Visual Plane
every picture states the point it is correct from
Mirrors that are not cameras

Where the focus went

If the reflected rays do not meet at a point, they meet each other in pairs, and the curve they are all tangent to is what the mirror has instead of a focus. On a sphere of 1.6 m vertex radius it is 184.1 mm long and has a cusp; near the axis it sits at R/2, which is why a small enough spherical mirror passes for a good one. This is the bright shape in the bottom of a coffee cup, computed rather than admired.
17 min read 14 figures Not a projection at allThe surface is a choiceProjected, not constructed

Worth reading first: A curved mirror has no eye.

The previous rung ends with an absence: a curved mirror’s rays do not pass through a common point. An absence is an unsatisfying place to stop, and it is also not the whole truth. The rays do not meet at a point, and they do meet each other — every pair of neighbouring rays crosses somewhere — and the collection of those crossings is a curve.

That curve is the caustic, and it is what a curved mirror has instead of an eye.

A spherical mirror's caustic — where the focus wentThe reflected rays are tangent to a cusped curve 184.1 mm long across a 1.24 m aperture. Near the axis it is at R/2 = 0.800 m, which is why a small enough spherical mirror passes for a paraboloid.79 mm of causticvertex radius 1.60 m, aperture 0.82 m79 mm of envelope
Fig. 1 Parallel rays striking a concave spherical mirror, and the curve their reflections are tangent to. It is 184.1 mm long and has a cusp on the axis. Near that cusp it sits at half the vertex radius, which is where the textbook focus is.

It is also the most familiar piece of optics anybody sees without looking for it: the bright cusped shape in the bottom of a cup of coffee lit from one side, and the wobbling bright net on the floor of a swimming pool. Both are caustics, and this one is computed from the reflection law and nothing else.

Envelope, not focus

The definition is worth taking carefully because it is easy to state loosely and the loose statement is wrong.

A family of lines has an envelope: a curve tangent to every member. Take two neighbouring members, find where they cross, and let them approach each other; the limit of those crossings traces the envelope.

That is how the caustic here is computed — by intersecting each ray with its immediate neighbour and taking the limit numerically — rather than from a closed form. The reason is not laziness. Computing it this way means the same routine works on a sphere, a paraboloid and a cone, and it means the paraboloid’s answer comes out as a single point rather than being defined to be one. A formula for the spherical caustic would have proved nothing about the paraboloid.

Two things follow immediately from the definition and neither is obvious from a picture.

No point of the caustic is on more than two rays. It is not a place where the light concentrates in the way a focus is; it is a place where the light concentrates because the rays are tangent to it, so they linger near it. That distinction is why a caustic is bright but not sharp.

And it is not a locus of intersections in general. Many rays cross each other far from the caustic. The envelope is specifically the limit of neighbours, which picks out the boundary of the region the ray family fills.

The cusp, and why textbooks get away with it

The spherical caustic has a cusp on the axis, and the cusp is at exactly half the vertex radius of curvature.

That is the textbook focal length of a spherical mirror, and it is where the whole paraxial account lives. Rays close to the axis do meet near a point, and the point is R/2R/2; rays further out miss it, by more the further out they are, and their crossings trace the two branches running back toward the mirror.

So the textbook is right in a limit and the limit is narrow. For the mirror drawn here, a bundle 4 cm wide focuses to within a few microns of R/2R/2; a bundle 1.2 m wide has a caustic 184 mm long.

A spherical mirror's caustic — where the focus wentThe reflected rays are tangent to a cusped curve 10.4 mm long across a 0.30 m aperture. Near the axis it is at R/2 = 0.800 m, which is why a small enough spherical mirror passes for a paraboloid.10 mm of causticvertex radius 1.60 m, aperture 0.30 m10 mm of envelope
Fig. 2 The same mirror with a much narrower bundle. The caustic collapses toward its cusp, and this is the regime the paraxial formula describes — the mirror has a focus in the sense a textbook means it, and the sense depends entirely on the aperture.

