A fitted radius is wrong before it is uncertain

The one shape that focuses puts a sphere and a paraboloid side by side and measures the difference. They are chosen to share a vertex radius of curvature, so they agree to second order at the axis, and the whole of what separates them is fourth order and beyond. That is what makes the comparison a comparison of two shapes rather than of two arbitrary mirrors.

This essay asks the question that comparison invites and does not answer. Given a mirror’s picture and nothing else, which shape is it?

The answer turns out to be a statement about experiments in general, and it is not the one a reader expects. The trouble with a small experiment is not that its answer is uncertain. It is that its answer is wrong — confidently, repeatably, with a residual that says nothing is amiss.

What “agree to second order” is doing #

The phrase does a great deal of work and it is worth unpacking before anything is measured, because the whole essay is a consequence of it.

Write both surfaces as a height above the tangent plane at the vertex, as a function of the distance $r$ from the axis. A paraboloid of vertex radius $R$ is exactly $r^2/2R$. A sphere of the same vertex radius is $R - \sqrt{R^2 - r^2}$, which expands as

$$\frac{r^2}{2R} + \frac{r^4}{8R^3} + \frac{r^6}{16R^5} + \dots$$

The first terms are identical. That is not an approximation holding for small $r$; it is an algebraic identity, and it is what “the same vertex radius” was chosen to mean.

So the entire difference between the two mirrors is $r^4/8R^3$ and beyond. At $r = R/10$ that leading difference is $R/80000$ — for a 1.6 m mirror, twenty microns. At $r = R/3$ it is $R/650$, which is two and a half millimetres. The difference grows as the fourth power of the aperture while the surfaces themselves grow as the second, so the ratio of what separates them to what they have in common grows only as the square.

That is the arithmetic behind every number below. A small aperture is not a slightly worse experiment than a large one. It is an experiment in which the quantity of interest is a part in a hundred thousand of the quantity being measured.

The observable #

A parallel bundle comes in along the axis, meets the mirror at some radius from it, and leaves at an angle. Sample that across the mirror and the result is a fan of directions — one number per radius, and the whole of what a picture of a reflected grid, or a Hartmann screen, or a photograph of a straight edge in the mirror, actually measures.

The fan is a smooth function of the radius and it depends on the shape. It is the same bundle the caustic is the envelope of, read one ray at a time instead of all at once — and reading it one ray at a time is what makes it fittable and what loses the information the envelope keeps. A paraboloid’s reflected angle is exactly proportional to the radius near the axis and departs from proportional in a way its own curvature dictates; a sphere’s departs differently, from fourth order on.

Fit a vertex radius inside a named family to a measured fan and there are two numbers to report: the fitted radius, and how much of the fan the fit failed to explain.

The pairing that matters #

Everything hangs on reporting both, so it is worth saying why one of them alone is uninformative.

The residual is the only quantity an experimenter has. It is computable from the data without knowing the truth, and it is what a reader looks at to decide whether a fit was successful.

The bias is what a reader wants and cannot have. It is the fitted answer minus the true one, and computing it requires the truth.

The whole content of this essay is that the two do not arrive together, and that the residual arrives late.

The numbers #

Fit a paraboloid to a sphere’s fan, over a sphere of radius 1.6 m, at six apertures.

At a half-aperture of 5 cm the fitted radius is 1.5995 m — a bias of 0.030% — and the residual is $2.8\times10^{-4}$ degrees. Against a measurement floor of a fiftieth of a degree, which is about a pixel at an ordinary focal length, that residual is seventy times too small to see.

At 20 cm the bias is 0.48% and the residual is 0.018°, still under the floor.

At 30 cm the residual is 0.060° and finally clears it. The bias there is 1.08%.

So the experiment that first tells the experimenter something is wrong is the one whose answer is already thirty-six times more wrong than the answer of the experiment that told them nothing.

Two exponents, and the ratio between them #

The six readings are two power laws, and separating them says exactly how badly the residual lags.

The bias goes as the square of the aperture. From 5 cm to 20 is a factor of four in $a$ and sixteen in the bias: 0.030% to 0.48%. From 20 to 30 is 2.25, and 0.48% to 1.08% is 2.25. Both exactly.

