A pane gives a product before it gives two numbers

What survives a pane of glass is one of the sharper results in this collection, and it is sharp because it is exactly two-sided. A plane-parallel slab leaves every direction untouched — the emergent ray is parallel to the incident one, to the last bit — and moves every point. So a photograph taken through a window has exactly the vanishing points it would have had without it, and nothing in it is where it was.

That essay takes the pane as given and measures what it does. This one turns the question round.

A reader has a photograph taken through a window, and a measurement of how far the window moved things. What is the window?

The displacement #

A ray meeting a slab of thickness $t$ at incidence $\theta_1$ refracts to $\theta_2$ inside, crosses, and refracts back out parallel to where it started. Its lateral displacement is

$$s = t \cdot \frac{\sin(\theta_1 - \theta_2)}{\cos \theta_2}$$

with $\sin\theta_2 = \sin\theta_1 / n$.

At small angles this simplifies in a way that decides the essay. Expand: $\theta_2 \approx \theta_1/n$, $\cos\theta_2 \approx 1$, and

$$s \approx t\left(1 - \frac{1}{n}\right)\theta_1.$$

The thickness and the index arrive multiplied together. Whatever a reader measures at small angles is a measurement of one number, and there is no arrangement of that measurement that returns two.

It is worth noticing how strong that statement is. This is not a claim that the two are hard to separate, or that the separation is noisy. At small angles the displacement is a function of one combination of $t$ and $n$, and a function of one number cannot return two — the deficiency is in the observable rather than in the measurement of it.

The combination is worth a name of its own, because it recurs in the field. $t(1 - 1/n)$ is the optical thickness deficit: how much less optical path the slab presents than the same distance of air, per unit of transverse angle. Every result about flat interfaces in this field turns out to be a statement about that quantity rather than about the thickness or the index.

What the valley looks like #

Fix the product $t(1 - 1/n)$ at 3.42 mm and vary $n$. Each choice of $n$ names a thickness: 13.2 mm at $n = 1.35$, 10.0 mm at 1.52, 8.0 mm at 1.75, 6.5 mm at 2.1.

Those are four physically very different pieces of glass. Over an eight-degree fan their displacement curves are 1.8 microns apart at worst — well below the resolution of a ruler laid on a photograph, and below the resolution of most things a ruler could be laid on.

A least-squares fit over that fan has a valley rather than a minimum: a whole one-parameter family of $(t, n)$ pairs explains the data equally well, and where the solver stops is decided by where it started rather than by anything in the measurement.

The condition number of the fit says the same thing in a colder way. Over the narrow fan the two parameters are correlated at 0.9999975 and the normal matrix’s condition number is $1.3\times10^{9}$. That is a fit with one degree of freedom and two names for it.

A valley is not a wide minimum #

The distinction is worth insisting on, because a fit reports the two identically and they mean opposite things.

A wide minimum means the answer is one point and the data constrain it loosely, so more data narrow it. Averaging helps. Error bars shrink as the root of the number of measurements, in the ordinary way.

A valley means the answer is a curve, not a point. More measurements along the same fan narrow the valley across its width and do nothing at all along its length, because there is no information in that direction to accumulate. Averaging a thousand narrow-fan measurements returns a thousand times more confidence in the product and no more in either factor.

That is the practical difference and it is a large one. It also explains why the failure is so persistent: every diagnostic a fitter routinely looks at — the residual, the fit’s own scatter, the agreement between repeats — improves exactly as it should while the reported thickness stays wherever the starting guess put it.

The instrument that would say so is the correlation between the parameters, and it is the one number a fit reports that nobody reads.

And the fan that separates them #

Open the fan to sixty-two degrees and the same four panes separate by 1.13 mm, which is a difference a ruler sees easily. The correlation falls from 0.9999975 to 0.975 and the condition number falls by four orders of magnitude.

So the degeneracy is severe and not exact, and the distinction matters enormously. A severe degeneracy is broken by better data — here, by a wider fan. An exact one is not broken by anything, which is the position a dome port is in, where widening the fan improves the estimate of the ratio and never separates the radius from the offset.

The two cases sit next to each other in this row on purpose. They look identical from inside a narrow experiment — two parameters, one observable, a solver that returns whatever it started near — and they are opposite in what to do about it.

Where the separation comes from #

It is worth knowing which term does the work, because it says how much fan is enough.