The size of the caustic is the classical quantity called spherical aberration, and having it as a length rather than a name is the point of computing it. Measured on this mirror it grows as the square of the aperture: doubling the width of the bundle multiplies the caustic’s extent by 4.13, then by 4.06, then by 4.13 again over three successive doublings. That is why slow spherical optics is fine and fast spherical optics is not — halving the focal ratio quadruples the aberration.

The comparison that makes it a measurement

A caustic 184 mm long is a number and not yet evidence. What makes it evidence is that the same routine, run on a mirror of the same vertex curvature but a different shape, returns something else entirely.

A paraboloid's focus, and the curve a sphere has insteadTwo mirrors with the same radius of curvature at the vertex, given the same parallel bundle. The paraboloid's reflected rays all pass through one point, 0.800 m from the vertex, to 4.8e-9 mm. The sphere's are tangent to a curve 184.1 mm long instead, and no point on it is on more than two rays.focus, R/2 = 0.800 m184 mm of causticvertex radius 1.60 m, aperture 1.24 m5e-9 mm against 184 mm
Fig. 3 Two mirrors with the same radius of curvature at the vertex, given the same parallel bundle. The paraboloid’s reflected rays all pass through one point at R/2, to 4.8e-9 mm. The sphere’s are tangent to a curve 184.1 mm long, and the two mirrors are indistinguishable near the axis.

The paraboloid’s envelope collapses to a point — 4.8e-9 mm across the whole aperture, which is arithmetic. So for one bundle of rays a curved mirror is an exact projection, and the shape that manages it is not the sphere.

Two controls keep that comparison honest.

The two mirrors agree near the axis. Both are built with the same vertex radius, so a narrow bundle cannot tell them apart, and the sphere’s paraxial focus is required to land at R/2R/2. Without that the comparison would be between two different mirrors rather than two different shapes.

And the flat mirror has no caustic at all. Parallel rays reflect parallel and no two of them cross, so asking for the envelope is refused rather than returning a very distant point. A routine that produced a curve for a flat mirror would be producing curves from nothing.

Every ray parallel to a paraboloid's axis, through one pointThe envelope collapses to a point at R/2 = 0.800 m, to 4.8e-9 mm across the whole 1.24 m aperture. For this one bundle of rays a curved mirror is an exact projection.focus, R/2 = 0.800 mvertex radius 1.60 m, aperture 1.24 m5e-9 mm of envelope
Fig. 4 The paraboloid alone. Every ray parallel to the axis passes through one point, and the “envelope” has no extent to report — which is what an exact focus looks like when it is measured by the same machinery that measures an inexact one.

Why the cusp is where it is

The cusp’s position has a one-line derivation and it is worth having, because it is the only part of the caustic that has a simple formula and it is the part everybody quotes.

A ray parallel to the axis strikes the mirror at angle α\alpha from the axis, measured at the centre of curvature. The normal there points at the centre of curvature, so the reflected ray leaves at 2α2\alpha to the axis, and it crosses the axis at a distance

R2cos⁡α\frac{R}{2\cos\alpha}

from the centre of curvature. As α→0\alpha \to 0 that is R/2R/2, which is the cusp; as α\alpha grows it moves toward the mirror, which is why the caustic’s branches run back that way rather than outward.

So the paraxial focal length is not an approximation that happens to be good near the axis; it is the limiting value of an exact expression, and the expression says exactly how the rest of the mirror departs from it. Everything else in this essay is the two-dimensional version of that one-dimensional statement.

The same expression shows why the sphere and the paraboloid cannot be told apart near the axis. Expanding R/(2cos⁡α)R/(2\cos\alpha) in α\alpha gives R/2+Rα2/4+…R/2 + R\alpha^2/4 + \ldots, and the paraboloid’s is R/2R/2 exactly, so the two differ at second order — which is the square law measured above, arrived at algebraically.

The caustic and the missing centre

The connection to the previous rung is direct and worth stating as an equivalence.

