Visual Plane
every picture states the point it is correct from
Field

Mirrors that are not cameras

A flat mirror is a second camera and a curved one is not a camera at all: continue the lines of sight behind a mirror ball and they pass through no common point, so a photograph of it is a projection of nothing from anywhere. What they have instead is a caustic — and one shape, the paraboloid, that does focus exactly, for one bundle of rays and no other.
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.

 184 mm of causticvertex radius 1.60 m, aperture 1.24 m184 mm of envelope

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.

 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.

 00.2500.5000.7501050100150angle of the direction from the camera's own axis (degrees)where it lands in the picture, as a fraction of the picture's radiusequisolidequidistantorthographica ball 24 cm across, camera 24 radii offequal-area within 0.82% · equidistant 21.5%

A mirror ball is an equal-area fisheye

Photograph a mirror ball from far enough away and its rule is ρ = R·sin(θ/2), which is the equal-area fisheye — not an approximation to it, the rule. Measured, the departure falls from 4.27% of the picture's radius at 3 radii to 0.01% at 2000, while the next-best named rule stays 21% out at every distance. And the ball reflects 100.0% of the directions there are, which no designed surface does.

 eye, on the axisa cone 29 cm across and 22 cm highthe design lies from 0.17 m to 0.47 m out

The cone that reads the floor

A conical mirror standing on a design turns it into a picture, and the map it performs is exact, one-to-one, and not a projectivity — a homography fitted to four of the marks returns those four to 7e-13 mm and puts the rest 2480 mm away, on a design 369 mm wide. The reason is that the cone turns the annulus inside out: the middle of the picture comes from the far edge of the floor and the rim from the near one.

All essays