The one shape that focuses

Two rungs of this field have been about absence. A curved mirror’s lines of sight do not meet at a point; what they have instead is a caustic with a measurable length. Both are statements about spheres, because a sphere is what a curved mirror usually is.

There is exactly one shape that escapes, and it escapes completely rather than by a margin.

The claim, stated carefully #

A paraboloid of revolution takes every ray parallel to its axis exactly through one point.

Three qualifications are load-bearing and each of them is the reason a different design exists.

Every ray, with no restriction on how far from the axis. That is what makes it different from the sphere, which is exact only in the limit of a vanishing aperture.

Parallel to its axis. A bundle arriving at an angle is not focused to a point at all — it is focused to a blur that grows with the angle, which is coma, and it is why a paraboloidal telescope has a small usable field and why wide-field instruments use other shapes.

And one point, which is the focus, at half the vertex radius of curvature.

So the sentence is not “a paraboloid is a good mirror”. It is that for one particular bundle of rays, a curved mirror can be an exact projection through a centre — the only case in this field where that is true and the case is not empty.

Measuring it with the same routine #

The claim is easy to assert and it is worth measuring, and what makes the measurement meaningful is that it uses no formula for either mirror.

The envelope of the reflected family is computed by intersecting each ray with its immediate neighbour and taking the limit — the definition of an envelope, applied numerically. For the sphere that produces a curve. For the paraboloid it produces a set of points whose bounding box is 4.8e-9 mm across.

That number is the arithmetic floor of the computation, and it is the right thing to report. Had the paraboloid’s focus been put in by hand — computed from $R/2$ and drawn — the figure would have been a picture of a formula. Coming out of the same envelope routine that produces the sphere’s caustic, it is a measurement.

The point it comes out at is checked separately against $R/2$, to nine digits, because a routine could in principle collapse the envelope to the wrong place.

The control that makes it a comparison #

A comparison between two mirrors is only about their shapes if everything else is held. Here that is one condition and it is not obvious which one.

The two mirrors are built with the same radius of curvature at the vertex. That means they agree to second order at the axis, so a narrow enough bundle cannot tell them apart, and the difference between them is entirely in the fourth order and beyond.

The machinery asserts that: the sphere’s caustic, taken over a bundle 4 cm wide, has its cusp at $R/2$ to a part in a thousand. Without that check, the “comparison” could have been between a mirror of focal length 0.80 m and one of 0.83 m, and the 184 mm would have been mostly a difference in focal length rather than in shape.

Matching the vertex curvature is also what makes the practical statement true. A spherical mirror is a paraboloid, to second order, and every optical shop that grinds a sphere and then figures it into a paraboloid is removing a fourth-order difference from a surface that already has the right focal length.

Why it is the parabola and not something else #

The property has a construction behind it that is worth seeing, because it explains the qualification about the axis.

A parabola is the locus of points equidistant from a point — the focus — and a line, the directrix. Consider a ray arriving parallel to the axis and striking the parabola at $P$. Its path length from the directrix to $P$ is the distance from $P$ to the directrix, which by the defining property equals the distance from $P$ to the focus.

So every incoming ray, measured from a common wavefront perpendicular to the axis, arrives at the focus having travelled the same total distance. Equal path lengths from a plane wavefront to a point is exactly the condition for a perfect focus, and the parabola’s definition is that condition written as a curve.

Two consequences fall out of that argument and neither is available from the algebra.

The bundle has to be parallel to the axis, because the equal-path property is measured from a wavefront perpendicular to it. Tilt the bundle and the path lengths stop being equal, which is why the field is narrow.

And the focus is a point at infinity’s image. A bundle of parallel rays is a point at infinity in the direction of the axis, so the paraboloid is a projection taking that one point to the focus exactly. It is the one place in this field where the vocabulary of vanishing points and the vocabulary of optics turn out to describe the same object.

What a sphere costs, in the units a designer uses #

The comparison is more useful as a rule than as a pair of numbers, so here it is in the form an optical bench would need.

The caustic’s extent grows as the square of the aperture at fixed focal length: across three doublings it multiplies by 4.13, 4.06 and 4.13. So the aberration depends on the focal ratio, and a spherical mirror is acceptable down to whichever ratio puts the caustic below the tolerance wanted.

For the mirror drawn here — vertex radius 1.6 m, so a focal length of 0.8 m — a bundle 20 cm across gives a caustic 4.5 mm long, one 40 cm across gives 18.6 mm, and one 1.24 m across gives 184.1 mm. The first of those is a mirror working at about f/4, and 4.5 mm of longitudinal spread is already enormous by any optical standard; the reason these look so bad is that they are quoted in millimetres of a metre-scale mirror rather than in wavelengths, which is where a real design would keep them.