A mirror has a centre of projection exactly when its caustic degenerates to a point. The lines of sight from a viewer are a ray family; if that family is concurrent, its envelope is the single point they are concurrent at; if it is not, the envelope has extent, and the extent is a measure of how badly the centre is missing.

So the two measurements in this field — the miss of the least-squares point, and the length of the caustic — are two readings of one fact. The first asks how far the rays are from the best compromise; the second asks what shape the rays actually organise themselves around.

The second is the more informative one, because a compromise point says nothing about the structure of the failure. A caustic says where the light goes.

A mirror ball 2.00 m across, and the point its lines of sight missThe backward continuations are drawn to the point that fits them best. They miss it by up to 2.46 mm — over 20 cm of mirror, so a photograph of this ball is a projection of nothing from anywhere.eyethe best fit — no ray goes through itno single viewpoint — the rays miss by 2.46 mmover 20 cm of a 2.00 m ball
Fig. 5 The first reading, from the previous rung. The lines of sight of a mirror ball, and the point that fits them best — which no ray passes through, and which is a compromise rather than a place.
How badly a mirror fails to have an eye, against how much it curvesA flat mirror's lines of sight meet exactly — 2.8e-12 mm, which is arithmetic. A ball 0.50 m across misses by 52.9 mm over the same 20 cm of mirror, and the miss falls to nothing as the curvature does.0204001234curvature of the mirror, 1/R (per metre)how far the lines of sight miss a common point (mm)flat: they meet20 cm of mirror, eye 1.40 m away3e-12 mm flat · 52.9 mm at 1/R = 4.0
Fig. 6 And the same reading across a family of curvatures. It falls to zero as the mirror flattens, which is the caustic collapsing.

Measuring the extent, and what the number is

The caustic is a curve, so “how long is it” needs a definition, and the one used here is the diagonal of its bounding box in the meridional plane. That mixes a longitudinal extent — how far along the axis the crossings are spread — with a transverse one, and for a spherical mirror the longitudinal part dominates, which is why the measured growth is a square law rather than the cube law the transverse aberration follows.

Quoting the bounding diagonal rather than one component is deliberate and worth defending. The number is being used for two things: as a size of the failure to focus, and as a comparison against the paraboloid. For the second, any measure that vanishes exactly when the envelope is a point will do, and the diagonal is the least arbitrary of them. For the first, a designer wants whichever component their detector is sensitive to, and the machinery returns the axis ranges separately so either can be taken.

What matters is that the same definition is applied to both mirrors. A comparison in which the sphere is measured longitudinally and the paraboloid transversely would be a comparison of two definitions.

A spherical mirror's caustic — where the focus wentThe reflected rays are tangent to a cusped curve 42.2 mm long across a 0.60 m aperture. Near the axis it is at R/2 = 0.800 m, which is why a small enough spherical mirror passes for a paraboloid.42 mm of causticvertex radius 1.60 m, aperture 0.60 m42 mm of envelope
Fig. 7 The same mirror at a third aperture, on the way to the square law. The three doublings between the narrowest bundle and the widest multiply the caustic’s extent by about 4.1 each time.

Not the same as a penumbra, and not the same as a blur

Three things look like a caustic in a photograph and are not, and separating them is useful because the repairs differ.

A penumbra is a shadow’s soft edge, caused by the source having width, and its size is the source’s own image. It has no tangency in it and is not an envelope of anything; it is a convolution.

Defocus blur is what a lens does to a point when the sensor is not at the image plane, and it is a disc rather than a curve. Stopping down shrinks it and moving the sensor removes it.

And a caustic is neither. It is present for a point source, with perfect surfaces, at every distance, in perfectly sharp geometry. Stopping down shrinks it — because the aperture is exactly what its size depends on — and nothing else does, short of changing the shape of the mirror.

That last is why a caustic is a geometric object on this site and belongs beside the vanishing point rather than beside a photographic defect. It is where the rays are, computed from the reflection law, with no physics of light beyond that.