The residual goes as the cube. $2.8\times10^{-4}$ to 0.018 degrees is a factor of 64, which is $4^{3}$; 0.018 to 0.060 is 3.33 against $1.5^{3} = 3.375$.

So the quantity a reader can see grows one power faster than the quantity they want to know, and the ratio of the two goes as $1/a$. The narrower the aperture, the larger the hidden error is compared with the visible one — which is the essay’s finding stated as an exponent, and it says the lag is structural rather than a feature of these six settings.

It also fixes the crossing’s own scaling. The residual clears a floor $\phi$ at $a \propto \phi^{1/3}$, and the bias there is $\propto \phi^{2/3}$ — so dividing the measurement floor by sixty-four moves the crossing aperture down by four and the bias at the crossing down by sixteen. Care buys a two-thirds power of the thing it is being spent on, which is the honest exchange rate for the whole exercise.

Care buys the diagnosis and not the answer #

That figure is worth pausing on, because it says what a more careful experiment does and does not do.

Lower the measurement floor by a factor of four and the crossing moves left: the residual clears a smaller floor at a smaller aperture, so the model announces itself sooner. Good.

What does not change is the fitted radius. At a 5 cm half-aperture the paraboloid fit returns 1.5995 m whether the measurements are good to a fiftieth of a degree or to a thousandth of one, because the bias is a property of the model and the geometry and has no noise in it at all. A more careful experiment over the same aperture returns the same wrong number, more precisely.

So precision and correctness come apart here in the most complete way available. Precision buys the ability to notice; it does not buy the answer. The only thing that buys the answer is a wider aperture, which is a different experiment rather than a better one.

The control, which is what makes the rest a measurement #

Fit the right family — a sphere to a sphere’s fan — over the same six apertures. The residual is $10^{-13}$ degrees at every one of them.

That matters more than it looks. Without it, the climb in the wrong family’s residual could be an artefact of the sampling, or of the fit’s conditioning at large aperture, or of anything else that grows with the amount of mirror in use. With it, the climb is the model and nothing else, because the identical machinery over the identical data returns arithmetic noise when the family is right.

This is the shape of check the whole collection runs and it is worth naming again: an assertion that has never rejected anything proves nothing, and a residual that has never been shown to go to zero on a correct model is not a measurement of a model.

Why the bias is negative, and what that means #

The fitted radius comes out smaller than the truth, at every aperture. That is not an accident of the arrangement.

A sphere’s sag is $R - \sqrt{R^2 - r^2}$, which expands as $r^2/2R + r^4/8R^3 + \dots$; a paraboloid’s is $r^2/2R$ exactly. So the sphere curves away faster than the paraboloid of the same vertex radius, and the extra curvature is all in the fourth-order term.

A least-squares fit inside the paraboloid family has one parameter with which to absorb an entire fourth-order discrepancy. The only thing it can do is tighten the second-order term — a smaller $R$ — and split the difference across the aperture.

Which is a general fact about fitting worth having in a sentence. An estimator handed a defect it has no parameter for reports the amount of the defect it does have that best imitates it. The same sentence describes a shadow reporting a curvature for a floor made of two planes, and the parallel is not an analogy — it is the same failure in a different subject, with the same cause.

The two cases differ in one respect that is worth naming, because it decides which is more dangerous. Here the truth is inside a neighbouring family — a sphere is a perfectly good surface that the paraboloid family happens not to contain — so the fitted answer is close to right and gets closer as the aperture shrinks. There the truth is outside every smooth family, because a step has no curvature anywhere, and the fitted number does not approach anything as the measurement improves.

A bias that shrinks with the experiment is recoverable by shrinking the experiment. A number with no referent is not, and no amount of care finds that out.

What the residual is actually measuring #

There is a temptation to read a small residual as “the model is right” and it is worth naming what the residual measures instead.

It measures the part of the data the model could not reach. If the discrepancy between two shapes lies mostly in a direction the model’s own parameter can move, the fit absorbs it and the residual is small. If it lies in a direction the parameter cannot move, the residual sees it.