Doubling the focal length at fixed aperture has the opposite effect and it is the same rule read the other way. The same 1.24 m bundle on a mirror of twice the focal length gives 90.5 mm instead of 184.1 — roughly half, since the aberration goes as the aperture squared over the focal length cubed and only one factor of the focal length is being changed here alongside the fixed aperture.

The shape of the rule is what transfers. Halving the focal ratio quadruples the aberration, so a spherical mirror is a slow mirror, and every fast one has to be figured.

What a paraboloid does with the rays it was not designed for #

The narrow field is the paraboloid’s price and it is worth quantifying, because “exact for one bundle” invites the question of how quickly exactness is lost.

Tilt the incoming bundle away from the axis by a small angle. The equal-path-length argument no longer holds, because the wavefront is no longer perpendicular to the axis, and the reflected rays no longer meet. What they form is a comet-shaped blur — coma — whose size grows linearly in the tilt angle and quadratically in the aperture.

Linearly is the important word. Spherical aberration is second order in the aperture and zero order in the field angle; coma is first order in the field angle. So a paraboloid is perfect on axis and degrades immediately off it, whereas a sphere is imperfect on axis and, being rotationally symmetric about every axis through its centre, degrades much more slowly with field.

That trade is why telescope designs are a catalogue rather than a winner. A paraboloid for a narrow field; a sphere with a corrector for a wide one; a pair of conics — hyperboloid and hyperboloid — where both matter and the extra surface can be afforded.

It is also the general shape of every result in this field. An exact statement about a curved mirror is exact for one configuration, and the interesting engineering is in how fast it stops being true.

The one exact projection in this field #

Setting the results of this field side by side gives a short and complete list.

A flat mirror is a projection, from the eye reflected in its plane, for every bundle of rays. That is the strongest statement available and it is why a mirror reads as a window.

A paraboloid is a projection, from its focus, for one bundle of rays — the one parallel to its axis. Exact, and useless for anything but that bundle.

A concave mirror with the eye at its centre of curvature is a projection, from the eye, for the bundle leaving the eye. Exact, and useless because the centre is where the viewer already is.

Everything else is not a projection at all. No point, no rule from directions to positions, and a caustic instead of a focus.

Two of those three exceptions are degenerate in the sense that the recovered centre is somewhere already known. The paraboloid is the only one that is both exact and informative, and its price is that it works for one direction.

Grinding, figuring, and why the sphere comes first #

There is a piece of workshop practice that this comparison explains, and it is a satisfying use for a geometric result.

A mirror is ground by rubbing two discs together with abrasive between them. That process has a natural product: the only surface that fits against another surface in every relative position is a sphere, so random rubbing converges to one, and it does so without anybody having to steer it. A sphere is what the process hands over for nothing.

Turning that sphere into a paraboloid means removing material from the middle relative to the edge — a fourth-order correction, since the two agree to second order — and it is done by figuring: local polishing, tested against a knife edge, in small amounts.

The measurement in this essay is the reason the second step is necessary and also the reason it is small. Necessary, because the caustic on a fast mirror is enormous. Small, because the two surfaces differ only in the fourth order, so the amount of glass to remove from a metre-class mirror is a few wavelengths — a fraction of a micron — and the whole difference between an unusable mirror and a perfect one is a layer thinner than a soap bubble.

That is a pleasing form for a geometric result to take: the number that says how bad the sphere is, and the number that says how little has to change, are two readings of the same fourth-order term.

The shapes nobody uses, and why #

The parabola is one member of a family and it is worth saying what the others do, because the pattern is that each conic focuses exactly between one particular pair of things.

An ellipsoid focuses one point to another point, exactly — its two foci. That is the whispering gallery, and it is why a lithotripter is an ellipsoid.

A hyperboloid focuses a bundle converging on one focus into a bundle converging on the other, which is what a Cassegrain telescope’s secondary mirror does.

A paraboloid is the limiting case where one of the two points has gone to infinity, which is exactly the parallel-bundle case.

And a sphere focuses a point to itself — its centre — and nothing else exactly, which is the degenerate exception the previous rung found by accident.

So the sphere is not a failed paraboloid. It is the member of the family whose two conjugate points have coincided, and its exact case is real and is the one nobody wants.

What to take away #

Exactness in this field is rare and conditional. One shape, one bundle, one point. Everything else has a caustic, and the caustic has a length that can be computed and designed against.

The sphere is right in a limit and the limit is the aperture. Quoting “the focal length of a spherical mirror is R/2” without an aperture is quoting the first term of an expansion as though it were the answer.

And the exact case is a projection in this site’s sense. A paraboloid takes a point at infinity to a point, with every ray passing through it, which is precisely what a pinhole camera does for a point of the world. That is the connection worth carrying out of this field: a focus and a centre of projection are the same object, and a mirror that has one is a mirror that is a camera.