Resolution — where it appears
Named by 16 essays across 6 fields — each of them below, with the objects they name alongside it.
The wall under the paint
An anamorph's design is a set of rays from a known eye, fixed before any paint is applied — so the marks are a calibration target whose rays are known exactly and whose shape is not. One ordinary photograph supplies the second ray for each mark and the wall comes back at the arithmetic floor, on a dished floor, a ridged one, and a floor with a step in it. What it says between the marks is nothing, by the sagitta.
A curved screen is eight flat ones
A projection matrix is a plane and nothing else, so a curved display cannot be rendered — it has to be driven as several planes and assembled. The gap between chord and arc is the whole error, it goes as the square of the angle each piece spans, and the piece count therefore goes as the inverse root of the tolerance — three for eight pixels, eight for one, fifteen for a quarter.
The evenness a curve buys
A curved screen is sold on evenness, and evenness turns out to be three quantities that disagree. On pixel pitch the curve wins from every seat; on the angle the glass is turned through it wins until three and a half metres along the sofa; on the plain distance from eye to glass — the reading the argument is usually made in — it gives up before half a metre.
Counting is a measurement
A tiled floor gives its area with no reference length at all — count the tiles and multiply. The count is an integer, so it is exact wherever it can be made, which is a completely different error law from the rectifier's smooth decay. And the distance at which it fails is set by the tile's depth edge, which foreshortens as one over the depth squared, so 18.7 m for a 62 cm tile, where the across edge alone would have allowed 217.
The camera that is a cylinder
A swing-lens camera turns its lens about its own entrance pupil and sweeps a slit across film bent into a circle concentric with it. Compute where the light lands, unroll the film, undo the pinhole's inversion, and the result is not similar to the cylindrical picture surface — it is the same map, to the arithmetic floor. What it pays instead is detail, and a shear on anything that moves.
One homography makes a shadow map the eye's picture
A shadow map for a lamp facing the eye needed 706 thousand texels however its rows were re-spaced, nearly five times the one texel per pixel no map can beat. Warp the whole map by a projective transformation, not only its rows, and it needs 154 thousand — within five per cent of the bound — and every texel, carried into the eye's picture, lands at one pixel. The reason is exact: the eye's picture of a floor and the lamp's picture of the same floor are one homography apart.
Keystone correction spends the panel unevenly
Turn a projector fifteen degrees and keystone correction throws away 18 per cent of its panel — the number a specification quotes. It is the smaller half of the price. What remains is spread unevenly: the near edge of the corrected picture gets a panel pixel for every source pixel, the far edge three-quarters of one, so the far edge carries 58 per cent of the source's detail. A tipped projector keeps more than a turned one, and a lens shift that places the same picture loses nothing at all.
An even spend of the panel is an uneven picture
A keystone correction that spends the panel evenly exists: render the picture straight into the panel pixels that reach the corrected rectangle, one sample each. At fifteen degrees that is 82.2 per cent of a frame's samples, and on the wall it delivers exactly what a full frame does — 0.99 of the source's detail at the near edge, 0.76 at the far — because the panel, not the source, was setting the detail everywhere. It removes no unevenness. On fewer samples it makes the picture worse where it was worst; the even picture is a uniform source of 57.6 per cent of a frame.
The even picture is fair to a centred room, and only to it
Judged by detail per degree of each seat's view, the corrected picture that is even on the wall is already the fairest possible for an audience centred on it: a layout drawn for that audience gains its worst seat nothing. For an audience on the far side it is also the best, because the far edge is where the panel is coarsest and the even picture already spends all of it there. Only an audience on the projector's side can be served better — thirty per cent better at a fifteen-degree turn — by putting the samples on the near edge, where the panel has them to give.
A walking viewer gains from a following layout only on the projector's side
Re-lay a turned projector's corrected picture frame by frame for one viewer walking across the room and they gain up to thirty-one per cent of detail per degree — on the projector's side of the picture, where the panel has samples to spare, and nowhere past eighty centimetres beyond its centre. A layout drawn once for the whole walk gains nothing at all. Half a second late, the following layout keeps four-fifths of the gain and at the worst moment serves the viewer six per cent worse than the even picture; four seconds late it is worse on average. And whatever it gives the walker it takes from a seated room: up to twenty-nine per cent of the worst seat's detail.
The stretch decides the band
A design band chosen by geometry — the rays that meet the object — includes rays that graze along it, and a grazing ray lands two design points twenty times further apart than the design says. Cap the stretch at four and a bare floor keeps 55 per cent of its design, a corner 80, and the top of a descending flight all of it.
Two walkers share a layout drawn for the worse of them
A projector turned to its wall can re-lay its corrected picture for one walking viewer and give them up to thirty per cent more detail per degree on the projector's side. For two, it draws the layout that serves the worse-served best. Walking the same way a metre apart, the one nearer the picture's centre keeps 99 per cent of its own gain and the other 77, falling to 44 at four metres apart. Crossing, they keep two per cent. A viewer on the far side vetoes the other only while it is the worse-served — and at different depths the shared layout takes from the walker it serves better, leaving them below the even picture.
A rule that weighs two walkers spares the one serving the worse would starve
A projector following two walkers can share one layout between them by serving whichever is worse-served. Weighed instead — the most detail in all, the most of both walkers' shares of their own best, or the largest of the smaller share — the shared layout gives up a little of the worse walker's gain and changes who pays for it. Walking two and five metres from the wall, serving the worse gives the near walker 11.7 per cent over the even picture and takes the far one below the even picture, out of resolvable detail in a fifth of the frames; keeping the smaller share largest gives the near walker 6.8 per cent and leaves both at least 61 per cent of their own best, never below what an eye resolves. The most detail in all is the worst rule of the four: crossing, it serves the worse walker 9 per cent below the even picture.
A walker charged at its worst place gains where it looks
A projector sharing one layout between two walkers, drawn for the worse-served of them, took the further walker below the even picture at its worst place — the picture's far edge, which it sees at a slant from five metres. Scored instead at the part of the picture straight in front of each walker, the same layout gives that walker 10 per cent more than the even picture, not 6 per cent less: the samples were taken from where it was not looking. The four rules for sharing then nearly agree, 11 to 12 per cent for the worse-served walker. Two things bring the charge back. A walker looking 30 to 40 degrees ahead along its walk looks at the edge the layout starved. And a walker looking at the picture's middle is served alike by every layout, because every layout the projector can afford takes its samples from there.
The distance at which two lamps part
Two lamps five centimetres apart are one lamp, and the drawing is right to say so. The separation at which they become two is proportional to how carelessly the picture was clicked — 3.6 centimetres at half a pixel, 7.2 at one, 28 at four — with no floor anywhere, so nothing but care stands between a reader and any separation at all.
A projector that is not at the dome's centre
A projector 0.40 of a dome's radius off centre puts its own picture up to 23.6 degrees from where it belongs, and the pre-warp that corrects it is exact for one seat and only one. Two metres from that seat costs 11.5 degrees of the same displacement, wherever the projector itself stands — because the correction never knew where the projector was in the first place.
Named alongside it
The objects these essays reach for when they reach for this one.
Sampling gridPicture surfaceKeystoneViewing distanceProjective mapViewing positionHomographyinstrument limitForeshorteningJacobianAnamorphosisArea scale