Over a small aperture, a sphere’s departure from a paraboloid is nearly proportional to what a change in $R$ would do — both are dominated by the $r^2$ term over that range — so the model can absorb it almost perfectly. Over a large aperture the two directions separate, and the residual is what is left in the direction the model has no handle on.

So the residual is not a measure of correctness. It is a measure of how much the model failed to disguise, which is a different quantity and is smaller by exactly the amount the model is flexible.

The same reading explains why a flat pane’s picture cannot separate its thickness from its index over a narrow fan, and why one view of a plane determines less than it appears to. In each case a family with a free parameter absorbs a discrepancy in the one direction it can move, and reports its success at doing so as a small residual.

What a second family would say #

There is a route the essay has been circling and it deserves its own paragraph, because it is what a careful experimenter actually does.

Fit both families to the same fan and compare their residuals. The sphere’s is $10^{-13}$ degrees at every aperture; the paraboloid’s is $2.8\times10^{-4}$ at 5 cm. Those differ by nine orders of magnitude at an aperture where neither residual is anywhere near a measurement floor.

So the comparison sees the shape long before either fit alone announces anything. It works because it asks a relative question — which of these two explains the data better — rather than an absolute one, and a relative question does not have to clear a floor. It has to clear the difference between two floors, which is a much smaller number when both models are fitted to the same data with the same machinery.

This is the reason a bench test is done against a reference surface rather than against a specification. The reference is not there to be accurate; it is there to make the question relative.

And that observation is the essay’s one genuinely cheerful result. The caustic separates the two shapes at an aperture where the fan cannot, because a caustic’s length is a fourth-order quantity read directly rather than a fourth-order discrepancy hidden inside a second-order fit. A sphere’s envelope over a 20 cm half-aperture is 18 mm long and a paraboloid’s is zero. There is no fitting involved and nothing to absorb the difference.

The aperture ratio is the real variable #

The last figure makes a point worth extracting. Nothing here depends on the aperture in metres; it depends on the aperture over the radius.

That is the same scaling law the caustic route obeys, and for the same reason: reflection is scale free, so a mirror and a mirror twice as large used over twice the aperture are the same optical object. Doubling the radius doubles every aperture in the table and changes no percentage in it.

The practical form is a rule of thumb with a number in it. Below about a tenth of the radius, no measurement of a fan can tell a sphere from a paraboloid at a pixel’s precision. That is why an optician’s test bench for a telescope mirror is long and why a curvature gauge that touches three points on a lens surface reports a radius and not a shape.

The paraboloid fitted to a sphere is not a wrong answer to a wrong question #

It is worth being fair to the fit. The number it returns is the vertex radius of the paraboloid that best explains the data, and that is precisely what was asked for.

The failure is at the join between the question and the reader. What a reader means by “the mirror’s radius” is a property of the mirror; what the fit returns is a property of the mirror and the family the fitter chose. Those coincide when the family contains the object and diverge silently when it does not.

So the honest report of any such fit has three parts rather than one — the number, the residual, and the family — and the third is the one that is almost never written down. A radius quoted without the shape it was fitted inside is not wrong; it is incomplete in a way that cannot be detected from the number.

Where this leaves a reader with a mirror #

Three sentences, in order of how much they cost.

A cusp measurement — the caustic route — gives the vertex radius with no shape assumption at all, because a cusp is a paraxial quantity and every shape that agrees to second order shares it. It is cheap and it says nothing about the shape.

A fan over a small aperture gives a radius inside whatever family was chosen, biased by an amount nothing in the data reports. It is cheap and it is confident and it is the one to distrust.

A fan over a large aperture gives both, and the second is what the extra aperture actually bought. The residual clearing the floor is not a nuisance; it is the measurement working.

And the fourth is the comparison above — two families, one fan, and a ratio of residuals rather than a residual — which sees at a tenth of the aperture either fit needs on its own.

Read against the rest of this row, the entry in the table is a dependency rather than a defect. A mirror ball borrows a near room point. A dome port borrows a length in the water. A pane borrows a wide fan or a quoted index. A reflected fan borrows the shape family, and the difference is only that the first three borrow something a reader can go and fetch and the fourth borrows something a reader usually does not know they have assumed.