A 12 cm source, an edge, and the band betweenThe penumbra is 10.3 cm wide by the projection — the source's width times the receiver-to-occluder distance over the source-to-occluder distance — and 10.2 cm by counting how much of the source each point can see. The two routes share no arithmetic.source, 12 cmthe occluder's edgefraction of the source visiblepenumbra 10.3 cmprojection: 10.29 cmsampled: 10.23 cm
Fig. 8 The first of the three, for contrast. A shadow’s soft edge is the image of the source projected through the occluder’s edge, and its width is set by the source’s size — a different construction with a different remedy.
A 1 mm gap in a canopy, at various heightsThe patch is the hole's shape near the hole and the sun's shape far from it, and the crossover — where it is half of each — is the hole's width over the sun's angular width: 0.13 m. Above that, every gap in the canopy is a pinhole camera imaging the sun, which is why they all go crescent-shaped together during an eclipse.00.2500.5000.75010123distance from the hole to the ground (m)how much of the patch of light is an image of the suncrossover at 0.13 mthe sun subtends 9.30 mradpatch 29 mm wide at 3 m
Fig. 9 And the crossover that makes the distinction concrete elsewhere on this site. Below a certain size a pinhole’s image is set by the hole’s own width and above it by the source’s; two constructions, two regimes, one picture.

Reading a caustic in a room

The construction is checkable without any equipment, which is unusual for anything in this field.

Fill a mug with coffee and put a lamp to one side. The bright cusped shape on the surface is the caustic of the mug’s inner wall, which is a cylinder rather than a sphere, so the curve is the two-dimensional version of the one drawn here — a nephroid when the source is far away and a cardioid-like curve when it is close.

Move the lamp closer and the cusp moves. The caustic here is computed for a source at infinity; a finite source produces a different envelope, and the machinery takes the source position as an argument for exactly that reason.

Narrow the illuminated part of the mug with a card and the caustic shrinks. That is the aperture dependence, seen with a piece of paper.

And the negative test, which is the informative one: line the mug with a strip of flat mirror and no caustic appears. A flat surface has none, because its reflected rays are parallel and never cross — which is the refusal the machinery makes and the reason the refusal is worth having.

A spherical mirror's caustic — where the focus wentThe reflected rays are tangent to a cusped curve 78.5 mm long across a 1.00 m aperture. Near the axis it is at R/2 = 1.200 m, which is why a small enough spherical mirror passes for a paraboloid.78 mm of causticvertex radius 2.40 m, aperture 1.00 m78 mm of envelope
Fig. 10 A gentler mirror at a moderate aperture. The caustic is shorter and the same shape — its size scales with the aperture and its form does not.
A paraboloid's focus, and the curve a sphere has insteadTwo mirrors with the same radius of curvature at the vertex, given the same parallel bundle. The paraboloid's reflected rays all pass through one point, 1.200 m from the vertex, to 7.9e-9 mm. The sphere's are tangent to a curve 121.2 mm long instead, and no point on it is on more than two rays.focus, R/2 = 1.200 m121 mm of causticvertex radius 2.40 m, aperture 1.24 m8e-9 mm against 121 mm
Fig. 11 And the shape comparison at that gentler curvature, where the sphere is a better approximation to the paraboloid and is still not one.

The source’s position is an argument

One structural point about the machinery, because it carries a claim.

The caustic drawn here is for a source at infinity — rays parallel to the axis. That is the case a telescope cares about and the case the textbook focal length is defined for. A source at a finite distance produces a different envelope, and the routine takes the source position as an argument rather than assuming infinity.

That is not tidiness. A caustic computed for the wrong source distance is a picture of a different situation, and the difference is largest exactly where the effect is most visible: a coffee cup lit by a lamp two hand-widths away has a caustic quite unlike the one a distant window makes, and both are commonly drawn as “the” caustic of a cylinder.

It also matters for the previous rung’s question. When the “source” is the viewer’s own eye at 1.4 m, the reflected bundle’s envelope is not the parallel-ray caustic at all — so the lines-of-sight measurement and the focusing measurement are computed from the same law with different arguments, and neither number substitutes for the other.

