The even picture is fair to a centred room, and only to it

An even spend of the panel is an uneven picture followed a projector turned fifteen degrees to its wall. Correcting its keystone throws away part of the panel and spends the rest unevenly: the panel’s pixels land close together on the near edge of the corrected picture and spread out on the far edge. Among the ways to lay the source out on what is left, the essay found that spending the panel evenly delivers exactly what a full frame does, and that the picture which is even on the wall — the same detail per millimetre everywhere — is a uniform source at the far edge’s density, costing 57.6 per cent of a frame.

The essay ended by pointing out that nobody looks at millimetres of wall. A projector in the viewer’s eye had found that a corrected picture on a flat wall is a correct flat picture from every seat, which is a statement about its shape. Its detail is another matter: a seat to one side sees one edge of the picture nearer and larger than the other, and what that seat can use is detail per degree of its own view, which varies across the picture differently from seat to seat. The question was which layout gives the worst seat its best worst place — whether the even picture on the wall is also the fair picture across the audience.

For most rooms it is. The exception is specific, and it points the opposite way from the one the earlier essay guessed.

Detail per degree, seat by seat #

A seat at depth d from the wall sees a millimetre of wall at distance r across a d/r² of a radian — the projector run backwards that a projector is a camera run backwards began from, with the seat as the camera: the nearer and more squarely it looks, the more of its view a millimetre fills. A layout gives each place on the wall a density of source pixels per millimetre, and the seat’s detail per degree at that place is that density times the millimetres its degree covers there. Each seat has a worst place — where it sees the wall largest relative to the detail there — and each layout has a worst seat.

The room is a set of seats, the kind of audience one picture and three people and the seats a screen will accept placed before a curved screen: four rows two, three and a half, five and six and a half metres from the wall, five seats across each, spread over 3.6 metres, centred on the corrected picture. The projector is the earlier essay’s, turned fifteen degrees, its corrected picture 1.92 metres wide. Four layouts are compared at one budget, the even picture’s own 57.2 per cent of a frame: the source laid evenly, the panel’s own density, the even picture, and a layout drawn for the audience, which gives each place the density that makes its worst-placed seat’s detail per degree the same everywhere, capped where the panel can give no more.

For the centred room the four curves nearly coincide, and the one that differs is worse. The even picture gives its worst seat’s worst place 26.39 pixels a degree; the source laid evenly, at this budget, is the same layout; the layout drawn specifically to maximise the worst seat gives 26.40. The panel’s own density — which the earlier essay found spends everything the panel has, rich at the near edge — gives 22.79, because its far edge is thinner than the even picture’s and the far edge is somebody’s worst place.

The worst seat is the front row’s centre, two metres from the wall. From there the whole picture, 1.92 metres across, is seen at nearly one scale: its middle is two metres away and its edges 2.2, so a degree covers 35 millimetres of wall in the middle and 42 at the edges. A seat whose view of the picture is nearly even wants a nearly even picture, and a room whose worst seat is its front centre wants what that seat wants.

Why every seat’s worst place is straight ahead of it #

The pattern has a short reason. A seat covers the least wall per degree where it looks most squarely and from nearest: at the point of the wall straight in front of it, where a degree covers its depth times π/180 of wall and no less. Anywhere else on the wall a degree covers more, by the square of the distance over the depth. So for any layout whose density does not change much across the picture, a seat’s worst place is the part of the picture closest to straight ahead of it, and its detail per degree there is the layout’s density times its own depth.

That makes the front row the worst row, always, and among the front row it makes the seats that face the picture’s coarsest part the worst seats. Under the even picture every place is equally coarse, so every front-row seat facing the picture ties, and the one at the centre, facing it squarely, is as bad as any. Under the panel’s own density the far edge is coarsest, so the worst seat is the front-row seat facing the far edge. A seat off to the side of the picture, facing no part of it, has its worst place at the picture’s nearest edge, seen as squarely as that seat can see anything in it.

