The caustic is the mirror's own ruler

Where the focus went computes a caustic as a defect. A spherical mirror does not send a parallel bundle through a point; the rays envelope a curve instead; and the length of that curve is spherical aberration expressed as a distance rather than named.

This essay reads the same curve as a measurement of the mirror.

The distinction matters more here than the phrase suggests, because of what the rest of this row has been finding. A mirror ball gives the ratio of its radius to its distance and nothing else. A dome port gives its decentring in units of its own radius. A pane of glass gives a product of its thickness and its index. Three instruments, three ratios, no sizes.

A caustic is different, and the difference is not subtle. It is centimetres on a table.

What the cusp is #

Take a bundle of rays parallel to the axis. Near the axis they cross at $R/2$ — that is the paraxial focus, and it is where every optics text puts the focal point of a mirror. Rays from further out cross closer to the mirror, which is spherical aberration, so instead of a point there is a curve.

The curve has a tip. The tip is at the paraxial focus, because that is where the innermost rays go and the envelope is continuous.

So the recovery is one subtraction. Measure from the mirror’s vertex to the tip of the bright curve, double it, and that is the radius of curvature.

Run it on a sphere of radius 1.6 m over a half-aperture of 50 cm and it returns 1.599624 m, which is 0.023% out. That residual is not noise. It is the offset between the true paraxial focus and the envelope evaluated at the innermost ray the computation actually starts from, which is at 2 cm rather than at zero — a real limitation of a real measurement, and it shrinks as the innermost ray comes in.

Why the tip and not the middle #

An envelope is a curve and a curve has many places on it. Choosing the tip is not an aesthetic preference; it is the only place on the caustic that names a quantity with a closed form.

Every other point of the envelope is the crossing of two neighbouring rays from some radius $r$ out on the mirror, so its position depends on $r$ as well as on $R$. Reading it would recover $R$ only if the reader also knew which part of the mirror those rays came from, which is exactly the information a photograph of a bright curve does not carry.

The tip is the $r \to 0$ end. It depends on the vertex curvature and on nothing else, so it is the one point of the curve that is a property of the mirror rather than a property of the mirror and a choice of where to look.

That is the same reason the site’s height recoveries read a cross-ratio against the horizon rather than against a convenient mark. A recovery wants the point of the construction that is fixed by fewer things, and finding which point that is usually takes longer than the arithmetic afterwards.

The paraboloid, where it is exact #

The one shape that focuses establishes that a paraboloid takes every ray parallel to its axis through one point exactly. So its caustic is not a curve; it is a point, and the recovery from that point is not approximate at all.

A paraboloid of vertex radius 1.6 m returns 1.6000000000001764, which is arithmetic. There is no aberration to bias the envelope because there is no envelope, only a focus.

That pairing — a per cent of a per cent on a sphere, twelve figures on a paraboloid — is the shape of the answer everywhere in this row where the answer is honest. The residual reports a real property of the object rather than a failure of the method.

Shape and size, separately #

Here is the property that makes the caustic the odd one out, and it is worth stating as a scaling law because that is what it is.

Take a mirror of radius $R$ with a half-aperture $a$, and scale both by the same factor. The reflected geometry is a similarity — Snell has nothing to do with it, reflection is scale-free, and the only lengths in the problem are the two that scaled. So the caustic is the same curve, larger.

Measured: at factors of 2, 0.5 and 7, the curve’s aspect ratio — its extent along the axis over its extent across — is unchanged to eleven decimal places, and every length in it is multiplied by exactly the factor.

Which means two independent readings are available from one bright curve:

• Its shape carries the dimensionless ratio $a/R$, how much of a mirror is being used.
• Its size carries $R$ itself, in whatever units the table is measured in.

Neither a mirror ball’s outline nor a dome’s bend nor a pane’s displacement offers both. They offer the first and stop.

