A walking viewer gains from a following layout only on the projector's side

The even picture is fair to a centred room, and only to it measured a turned projector’s corrected picture by what each seat can use: detail per degree of its own view, at the place where that seat is worst served. A projector turned fifteen degrees to its wall throws away part of its panel and spends the rest unevenly, rich on the edge nearest it and coarse on the far edge, and the corrected picture that is even on the wall spends the far edge’s density everywhere. For a room centred on the picture, or sitting to the far side, that even picture is the fairest layout possible: the worst seat looks hardest at the far edge, and the even picture already gives it everything the panel has there. Only a room on the projector’s own side could be served better — thirty per cent better at fifteen degrees — by putting the near edge’s surplus where its worst seats look.

Those layouts were drawn once, for seats that do not move. The essay ended on a viewer who does: a visitor walking along a gallery wall, a presenter crossing in front of their own slides. A projector that knows where such a viewer stands could re-lay the picture for them every frame, and the question was how much of the seated room’s thirty per cent a walker keeps when the layout follows them, and whether a layout that follows late gives back more than it gains.

It keeps all of it, on the projector’s side, and none anywhere else. Following late costs both ways. And the walker’s gain is the seated room’s loss.

Following pays only where the panel has samples to give #

The projector is the earlier essays’: fifteen degrees to its wall, its corrected picture 1.92 metres wide, every layout held to the even picture’s budget of 57.2 per cent of a frame’s samples. One viewer walks across the room three metres from the wall at a metre a second, from 3.6 metres to the projector’s side of the picture’s centre to 3.6 metres beyond it on the far side. At every tenth of a second, three layouts are compared by the detail per degree they give the walker at the walker’s own worst place: the even picture; a layout drawn once for every position on the walk, which gives the walk’s worst moment its best; and a layout redrawn for where the walker stands.

The following layout is drawn carefully. For each position it takes the layout that maximises that viewer’s worst place, and where that layout does no better than the even picture it uses the even picture. The care matters because for a viewer on the far side many layouts tie: the walker’s worst place is the far edge, the panel has nothing more to give there, and a layout drawn for that viewer alone gives them the same as the even picture while taking samples from the near edge, which they see from far away. A projector that follows one viewer should not starve the rest of the picture for no gain, and the even picture is the layout among the ties that starves nothing.

The result has exactly the shape the seated room’s had. At the projector’s-side end of the walk the following layout gives thirty-one per cent more detail per degree than the even picture: the walker sees the near edge closest, the near edge is where the panel is rich, and a layout that spends the surplus there serves them. As they cross towards the far side the gain falls, and past eighty centimetres beyond the picture’s centre it is zero — their worst place has become the far edge, which the even picture already spends everything on. Averaged over the walk, following gains eleven per cent.

The layout drawn once for the whole walk gains nothing. It is the layout that maximises the walk’s worst moment, and the walk’s worst moment is on the far side, where the even picture is already optimal. A single compromise over a walk that reaches both sides of the picture is the even picture.

Where the gain lands, and where it does not #

The gain is largest at the end of the walk, and that is not where the walker most needs it. Three metres from the wall and 3.6 metres to the projector’s side, the walker sees the whole picture obliquely and from well away, and even the even picture gives them 70.7 pixels a degree at their worst place — more than the sixty or so an eye resolves, the threshold a pixel is not a point uses. Following lifts that to 92.2, a gain no eye can see.

The walk’s worst moment is straight in front of the picture, where the walker is nearest it and sees its middle squarely: 39.6 pixels a degree under the even picture, well below what an eye resolves. Following lifts it to 40.9, three per cent, because a viewer standing in front of the picture sees it at nearly one scale, as the seated room’s front centre did, and a viewer who sees the picture evenly is best served by the even picture. Two metres from the wall the worst moment gains nothing at all; five metres back, the even picture already gives more than sixty everywhere on the walk and following can only add detail nobody resolves.

Between those ends the gain is real and visible. A metre or two to the projector’s side of the picture, where the walker sees the near edge closest and the even picture gives them fifty or sixty pixels a degree, following lifts them by up to thirty per cent, across the line an eye can resolve. That stretch is where a following projector earns its keep: a presenter standing beside their own slides on the projector’s side, a visitor stopped at the near end of a long projected wall. The visitor standing in front of the middle of the picture, who is worst served of all, is given almost nothing, and the reason is the same as for the seated room: a projector in the viewer’s eye is drawn correctly from every seat, and the even picture is already the fair one for a seat that sees it squarely.

How far back the walker is #

The gain’s shape holds at every depth: nothing on the far side, rising towards the projector’s side. Its size falls with depth. Two and three metres from the wall the peak is the same, thirty per cent and a half, because at both depths the walker at the end of the walk sees the near edge so much nearer than the far edge that the panel’s limit on the near edge binds; the deeper walk gains it over less of the path, eleven per cent on average against fourteen. Five metres back the walker sees the picture nearly evenly from anywhere on the walk, their worst place moves to the far edge sooner, and the peak gain is seventeen per cent.

This is the earlier essay’s arithmetic carried along a path. A viewer’s worst place is the part of the picture they see most squarely and from nearest, because there a degree covers the least wall. A walker on the projector’s side looks most squarely at the near edge; a walker past the picture’s centre looks most squarely at the far edge. The following layout is only ever as good as the panel lets it be at the place the walker is looking hardest, and the panel’s surplus is all on one side.

A late layout serves somebody who has left #

A projector that follows a viewer has to find them first: a camera, a tracker, a frame of latency in the pipeline. The layout it draws is for where the viewer was, not where they are.

