Keystone — where it appears
Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.
The screen is a picture surface too
A curved television is one of the six named picture surfaces sitting in a living room, and from its own axis it delivers azimuth in proportion to the picture, exactly. What it is shown is a rectilinear picture from a sofa, and the difference is not a matter of degree — a flat screen from any seat shows a homography of the intended picture, so it is a correct picture of a transformed scene, and a curved one shows a map that is not a homography from any seat at all.
Straightening does not move the eye
Correct a photograph's converging verticals and what comes out agrees with a level camera at the same point — one the correction was never shown — to 3e-13 px, with the verticals parallel to 0e+0°. The cross-ratio of four points along a ground line reads 1.3333 before and after, so the corrected picture measures exactly what the original measured, from exactly where the original was taken and nowhere else.
A projector in the viewer's eye
A projector paints a wall along its own rays, so from the projector's own position the wall's shape is invisible — exactly, on a cylinder, on a dome and on a plane alike. Move a hand's breadth away and the residual is pixels. The one place a projector can stand and ignore the shape of what it is throwing onto is the eye of the person watching.
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.
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.
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.
Picture surfaceResolutionSampling gridViewing distanceHomographyProjective mapViewing positionRectificationcentre of projectionCylindrical projectionJacobianAnamorphosis