The consequence for a reader is a practical one and it cuts against how these things are usually written up. A photograph of a caustic with no scale bar in it still carries a real number: the aperture ratio, from the curve’s shape. It carries the radius only if something in the frame has a known size. So the two readings degrade differently, and a picture that has lost its scale has lost exactly one of them.

Why it is a length and the others are not #

The general principle is worth extracting, because it applies well beyond mirrors.

An instrument that returns angles at a single point cannot return a length. Every angle is a ratio of two lengths and the ratio is what survives; scale the whole configuration and the angles are identical, which is the one thing a single view cannot give.

An instrument that lays a mark somewhere returns a length, because the mark’s position is measured against the object rather than against the eye. A caustic is exactly that: the mirror throws its focus onto the table, and the distance from the mirror to the bright tip is a distance in the room.

The same distinction sorts the rest of the collection. A vanishing point is an angle and gives a direction. A shadow’s tip is a mark and gives a position — which is why the lamp comes out of a picture with a height rather than only a direction.

The route that does not work, and why it is worth keeping #

The first draft of this measurement had a second route. The caustic’s extent along the axis is longitudinal spherical aberration, which for a spherical mirror over a half-aperture $a$ ought to go as $a^2/(4R)$ — so measuring the extent should also give $R$.

It does not, and the reason is instructive. The caustic’s axial extent is not the axial-crossing spread; it is the extent of an envelope, which reaches further out than the places rays cross the axis. Measured against $a^2/(4R)$ the answer came out 65% wrong at every aperture, and the error did not fall away as the aperture grew.

The lesson is not about mirrors. A closed form quoted for one quantity does not transfer to a neighbouring quantity that shares a name. The axial spread and the envelope’s extent are both “the size of the caustic” in ordinary speech and they are two different numbers.

So the route was dropped rather than calibrated. Fitting a coefficient to make it agree would have produced a second recovery that worked and meant nothing.

That is the right decision and it is worth saying what would have made a different one right, because the measurement contains the evidence for it. The 65% did not change with the aperture. A constant ratio between two quantities across a sweep is not what a coincidence looks like; it is what two quantities sharing a leading order look like. The axial-crossing spread is $a^{2}/4R$ to leading order in $a/R$, and if the envelope’s extent is also $c\,a^{2}/R$ for some universal $c$, then the ratio is $4c$ at every aperture — which is what was measured.

So the honest repair is not to fit $c$ but to derive it. The envelope point for a ray striking at $\alpha$ is the tangential focus of that pencil, at $R\cos\alpha/2$ along the reflected ray from the surface, and its axial coordinate is a function of $\alpha$ that can be expanded exactly as the axis-crossing formula was. Its leading term is $c\,a^{2}/R$ with $c$ a number that comes out of the expansion rather than out of the data, and a recovery built on it would be a second independent route to the radius — measuring the curve’s length where the first route measures its tip.

Two routes to one number is what this collection asks for everywhere, so the abandoned route is a shortfall rather than a dead end, and it is recorded as one. What it is not, and what the draft version was, is a recovery whose constant was chosen to make the answer come out — which is the same fault a fitted radius records in a different form, where a shape assumed rather than derived produces a confident number about the wrong family.

The distinction is worth carrying because it is easy to state and easy to miss in practice: a coefficient obtained from the geometry is a measurement, and the same coefficient obtained from the measurements it is about is a restatement. The 65% is currently the second, and it is one expansion away from being the first.

The refusal #

A flat mirror’s reflected rays are parallel. Parallel rays have no envelope, so there is nothing for the routine to return, and it refuses rather than reporting a very distant focus.

That refusal is the check that makes the rest of the essay a measurement. A recovery that always returns a number is a recovery whose successes prove nothing, and the flat mirror is the input where a number would be wrong in the most seductive way: a very large radius is exactly what a nearly-flat mirror has, and a very large radius for a perfectly flat one is not an answer but a division by zero wearing an answer’s clothes.

The same structure appears throughout this collection — a shadow that reports a curvature for a floor that has none is the same failure with the refusal missing, and it is what happens when the degenerate case is inside the model’s range instead of outside it.