A tenth of a second late, which a tracker running at a common camera rate manages, keeps ninety-five per cent of the gain. Half a second keeps four-fifths walking away from the projector’s side and three-quarters walking towards it. Two seconds keep a third and a seventh. Four seconds late the following layout is worse on average than never following at all: it gives back a third more than it gains walking away, and a tenth walking towards.

The averages hide the part a viewer would notice. From a quarter of a second late, the walk’s worst moment is worse than the even picture: by 1.5 per cent at a quarter, 5.8 per cent at half a second, thirteen at a second. A late layout is the best layout for a place the viewer has left, and every place’s best layout takes samples from where that place does not look to give them where it does. Walking away from the projector’s side, the late layout keeps the near edge rich after the walker has crossed the picture and now looks hardest at the far edge, which the late layout has thinned to pay for it. Walking towards the projector’s side, the late layout is tuned for a seat a little further out, whose worst place is not quite the walker’s.

So the second half of the earlier essay’s question has an answer with a threshold in it. A layout that follows within a few tenths of a second is a clear gain; one that follows a second or more late is not worth having, because the moments it makes worse are the moments the viewer is crossing the picture, which is when they are looking at it.

What the walker’s layout costs a seated room #

A projector that re-lays its picture for one walker re-lays it for everyone. The earlier essay’s room — four rows from two to six and a half metres, five seats across each, centred on the picture — sees the same picture.

While the walker is on the far side the following layout is the even picture and the room loses nothing. As the walker moves to the projector’s side, the layout moves samples to the near edge for them, and the budget takes those samples from the far edge. The room’s worst seat is its front centre, two metres from the wall, which sees the whole picture at nearly one scale and needs the far edge as much as any other part of it. At the projector’s-side end of the walk the room’s worst seat loses twenty-nine per cent of its detail per degree while the walker gains thirty-one.

The two numbers are nearly mirror images, and that is not a coincidence of these seats. A fixed budget spent on one viewer’s worst place is taken from somewhere, and on a turned projector the only place with anything to spare is the near edge, which the walker gets, and the place the near edge is paid for is the far edge, which the room needs. One picture and three people found the same conflict for a curved screen drawn for one seat at the expense of the others; here it is in samples rather than geometry, and it moves.

How much a turn gives a walker #

The surplus the following layout spends is the near edge’s density over the far edge’s, which the turn creates. At five degrees the panel is nearly even and following gains nine per cent at its best and four over the walk. At fifteen, thirty and eleven. Past twenty degrees the gain levels off, thirty-two to thirty-four per cent at its best, because the near edge’s own density stops rising as fast as the far edge’s falls — the same levelling the earlier essay found for a seated room on the projector’s side.

So the following layout is worth most behind a projector turned well away from its wall, which is the projector that most needed correcting in the first place. A projector square to its wall has nothing to give a walker, whatever it knows about them.

What following a viewer can give #

Put together, the measurements answer the question the seated room left. A walker keeps all of the seated room’s projector-side gain, thirty-one per cent at fifteen degrees, while they stand where it can be given, and none of it anywhere else; averaged over a walk across the whole room, eleven per cent; at the walk’s worst moment, standing in front of the picture, three. A layout drawn once for the whole walk is the even picture. A layout that follows within a few tenths of a second keeps most of the gain; from a quarter of a second late it makes the walker’s worst moment worse than the even picture would, and from about four seconds late it loses on average. And every sample it gives the walker it takes from a seated audience’s worst seat, nearly one for one.

The general point is the earlier essays’ carried one step further, and the screen sets the distance is the reminder that where a viewer stands decides more than any layout does: walking from three metres back to five does more for the walk’s worst moment than any following layout can. Keystone correction spends the panel unevenly found the turned projector’s panel rich on one edge and poor on the other; an even spend of the panel is an uneven picture found that the even picture on the wall spends the poor edge’s density everywhere; the seated room found that only an audience on the rich side can be given more. A walker is an audience on the rich side for part of their walk. Knowing where they are lets the projector give it to them then, and only then, and at a price paid by anyone else in the room.

A projector is a camera run backwards began this sequence by turning the camera’s geometry around. The following layout turns one more thing around: a camera that tracks its subject keeps the subject sharp, and a projector that tracks its viewer keeps the viewer’s worst place as sharp as the panel allows — which, behind a turned projector, is a sharpness that depends on which side of the picture the viewer is standing.

What the walk takes for granted #

The walker’s worst place is what matters to them. Detail per degree is measured where each viewer is worst served, as the earlier essay measured it. A walker reading a caption on the far edge cares about the far edge whatever their worst place is, and a layout that follows their gaze rather than their position would be a different instrument.

The budget is fixed at the even picture’s. Every layout spends 57.2 per cent of a frame’s samples. A projector that spent its full panel would have more to give everywhere, and the earlier essays found that at a full frame the layouts converge on the panel’s own limit; following would then gain less.

The tracker knows the walker’s position exactly, only late. A position read with error moves the layout to a seat slightly off the walker’s, which is like a lag without a direction; its cost would follow the lag figure’s early part, since a small error is a small lag.

The room is seen in plan. Every viewer is at the picture’s height and every layout varies across the picture only, as in the earlier essays.

Still open: whether two walkers can share one following layout #

One viewer can be followed. A gallery has several visitors, and a projector that knows where all of them are can draw a layout for the group’s worst place at each moment — the seated room’s layout, recomputed as the group moves.

The measurement that settles what that is worth puts two viewers on the walk at once, walking in the same direction a stated distance apart and in opposite directions crossing in the middle, re-lays the picture each frame for the worse-served of the two, and asks how much of the single walker’s gain each keeps — whether two walkers on the projector’s side share it nearly whole, and whether one walker on the far side, who can be given nothing, takes away everything the other could have had, since the far edge they need is what the near edge’s surplus would be paid for.