What it is for

Three uses, which is more than an envelope usually earns.

It says what aperture a spherical mirror can be used at. Choose the accuracy wanted, compute the caustic’s length, and stop the mirror down until the length is below it. That is a design calculation with one number in it.

It says where the light actually is. For a concentrator — a solar trough, a reflector lamp — the caustic is where to put the absorber, and it is not the paraxial focus unless the aperture is small.

And it says what a picture in a curved mirror is a picture from. Nowhere, but nowhere has a shape, and the shape is the caustic. A reconstruction that treats a mirror ball’s reflection as a second view will place the phantom camera at the least-squares point; knowing the caustic says how wrong that is and in which direction.

A spherical mirror's caustic — where the focus wentThe reflected rays are tangent to a cusped curve 195.0 mm long across a 1.10 m aperture. Near the axis it is at R/2 = 0.600 m, which is why a small enough spherical mirror passes for a paraboloid.195 mm of causticvertex radius 1.20 m, aperture 1.10 m195 mm of envelope
Fig. 12 A tight mirror at nearly full aperture, where the caustic is a large fraction of the focal length. There is no sense in which this mirror has a focus, and the drawing is what “no focus” looks like when it is drawn rather than said.
A spherical mirror's caustic — where the focus wentThe reflected rays are tangent to a cusped curve 137.1 mm long across a 0.80 m aperture. Near the axis it is at R/2 = 0.450 m, which is why a small enough spherical mirror passes for a paraboloid.137 mm of causticvertex radius 0.90 m, aperture 0.80 m137 mm of envelope
Fig. 13 A short-focus mirror at a moderate aperture, where the caustic is a large fraction of the focal length and the word focus has stopped meaning anything.
The shadow of a ball, under a lamp 2.40 m upThe lamp's tangent cone cuts the ground in an ellipse. The top of the ball is at 0.77 m and the lamp at 2.40 m: above that height the shadow closes, below it the shadow reaches the horizon, and the ball's distance from the lamp appears nowhere in the condition.the lamp is above the top of the ball — an ellipsecorrect from 22 cm, at 160 mm wideaxis ratio 0.5445 · discriminant -1.16e+0
Fig. 14 And the neighbouring construction in the light field, where a curved object and a straight ray produce a conic rather than a caustic: the shadow of a ball is an exact conic section, because the shadow’s boundary is a cone cut by a plane. Same ingredients, different question, exact answer.

Where this goes next

eyethe best fit — no ray goes through itno single viewpoint — the rays miss by 2.46 mmover 20 cm of a 2.00 m ball A curved mirror has no eye A flat mirror is a second camera — reflect the eye in the plane and every line of sight passes through the reflected point, to 2.8e-12 mm. Curve the mirror and the point is gone. Over 20 cm of a mirror ball two metres across, the lines of sight miss their own best-fitting point by 2.8 mm, and by 52.9 mm on a ball half a metre across. What replaces the eye is not a worse eye; it is nothing. focus, R/2 = 0.800 m184 mm of causticvertex radius 1.60 m, aperture 1.24 m5e-9 mm against 184 mm The one shape that focuses A paraboloid takes every ray parallel to its axis exactly through one point — 4.8e-9 mm of envelope across a 1.24 m aperture, which is arithmetic. A sphere with the same vertex curvature spreads the same bundle over 184.1 mm. So there is exactly one curved mirror that is a projection through a centre, it is a projection for exactly one bundle of rays, and every telescope in the world is built out of that sentence. source, 35 cmthe occluder's edgefraction of the source visiblepenumbra 17.5 cmprojection: 17.50 cmsampled: 17.41 cm The penumbra is the lamp's image The soft edge of a shadow is a picture of the light, projected through the occluder's edge as through a pinhole. That gives its width without any integration — and it is why the dapples under a tree go crescent-shaped during an eclipse.

Reads more easily once this is understood

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ApertureAstigmatismCausticcentre of projectionCuspEnvelopeFocusNot a projectionParaboloidRay tracingReflectionSpherical aberration