Expand the displacement to third order in $\theta_1$. The leading term is $t(1 - 1/n)\theta_1$, which is the product. The next is a $\theta_1^3$ term whose coefficient depends on $n$ differently — it carries $1/n^3$ rather than $1/n$ — so it is not a multiple of the first, and a measurement that can see it can separate the two parameters.

The size of that term relative to the first goes as $\theta_1^2$. At eight degrees, $\theta_1^2$ is about 0.02, so the separating information is two per cent of a signal that is itself microns. At sixty degrees it is more than one, and the two terms are comparable.

That is the whole story of the fan, and it is a general one. Two parameters that enter a leading term as a product separate in the next term or not at all, and how much of the range is needed is set by how fast the next term grows.

The separation is a cube, and that is the whole design rule #

The third-order argument can be carried out rather than gestured at, and doing so replaces “wider” with a number and explains why the fan has to be so uncomfortably wide.

Expanding the displacement to third order in the incidence angle, with $a = 1 - 1/n$,

$$\frac{s}{t} = a\,\theta + C_{3}\,\theta^{3} + \ldots, \qquad C_{3} = \frac{1}{6}\left(\frac{1}{n} - \frac{1}{n^{3}} - a^{3}\right) + \frac{a}{2n^{2}}.$$

Along the family the product $ta$ is held fixed at 3.42 mm, so the leading term is identical for every member by construction and the entire difference between two panes is $ta\,\theta^{3}\,\Delta(C_{3}/a)$. For the family’s extremes, $n = 1.35$ and $n = 2.1$, the ratio $C_{3}/a$ is 0.478 and 0.185, so

$$\Delta s \approx 1.0\ \text{mm} \times \theta^{3},$$

with $\theta$ in radians. At eight degrees that is 2.7 microns; at forty, 0.34 mm; at sixty-two, 1.27 mm. The measured separations are 1.8 microns, tenths of a millimetre, and 1.13 mm — the cubic law reproducing all three across a factor of five hundred in the answer, and running slightly high at the wide end because a third-order expansion is optimistic where the angle is no longer small.

A cube is a brutally unhelpful exponent for the reader and a very helpful one for the designer, and both halves matter.

For the reader of an existing photograph it means the information is not merely faint at small angles, it is faint as the cube. Halving the fan divides the separation by eight. There is no amount of care, precision or averaging that recovers a quantity suppressed that fast, which is the formal version of the essay’s conclusion that the information is not in the pictures people take.

For anyone able to choose the geometry it means the opposite. Going from twenty degrees to forty multiplies the separation by eight, and from twenty to sixty by twenty-seven. So a single steeply oblique view is worth more than any number of frontal ones, and the design instruction is a single number rather than a strategy: reach a fan whose half-angle cubed times a millimetre exceeds the measurement precision. At a tenth of a millimetre that is about thirty degrees by the cubic estimate and rather more in truth, which is where the forty degrees quoted below comes from.

The same exponent explains why the narrow-fan correlation is so extreme. The off-diagonal term of the normal matrix is set by how much the two parameters’ effects differ over the sampled range, and that difference is $O(\theta^{2})$ relative to the signal, so the correlation approaches one as $1 - O(\theta^{4})$. At eight degrees $\theta^{4}$ is $3.8 \times 10^{-4}$, and the reported correlation of 0.9999975 is that number squared away from one — which is why a condition number of $10^{9}$ arrives from a geometry that looks perfectly reasonable.

What the vanishing points do not say #

There is an obvious idea for breaking the degeneracy that does not work, and ruling it out is worth a paragraph because it is the first thing a reader of this collection would reach for.

Use the vanishing points. They are exactly recoverable, the camera comes out of them, and a recovered camera is a strong constraint.

It says nothing. The pane preserves directions exactly, so the vanishing points are the vanishing points the picture would have had with no glass at all — they carry no information about the glass whatsoever, not a little, none. That is the content of the earlier essay’s headline read from the other side: the quantity that survives the pane is precisely the quantity that cannot measure it.

So the recovery has to work on finite points, which are the ones that moved, and finite points are where the scale trouble in this collection always lives.

The practice that hides the problem #

In practice nobody fits both numbers, and it is worth being explicit about what is done instead, because it is a choice presented as a fact.

The index is quoted. Crown glass is 1.52, acrylic is 1.49, water is 1.333, and a measurement that takes one of those as given has one unknown left and returns it cleanly.

That is entirely reasonable when the glass is known. It stops being reasonable the moment the material is uncertain, and the failure is silent: a fit that assumes 1.52 for a pane of index 1.75 returns a thickness wrong by the ratio of the two $(1 - 1/n)$ factors, which is 24%, with a residual that says nothing at all.

