too — so a mirror ball has a paraxial focus in exactly the way a water ball does. Why, then, does the mirror essay find no centre at any aperture while this one finds one at small aperture?

Because the two essays are asking about different bundles. The mirror essay fits a centre to the lines of sight from a fixed eye looking at a curved mirror, which is a bundle of rays arriving from all over the mirror at once — the whole aperture, by construction, because that is what looking at a mirror ball means. This essay fits a centre to a bundle restricted to a stated fraction of the aperture, and the restriction is the variable.

Run the mirror computation on a small patch and it too returns a centre. Run this one on the whole ball and it too returns nothing. The two results agree; what differs is the arrangement each is a statement about, and the honest summary of both is a single sentence: a curved surface has a centre over a region small enough, and the size of “small enough” is what the aperture sweep measures.

That matters because it is the same sentence the eye is a place, not a point arrives at from the sensor side. Every projection through a centre on this site is an idealisation of an arrangement with a size, and the useful question is never whether the idealisation is exact — it is not — but over what region it holds and what leaving that region costs.

What the model does not have #

One sphere, one refractive index, one wavelength, and rays that meet it. Each is a real restriction.

A drop of water is not a sphere unless it is very small or falling, and a drop resting on a leaf is a spherical cap with a flat base, which is a different arrangement with a different answer. Two indices — a glass shell around water, say — is a two-surface system and belongs with the dome ports, where the dome knows its offset in units of itself does the arithmetic. Rays arriving steeply enough are totally internally reflected at the back surface and never emerge at all, which the trace reports as a refusal rather than as a bent ray.

And nothing here is about colour. The refractive index of water depends on wavelength, so a real ball has a different focus for each colour, and the spread that produces is comparable to the spread measured above at small apertures. That is dispersion, it is optics, and this collection computes geometry.

What is measured here #

Three numbers and a control.

Nine rays through a ball of water at seven tenths of its radius cross the axis over 18.57 millimetres, on a ball whose radius is fifty — thirty-seven per cent. The same rays continued and fitted to a common point miss it by 1.31 millimetres, which is 2.6 per cent of the radius. The same fit on a fan at two per cent of the aperture misses by 0.024 micrometres, a factor of fifty-four thousand, which is what says the first number is a measurement of the aperture rather than of the solver. And the paraxial focus predicted by the thin-lens formula, 50.08 millimetres behind the glass, is where that tight fan converges.

The short version #

A curved refracting surface has a centre of projection at zero aperture and none at any aperture where it gathers light. A ball of water fifty millimetres across, used out to seven tenths of its radius, spreads its axis crossings over eighteen and a half millimetres and misses its own best-fitting point by 1.31; used out to two per cent, it misses by twenty-four nanometres.

That is a sharper statement than the curved mirror’s, which has no centre at any aperture at all, and it comes with the crossing point attached. The bundle that has lost its point still has an envelope, and the bright arc a glass of water throws on a table is where a reader can see it.