Caustic — where it appears
Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
A picture through water has no viewpoint
Continue the rays of a refracted picture into the water and fit them to a common point. They miss it by ten millimetres. The same fit with the water taken away misses by zero, which is what makes ten millimetres a measurement rather than a number.
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.
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.
The caustic is the mirror's own ruler
Four of the five instruments in this row return a ratio and refuse a size. The bright curve a mirror throws is the exception, and the reason is that it is a length lying on the table rather than an angle in a picture. Its tip is the paraxial focus, so a ruler laid from the mirror's vertex to it returns the radius of curvature — 1.5996 m against 1.6, and exactly right on a paraboloid, where there is no aberration to bias the envelope.
A ball of water has no eye either
A flat interface is not a projection through a centre and misses by ten millimetres. A sphere of water misses by more than that on a ball the size of a plum — 1.3 mm on a fifty-millimetre radius, and the axis crossings spread over 18.6 mm at seven tenths of the aperture. But at two per cent of the radius the same fit returns 24 nanometres, so a ball does have a centre — one at zero aperture and none by the time it is gathering any light.
The pivot that is not the eye
A camera bolted to a tripod turns about the screw under its baseplate, and the light crosses somewhere else. Every ray of the panorama that results passes the pivot by e sin γ — the offset times the sine of how far off its own frame's axis the ray points — so the picture has a radius where a projection would have a point.
Named alongside it
The objects these essays reach for when they reach for this one.
centre of projectionEnvelopeRay tracingSpherical aberrationApertureFocusleast-squares intersectionNot a projectionParaboloidReflectionAstigmatismCusp