Which is the same failure a wrong shape family produces on a mirror, arriving from a different direction. A model with a parameter fixed by assumption returns the other parameters conditioned on that assumption, and the residual measures how well the remaining freedom disguised the error rather than whether the assumption was true.

How large the fan has to be, as a number #

It is worth turning the third-order argument into an instruction, because “wider” is not one.

The separating term is smaller than the leading one by roughly $\theta_1^2$, so to see a fractional difference $\epsilon$ between two panes of the family, the fan has to reach about $\sqrt{\epsilon \cdot k}$ radians, with $k$ a factor of order one that depends on how far apart the two indices are.

Put the numbers in. To separate $n = 1.35$ from $n = 1.75$ at a measurement precision of a tenth of a millimetre, on a pane whose displacement is a few millimetres, the fan needs to reach past about forty degrees. Below that the difference is inside the measurement; above it, it is not.

Forty degrees off perpendicular is a sharply oblique view of a window and it is not what anybody photographs by accident. Which is the honest summary of the whole essay: the information is available and it is not in the pictures people take.

Two panes, and why that does not help either #

Suppose the reader can photograph the same scene through two different panes. Does that separate anything?

It separates the two panes from each other and neither from itself. Each picture supplies its own product, so two pictures give two products and four unknowns, which is worse than where it started.

That is worth contrasting with the pair of pictures that does work on a ball, where two views from different places genuinely add a constraint. The difference is that moving the camera changes the geometry there and changes nothing here — the pane’s effect depends on the angle at the glass, not on where the camera is, so two cameras looking through the same pane at the same obliquity see the same displacement.

What would help is two pictures through the same pane at very different obliquities — which is the wide fan again, gathered across pictures instead of within one. The information is in the angular range, and it does not matter whether the range is covered by one wide photograph or several narrow ones aimed differently.

This is a useful thing to know because a narrow lens is common and a wide fan through a window is not. A sequence of ordinary photographs of the same window edge, taken from steeply different angles, carries the same separating information as one fisheye frame.

What the aquarium does differently #

The field’s other flat interface is worth putting beside this one, because it separates the two parameters for free and the reason is instructive.

Measured down from the waterline and the port that is not there both concern an interface with water behind it rather than air. There the ray does not return to its original direction — it stays bent, because the media on the two sides differ — so the observable is an angle rather than a displacement, and an angle depends on the index alone with no thickness in it at all.

So the aquarium’s flat port hands over the index cleanly and says nothing about the glass’s thickness, and the window hands over a product of the two. Two arrangements of the same materials, two completely different identifiability structures, and the difference is only whether the media on the two sides match.

That is the sort of thing that is obvious once stated and is very easy to carry the wrong intuition about, because both arrangements are “a flat piece of glass” in ordinary speech.

What this row has now said five times #

The pane is the fifth instrument in a row that has asked one question of each of them: given the picture, what is the surface?

A mirror ball returns $r/d$ and borrows a near room point for the size. A caustic returns the radius as a length and borrows nothing. A reflected fan returns a radius inside a family and borrows the family. A dome port returns its decentring over its radius and borrows a length in the water. A pane returns $t(1 - 1/n)$ and borrows either a wide fan or a quoted index.

Four ratios and one length, and the length is the one instrument that puts its answer on the furniture.

The rule, and the exception that proves it #

The rule is that an arrangement made of reflection, refraction and straight rays has a similarity acting on it, so its observables are functions of dimensionless ratios and its dimensioned quantities are invisible.

The pane’s case shows the rule needs one refinement. The degeneracy here is not the similarity — a pane of double the thickness at the same index gives double the displacement, so the thickness is perfectly visible. The degeneracy is between the thickness and the index, which is a different kind of thing: two parameters entering one term as a product, in a way that a higher-order term undoes.

The same two-mode reading sorts the rest of the collection’s degeneracies. The scale a single view cannot give is the first kind and no data fixes it. The two readings of a drawn fold are a third kind again — a discrete ambiguity rather than a continuous one, which needs a different observation rather than a wider one.

So there are two failure modes wearing the same clothes, and separating them is the first thing to do with any such fit. If the parameters cancel out of a similarity, no data helps. If they merely multiply in the leading term, more range does. The narrow fan cannot tell those apart, which is why a narrow experiment reporting a strong correlation should be read as a question rather than as an answer.