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

Arcminute — where it appears

A sixtieth of a degree, which is roughly the finest detail a good eye resolves and the angle a millimetre subtends at three and a half metres. It is the honest unit for an error a viewer receives, because what reaches an eye is a direction rather than a distance.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

in section, two of the three facesworst 0.0e+0°

The corner that answers every eye

Three mirrors at right angles compose into the point inversion, so a ray entering leaves antiparallel to itself whatever direction it arrived from — a picture surface with no correct viewpoint because every viewpoint is correct. Tilt one face half a degree and the worst returning ray is out by exactly one degree, twice the error; the best is out by a twelfth of it, which is why a tolerance quoted from one measured ray is a statement about that ray.

mirrors · Twomirrors 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 -2024-0.500-0.25000.2500.500where the viewer sits, as a fraction of the sitting distance (log₁₀)the two eyes' vertical disagreement, in arcminutes (log₁₀)15′ fusion limitthe raw differencewhat a common frame leavesthe sitting distancecurved monitor, against the chairlimit at 29 cm

The distance at which the eyes part

The two eyes' disagreement on a curved screen was measured at each screen's own sitting distance and reported as a null result. The sitting distance is a parameter and the chair moves — swept, the raw difference falls like the cube of it and the residual like the fourth power, and a viewer twenty-nine centimetres from a curved monitor crosses the fusion limit the null result was quoted against.

screen · Stereodisplay 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 curvatureScreenAngular sizeMatched surfacePicture surfaceCylindrical projectionEquirectangularHomographyToleranceAsymptoticsAudienceBinocular disparity

All concepts