The layout drawn for an audience simply follows this: at every column of the picture it asks which seat covers the least wall per degree there, and gives that column the density that seat needs. For a centred room that seat is, at every column, the front-row seat facing it, at the front row’s depth — the same number everywhere — and the fair layout is flat. It is the even picture.

Four layouts of one budget #

The layout drawn for an audience is not always the even picture. It depends on where the audience’s worst-placed seats are, which the next figure shows for an audience that sits to one side.

The dashed line is the limit: the panel can give 0.987 source pixels a millimetre at the projector’s side of the picture and 0.756 at the far side. The even picture sits at the far edge’s density across the whole wall. The panel’s own density follows the limit down, scaled to the budget. The layout drawn for an audience sitting to the projector’s side of the room follows the limit on the near edge and falls well below the even picture on the far edge, to 0.467: that audience sees the near edge closest and the far edge from well away, so its worst-placed seats are near the near edge, and that is where the samples go.

Where the audience sits #

Moved to the far side of the room — three seats in each row, 1.8 to 3.6 metres beyond the picture’s far edge — the audience is served by the even picture at 31.26 pixels a degree, and nothing better exists: the layout drawn for them gives 31.26 too. The earlier essay’s guess was that a projector off to one side should render for the far-side seats, where its picture is both softest and nearest. The far-side seats are indeed limited at the far edge, where they sit closest; but the far edge is where the panel is coarsest, and the even picture already spends every sample the panel has there. There is nothing more to give, so nothing to gain by giving it. The panel’s own density, thinner at the far edge still, costs them fourteen per cent.

Moved to the projector’s side, the same audience is worst served near the near edge, which it sees closest. There the panel has samples to spare — the even picture uses 0.756 of the 0.987 a millimetre it could — and a layout that spends them there lifts the worst seat from 31.26 to 40.79 pixels a degree, thirty per cent. Even the panel’s own density, which was not drawn for anybody, gives this audience 35.22, twelve per cent better than the even picture. For a single viewer the figures are the same shape: 33.8 from the even picture on either side, 44.1 laid for a viewer on the projector’s side, still 33.8 laid for one on the far side.

The asymmetry is the whole finding. A layout can only give a seat what the panel has to give where that seat looks hardest. The turned projector’s panel is rich on its own side and poor on the far side, so seats on its own side can be served better and seats on the far side cannot.

The turn decides how much it matters #

At a turn of five degrees the panel is nearly even and so is everything else: a layout drawn for a projector-side audience gains it nine per cent. At fifteen degrees, thirty; at twenty, forty-three; at thirty, forty-six, where the gain levels off because the near edge’s own density stops rising as fast as the far edge’s falls. The layout drawn for a centred audience never gains more than a third of a per cent at any turn. And the panel’s own density — the layout an unconsidered correction might fall into — costs a far-side audience five per cent at five degrees and a quarter at thirty.

So the question the earlier essay asked about the even picture — is it fair? — has a plain answer with one exception. For a room centred on the picture, or sitting towards the far side, the even picture is the fairest layout there is, at every turn. For a room sitting towards the projector’s side, it leaves between a tenth and nearly a half of the worst seat’s detail unused, and the more the projector is turned, the more it leaves.

The front row decides the level #

The layouts decide the shape of the detail across the room. What decides its level is how close the audience sits.

The worst seat is always in the front row, and its detail per degree grows in proportion to its distance from the wall: 15.8 pixels a degree at 1.2 metres, 26.4 at two, 52.8 at four, 79.2 at six. The proportion is exact to the three figures shown, because every degree of the front row’s view covers wall in proportion to its distance, and the picture’s density on the wall is fixed by the projector. An eye resolves about sixty pixels a degree, and a pixel is not a point is the reminder that what it resolves is the pixel’s area, not its centre. The even picture of this fifteen-degree projector reaches that for its front row only from about 4.5 metres back; a room whose front row is closer sees the picture’s pixels, whatever the layout.

