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

Mirror — where it appears

A reflecting surface, which is a second camera when flat and a projection from nowhere at all when curved. The flat case needs no new machinery at all and the curved case needs all of it, which is why the two have separate fields here.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

mirrors · Curvedmirror 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.

mirrors · Mirrorball 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.

mirrors · Conemirror horizoncorrect from 20 cm, at 160 mm wide44° across

Measured down from the waterline

Whatever stands so far above the water, draw its reflection the same distance below. Through a vertical picture plane that is not an approximation — it is the reflection, to the arithmetic floor. Tilt the camera twenty-two degrees and it is eleven pixels out. Draw a gull, which touches the water nowhere, and guessing its waterline point wrong by two metres of depth costs fifty.

wrong · Mirror 25102050131030how far away the object really is, in metreshow far away it appears, in metresa flat mirrortwo metres of radiusboth axes logarithmic · the eye 0.8 m from the glassceiling 1.30 m

Closer than they appear, by a factor with a number in it

A wing mirror of one metre radius held eighty centimetres from the eye reports 1.30 metres for an object at sixty-four, because the image of anything distant sits half a radius behind the glass. The size such an object subtends reads as a distance 2.60 times the true one, and the factor is exactly one plus twice the eye's distance over the radius — so the warning is a number, and it is larger for the mirror that is further away.

mirrors · Curvedmirror

Named alongside it

The objects these essays reach for when they reach for this one.

ReflectionPicture planeRay tracingVirtual imageAstigmatismcentre of projectionfield of viewNot a projectionAnamorphosisAngular sizeArea scaleCaustic

All concepts