Every essay — page 19
What survives Constructing a view Where to stand Surfaces that are not flat The other systems Light and mirrors Mirrors that are not cameras Measuring from one picture Through water and glass The real instrument The second eye What a pair is for Many pictures at once The eye that moves Systems that kept the measure What each system gave up What a machine computes The rectangle behind the lens The second projection Drawn confidently
The second projection
A picture that has been made still has to be shown, and the display is a projection with a correct point of its own. Here the assumed figure width every earlier essay quoted is replaced by an actual chain — sensor, focal length, screen — and the answer is that almost nobody is standing where the picture says to.
One picture and three people
A curved screen can be pre-warped for one seat, and the search over which seat to choose returns the middle one to a quarter of a per cent — there is nothing to be clever about. What the correction buys the sofa as a whole is six per cent, and the worst seat grows at fourteen pixels for every metre of audience, with no width at which it is zero except one person.
What the two eyes are sent
A reader's eyes are two seats sixty-three millimetres apart, so a curved screen delivers each of them a different map — and the part of the difference no homography absorbs is binocular evidence of the glass. Turned into a depth it comes back as the screen's own sag, 49 millimetres against 47 on a television, by a route that never saw the radius.
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.
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.
A wider screen keeps its seat under one rule
A renderer given a wider screen must choose between growing the horizontal field and shrinking the vertical one. Players prefer the first by a large margin. It keeps a picture's station point exactly where it was, so a viewer who was too far back stays too far back by the same factor while the edges reach further out. The despised rule moves the seat back, and on a very wide screen it arrives at the player's own chair.
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.
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.
A stereo picture is drawn for a level head
Every stereo pair is drawn for two eyes level with each other — a point's two images share a row and differ only across it. Tilt the head 10° in front of a television and that difference turns partly vertical, 14.46 arcminutes for anything drawn at infinity, and the two sightlines to a point stop meeting. At a desk monitor the same fifteen-arcminute limit arrives at 2.58°.
A curved screen tilts a stereo pair both ways at once
A stereo pair drawn for a flat screen and shown on a curved one gets a vertical disparity from the curve alone, with the head level — opposite in sign either side of the middle, and 31.8 arcminutes at the corners of a curved desk monitor for anything drawn at infinity. A rolled head adds its own, of one sign everywhere. The two add in one pair of corners and cancel in the other, so the curved screen is kinder than a flat one in two corners and harsher in the two that decide.
Drawing for the curve removes the curve, not the nearness
A stereo pair drawn for a flat screen and shown on a curved one sends a level head 31.8 arcminutes of vertical disparity before anyone tilts anything. Draw the pair for the curve and that vanishes exactly. What remains when the head rolls is the roll's own disparity, and on the curve it is 8.6 per cent worse than on a flat screen — not because of the curve's shape but because the arc brings its edges nearer the eyes, and a fixed separation seen from nearer is a larger angle.
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.
The eye that moves
A handscroll is drawn by an eye that travels, imaging one line at a time — orthographic along the roll and perspective across it, so a mile of river holds its scale while a single pavilion still recedes. Its rays miss their own best centre by metres, and the miss is exactly the length of track you unroll.
A scroll is a camera that moves
A Chinese handscroll is not a picture with a wandering viewpoint or a picture with no viewpoint. It is the image of an eye that travels along a track and records one vertical line at a time, and that object has an exact geometry — orthographic along the roll, perspective across it.
A straight line in a scroll is a hyperbola
Under a pushbroom the image of a straight world line is a Möbius function of the paper coordinate, which is a rectangular hyperbola. It is straight exactly when the line holds its depth — so a curve in a handscroll is a depth signal rather than a stylistic one, and the sag is computable in pixels.
The centre a scroll does not have
Fit a common point to the rays of one section of a handscroll and it misses by metres. The miss is not a residual to be tightened — it is exactly the standard deviation of the eye's own track, it grows linearly with how much is unrolled, and it goes to zero only for a section of no width.
A map along, and a picture across
The midpoint of a segment lying along a scroll's length images to the midpoint of its image, exactly. The midpoint of a receding one lands 22% of the way off. One projection, two answers — and the direction that keeps measure keeps something a perspective picture never offers.
A scroll is not a panorama
Both draw straight world lines as curves, and one of them is a projection. A rotating eye keeps its centre exactly however far it turns; a translating eye has none at all. Curvature and centrelessness are independent properties, and conflating them is the standard mistake about both objects.
Three distances in one landscape
A landscape assembled from a low station for the near ground, a level one for the middle and a high one for the far gives its furthest band 6.9 times the picture one camera would allow it — because the image of a fixed depth interval falls as one over depth squared, and its own station gives that back. What it costs is a jump in the rate at which depth runs, 2.50 at the first join and 2.75 at the second, and three views from one height leave no seam at all.
A landscape that changes its rule halfway up
A reader handed only the marks of a three-station landscape recovers each band's own camera height without being told any of them — 1.60 m, 4.00 m, 11.00 m, to 3.6e-15 m. What that same reader cannot recover across a join is a common ground: the next band's own marks read as a ground point 2.40 m away from the true one at the first seam, 7.00 m at the second.