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

Thin lens — where it appears

A lens modelled as a single refracting plane with one focal length, so that an object at distance a images at b with 1/a + 1/b = 1/f. It is enough for every geometric statement about apertures here, and it cannot show two pupils of different sizes.

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

a point at 1.5 mthe pupilthe sensorwhere it focusesf/2.8, focused at 3 m5.8 px across

The centre has an area

Every theorem of perspective follows from a projection through a point, and no instrument has one. Give the pupil a radius and each world point images as a disc — whose centre is the pinhole's mark to 5.7 × 10⁻¹⁴ millimetres, at every aperture, every distance and every field angle. The geometry survives exactly; only the sharpness is spent.

sensor · Aperture the entrance pupilthe stopthe glassthe chief rays, from four object distances3.6e-15 mm apart

The hole a scene actually sees

The stop is not the centre of projection. Model a 50 mm lens with its stop 18 mm behind the glass and the chief rays from every object distance cross the axis at one point 28.1 mm on the other side of the lens — 10.1 mm from the stop and 1.56 times its size — to 3.6 × 10⁻¹⁵ mm. That point is the entrance pupil, and it is where a picture is a projection from.

lens · Pupil 34363840-0.50000.5001distance the lens is focused at (m, log scale)horizontal angle of view (degrees)38.99 at 3 m37.76 at 1 m35.90 at 0.5 m39.60° at infinity35.90° at 0.5 m

Focusing is a zoom

A 50 mm lens focused at half a metre is not a 50 mm camera. It stands 55.56 mm from the sensor, its picture is a pinhole picture at that distance, and it covers 35.9° where the same lens at infinity covers 39.6°. Recover the camera from the picture and it reports 55.56 mm. Read the picture with the engraved 50 mm instead and a right angle comes back as 96.0°.

sensor · Format 0.313100510how far away the subject is, in metreshow far the pupil moves forward, in millimetres0.50 mthe whole walk with field anglea 50 mm lens, focused as one pieceequal at 0.50 m

Focusing moves the pivot past its best place

Focusing a fifty-millimetre lens to one metre carries its entrance pupil 2.63 millimetres forward of the camera body, and to half a metre 5.56 — which is more than the whole 5.53 that the pupil walks with field angle, so past a subject at 502 millimetres the focus decides where the pupil is. A panorama head aligned at infinity and used at a metre leaves 7.46 arcminutes along its seam; aligned at four metres it leaves 2.21, better than pivoting at the pupil at all.

lens · Pupil 0.30.71.537200.11101001000distance the lens is focused at (m, log scale)how far the recovered distance can be out (% , log scale)vanishing points ±0.25 pxvanishing points ±1 px±0.25 px: 0.7 % at 1 m → 28 % at 30 m50 mm lens

A close picture carries its own distance

A 50 mm lens focused at one metre stands 2.632 mm beyond its focal length, and a picture records where the lens stood. Recover that from two vanishing points, put in the engraved focal length, and the focus distance comes back — to 0.7 per cent at 1 m, 3.7 at 5 m, and as an interval 28 per cent wide at 30 m. A single picture has given a scale. What it has given is where the lens was focused, and the subject can be anywhere in a sharp band two to fifteen times wider.

sensor · Format

Named alongside it

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

Focal lengthcentre of projectionChief rayEntrance pupilinstrument limitApertureCamera calibrationCircle of confusiondepth of fieldExit pupilParallaxPinhole

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