That puts the thirty per cent in proportion, and it is the same lever the screen sets the distance found for a picture’s geometry: where the viewer sits decides more than how the picture was made. Laying the picture out for a projector-side audience is worth as much, for that audience’s worst seat, as moving the front row back by thirty per cent of its distance: from two metres to 2.6. A room that can move its chairs has a cheaper remedy; a room with fixed seats on the projector’s side has only the layout.

What twenty-six pixels a degree looks like #

The level is worth reading as a viewer would. At twenty-six pixels a degree, a single source pixel subtends a little over two minutes of arc from the worst seat — twice what an eye can resolve at good contrast. The front row centre of this room sees the picture’s pixel grid, faintly, and sees fine text in the picture as slightly soft. That is a property of the projector’s turn as much as of the room: square on, the same projector would lay 0.96 source pixels a millimetre across a two-metre picture, and the front row at two metres would receive 33.5 pixels a degree, about a quarter more.

So the correction has two costs for the worst seat, and the layout can only address one of them. The turn throws away a share of the panel and makes the far edge coarse, which caps what any layout can give anyone looking at the far edge; that cost is paid in the projector’s placement and is the one keystone correction spends the panel unevenly measured. The layout then decides who absorbs the remaining unevenness. For a centred room nobody needs to, because the even picture lays the far edge’s coarseness evenly and the front row sees it evenly. For a room on the projector’s side, the layout can hand that room the near edge’s surplus.

What the audience asks of the picture #

Put together, the earlier essay’s even picture turns out to be the right default for a reason it did not give. It was built to spend the far edge’s density everywhere so that no part of the wall is worse than another. The measurement here says that for most audiences the worst part of the wall, for the worst seat, is either the whole picture seen from its front centre or the far edge seen from beside it — and in both cases the even picture already gives it everything the panel can. It is fair to the room for the same reason it is even on the wall: the far edge sets the limit, and nothing is wasted matching it.

What the even picture wastes is the near edge’s surplus, and that surplus is only useful to an audience that looks at the near edge hardest. Keystone correction spends the panel unevenly found the near edge oversampled; for a room seated on the projector’s side, that oversampling is exactly where its worst seats look, and a layout that keeps it there serves them. For every other room, spending the near edge’s surplus buys detail nobody at the worst seat can use.

What was assumed #

The room is seen in plan. Every seat here is at the picture’s height, and detail per degree is counted across the picture only. A tilted projector — a ceiling mount pitched down — spends its panel unevenly from top to bottom as well, and the rows of a raked room see the picture’s top and bottom at different scales; the same argument applies in that direction and was not measured.

Detail is counted in the worst direction. A layout’s density on the wall is taken as its smaller singular value, the resolution in the direction it is worst. The picture’s rows and columns are resolved differently on a turned wall, and a viewer reading text across a line cares about one direction more than the other.

The budget is the even picture’s. Every layout is compared at 57.2 per cent of a frame. At a full frame’s budget the layouts can all reach the panel’s limit where they want it, and the differences shrink to those set by the panel alone — the uniform source then caps at the panel everywhere, which is what the earlier essay found.

An eye resolves sixty pixels a degree. Acuity varies between people and with contrast; the figure is the conventional one-arcminute threshold, used only to place the front row’s distance, not to decide which layout is fairer.

Still open: whether a moving audience wants a moving layout #

The layouts here are drawn once for a seated room. A viewer who walks across the room — a visitor in a gallery, a presenter at the side of the picture — changes which place is worst for them as they move, and a projector that knows where they are could re-lay the picture for them frame by frame.

The measurement that settles what that is worth takes one viewer walking across the room at a stated distance, lays the picture out for their current position each frame, and compares the detail per degree they receive with the even picture’s, and with a layout drawn once for the whole of their path. The layouts above say what to expect at the two ends — nothing on the far side, thirty per cent on the projector’s side — and the question with a number in it is how much of that the viewer keeps when the layout follows them, and whether a layout that lags their position by a stated time gives back more than it gains.