What a reader with a coffee cup can actually do #

The measurement is not hypothetical, which is worth saying plainly because a caustic is the one piece of optics most people have already seen.

A cup of coffee in sunlight shows a bright cardioid-like curve on the surface. The cusp of that curve sits at half the cup’s radius from the wall — the near wall, the one the light is coming through — because the inside of a cylindrical cup is a concave mirror with the cup’s radius as its radius of curvature.

So a ruler across the cup and a ruler to the cusp measure the same thing twice, and the two agree. That is a round trip a reader can run at breakfast, and it works for the reason the whole essay gives: the bright curve is on the coffee, in centimetres, and not in the eye, in degrees.

The aperture is the part that varies, and it varies enough to matter. A cup lit from one side is a mirror used over three quarters of its own radius, which is an enormous aperture ratio by any optical standard — a telescope uses a tenth of that. So the paraxial reading is a per cent out rather than a hundredth of a per cent, and the figure below prints what it actually got rather than the ideal.

That is the honest form of the earlier claim. The cusp names the radius, and how well it names it depends on how close the innermost ray of the bundle actually gets to the axis. At a telescope’s aperture ratio the answer is exact for any practical purpose; at a coffee cup’s it is good to about a per cent; and a routine that quoted the first figure for the second would be quoting an ideal rather than a measurement.

What changes when the source is not at infinity #

Everything so far uses a bundle parallel to the axis, which is what a distant source produces. A near source is a different measurement and it is worth saying what changes, because the coffee cup’s source is the sun and a reading lamp’s is not.

The envelope still exists and still has a tip. The tip is no longer at $R/2$; it is at the image distance the mirror equation gives for that object distance, so recovering $R$ from it needs the source’s distance as a second input.

Which puts the near-source case back in the same position as the rest of this row. Two unknowns, one observation, and a length has to come from outside the instrument — here, from knowing where the lamp is. The parallel-bundle case is special precisely because the source’s distance has dropped out of the problem, and it has dropped out because it was infinite.

What the essay does not settle #

It settles the radius and it assumes the shape. Everything above takes the mirror to be a sphere or a paraboloid and reads a vertex radius inside that family.

A fitted radius is wrong before it is uncertain is what happens when that assumption is put under the same scrutiny, and its answer transfers directly: a fit inside a family returns a confident number for an object outside the family, and the residual does not report the difference until the aperture is large.

Here that limitation is visible in a specific place. The cusp’s position is a paraxial quantity — it depends on the vertex curvature and on nothing further out — so it is the same for a sphere, a paraboloid and anything else that agrees with them to second order at the axis. Reading a cusp tells a reader the vertex radius of the mirror’s own osculating sphere. It does not tell them the mirror is a sphere, and it never could.

The row, restated #

Five instruments, one question, and one of them answers without borrowing.

The rule sorts one more thing, and it is the one this row started from. A recovery that borrows a length is not worse than one that does not — it is a recovery with a dependency, and knowing which dependency is the whole of what makes it usable. A mirror ball needs a near room point. A dome port needs something of known size in the water. A pane needs a wide fan of sightlines or a quoted index. Each of those is a sentence a reader can act on, and none of them is available until somebody has asked what the instrument returns on its own.

The general form is worth carrying out of the row. An instrument that measures at the eye returns ratios; an instrument that leaves a mark returns lengths. Everything else in this collection sorts by that rule — a vanishing point against a shadow’s tip, an outline against a caustic, an anamorph’s design against the marks it puts on the floor — and the sorting was there before anybody looked for it.

It is a rule with a sharp edge, which is what makes it worth stating rather than admiring. An instrument that leaves a mark returns a length in the units of the surface it marked, so the surface has to be somewhere the length matters: a caustic on the tabletop measures the mirror because the tabletop is where the mirror sits, and the same caustic thrown onto a wall across the room measures the mirror just as well and answers a question nobody asked. The mark is the ruler, and a ruler laid somewhere irrelevant is not a measurement of anything.