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

Angular size — where it appears

How wide something looks from where the eye is, measured as an angle rather than as a length. It is what a picture of an object actually records — the ratio of size to distance and nothing else — so a photograph gives an angular size exactly and a size only once a distance is supplied.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

the eyecorrect from 9 cm, at 160 mm wideoutlines agree to 6e-12 px

One picture of a ball

The outline of a ball in a photograph fixes the cone of rays that graze it, so the cone's axis is the direction of the ball's centre and its half-angle is the arcsine of radius over distance — both exactly, and neither of them separately. A ball a hundred and thirty-seven times larger, a hundred and thirty-seven times further away, draws the identical conic. And the drawn ellipse's own centre is not the image of the ball's.

metrology · Ballrecovery 020406012345the baseline between the two eyes (metres)angle between the cone axes (°)42.6°two cones, one ballcentre to 9e-7°

Two pictures of a ball

Two outlines give two tangent cones, and where their axes cross is the ball's centre — with the radius following from either half-angle. There is no correspondence problem, because a ball's outline in one picture and its outline in another are guaranteed to be the same ball; and there is a degeneracy, which is the two eyes and the ball in a line, and the solver refuses it rather than returning something plausible.

metrology · Ballrecovery 025507505001e+31.5e+3how far along the sofa the viewer sits, in millimetreshow far a mark is from where it was drawn to be seen, in arcminutesthe matched picturean ordinary flat picturethey crosscurved television, along the sofacrossing at 400 mm

Matching buys one seat

Feeding a curved screen its own picture surface makes the picture exact at one point and worse everywhere else than the flat picture it replaced. Forty centimetres along the sofa the two cross, and past that the matched picture is the worse of the pair — because a flat picture is mediocre everywhere and a matched one is perfect at a point and falls away from it faster.

screen · Matched 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 the centre of curvature, 4.0 mthe sitting distance, 2.6 mno single viewpoint — the rays miss by nothing — this is a plan of a room319′ out where people sit

Drawn for the cylinder, shown on the cylinder

Four essays in this collection have named the arrangement where a picture's surface and a screen's surface are the same surface, and none has run it. It is exact — a tenth of a millionth of an arcminute — and the exactness is not the finding. Three separate things have to be true at once, and a curved television's own maker prints a sitting distance where the first of them is out by five degrees.

curved · Matched 01002000.80011.20the condition, as a fraction of what makes the picture exacthow far a mark is from where it was drawn to be seen, in arcminutesthe seatthe horizontal scalethe vertical lawcurved television, one condition at a timeall three linear

Three conditions, and three prices

A matched picture needs the seat, the horizontal scale and the vertical law all at once. Each is broken alone here with the other two held, and all three turn out to be linear in the mismatch — the unforgiving case, with no margin at all. The vertical law costs a third of an arcminute on a television and eighty-four on a dome, because it is the difference between an angle and its tangent and that difference is cubic in the picture's vertical field.

curved · Matched cylinderexactequirectangular0.18′equidistant fisheye0.73′equal-area fisheye0.82′stereographic0.99′flat plane1.79′curved television, from its matched seateach surface fitted to it

The surface a screen wants

Six picture surfaces laid on one screen and viewed from the seat its curvature names, each with its own extents fitted so the ranking is about shape rather than scale. Each screen's own surface is exactly right on it and nothing else is — and on a curved television the runner-up is a fifth of an arcminute behind, which nobody can see.

curved · Matched

Named alongside it

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

Centre of curvatureArcminuteMatched surfacePicture surfaceScreenCylindrical projectionEquirectangularCalibrationDemonstrationHomographyQuadricReconstruction

All concepts