Two walkers share a layout drawn for the worse of them

A walker is followed only on the projector’s side put one viewer in front of a projector turned fifteen degrees to its wall. The keystone correction leaves the projector’s panel rich in samples at the picture’s near edge and thin at its far edge, and every layout must spend the same budget of samples. A layout redrawn every tenth of a second for where the walker stands gives them up to thirty-one per cent more detail per degree at their worst place than the even picture does — all of it on the projector’s side, where the walker looks hardest at the near edge the panel can feed. Past the picture’s centre their worst place becomes the far edge, which the even picture already spends everything on, and following gains nothing.

That essay ended on the obvious next case. 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 question was how much of the single walker’s gain each keeps when there are two — whether two walkers on the projector’s side share it nearly whole, and whether one on the far side, who can be given nothing, takes away everything the other could have had.

Both, with a qualification on the second that turns out to be the interesting part.

A layout for the worse-served #

The projector, the wall and the budget are the earlier essay’s. Two viewers walk three metres from the wall at a metre a second. At every tenth of a second the projector draws one layout for both: the layout that gives the worse-served of the two the most detail per degree it can at that viewer’s own worst place, or the even picture where no layout does better than it for that viewer. That is the rule the even picture is fair to a centred room used for a seated audience, applied to two seats that move. For each walker the result is compared with two things: the even picture, and what a layout following that walker alone would have given them.

Walking the same way a metre apart, from the projector’s side towards the far side, the two share well while both are on the projector’s side. For the first second the second walker has not started and both stand near the projector’s-side end, where each alone would gain thirty-one per cent and the pair layout gives both thirty-one. As they move on, the one in front is always nearer the picture’s centre, where the even picture serves a viewer worst — straight in front of the picture a viewer sees its middle squarely and from nearest, and gets only forty pixels a degree there. So the leader is the worse-served of the two, the shared layout is drawn for the leader, and the leader keeps 99 per cent of what following it alone would give. The trailer, a metre nearer the projector’s side and wanting a layout that spends more on the near edge, is given the leader’s: it keeps 77 per cent of its own gain, 9.2 per cent on average over the walk where alone it would have had 13.

The further apart, the less the second keeps #

The cost to the walker who gives way grows with the distance between them.

Half a metre apart the trailer keeps 86 per cent; a metre, 77; two metres, 63; three, 50; four, 44. The leader keeps all of its gain at every gap. Close together the two want nearly the same layout, because their worst places are nearly the same part of the picture; four metres apart they want very different layouts, and the pair layout is the leader’s. Walking the other way — from the far side towards the projector’s side — the roles swap: the trailer is then nearer the centre and worse-served, and the leader gives way.

There is an asymmetry in that, worth stating as a rule. A shared following layout serves the walker nearest the picture’s centre and charges the one further out to the projector’s side. The walker further out had the most to gain and had the least need: the even picture already gives a viewer at the projector’s-side end seventy pixels a degree, more than an eye resolves, which a pixel is not a point puts at about sixty. Serving the worse-served first takes from the viewer whose loss is least visible.

Crossing, nobody gains #

Two walkers crossing in front of the picture — one from each end — are the case the earlier essay’s question singled out.

Followed alone, each walker gains up to thirty-one per cent while on the projector’s side and ten and a half per cent averaged over the walk. Sharing, each gains three-tenths of a per cent on average and keeps two per cent of its own gain. For the first and last stretches of the walk one walker is out on the projector’s side and the other out on the far side, and the walker on the far side is the worse-served: its worst place is the far edge, which the even picture spends everything on and the panel can give no more. A layout that helped the walker on the projector’s side would take samples from the far edge to pay for it, and the far-side walker would lose. So the shared layout is the even picture, exactly, until the two are both near the centre, where neither has much to gain.

Who vetoes whom #

That last answer suggests the far-side walker always vetoes, and the earlier essay’s question expected as much. The map of every pair of positions says it is more particular than that.

Along the diagonal the two stand together and the gain is one walker’s: thirty-one per cent at the projector’s-side end, falling through twenty-four and sixteen to nothing past the picture’s centre. The whole quarter of the map where both stand on the projector’s side shades evenly with it: two viewers there share a layout that helps the worse of them by nearly as much as either would be helped alone.

The other quarters are not uniformly blank. Where one walker stands on the projector’s side and the other on the far side, there are pockets of gain — up to twenty-seven per cent — and they sit where the far-side walker is far out and the other is nearer the centre. The even picture’s detail at a viewer’s worst place depends on how far they stand from the centre, not which side: seventy pixels a degree 3.6 metres out on either side, forty in front. A viewer far out on the far side is therefore better served by the even picture than one near the centre on the projector’s side, and it has detail to spare: the shared layout can take some from the far edge, lowering the far-side walker towards the other, and give it to the near edge for the walker who needs it. The far-side walker vetoes only while it is the worse-served of the two. What decides it is not the side but the ranking.

At different depths, fairness is a transfer #

Every walker so far was three metres from the wall. Two people rarely walk at the same distance.

With the leader two metres from the wall and the trailer five, the shared layout keeps all of the leader’s gain and gives the trailer minus eighty-nine per cent of its own — that is, it serves the trailer worse than the even picture would. Swapped, five metres and two, the leader loses seventy-eight per cent of its gain while the trailer keeps eighty-seven.

The reason is the same ranking, made larger by depth. A viewer five metres back sees the picture nearly evenly and is served well by the even picture everywhere on the walk; a viewer two metres from the wall sees the picture’s near part much more squarely and much closer, and is the worse-served almost throughout. The layout drawn for the worse of the two moves samples to where the near viewer looks hardest and takes them from where the far viewer does, and the far viewer, who started above the near one, ends below where the even picture would have left them. Serving the worse-served is not free for the others; with two viewers at different depths it is a visible transfer from one to the other.

Why the worse-served is the one nearer the middle #

The ranking that decides every result here comes from two facts the earlier essays measured separately, and it is worth putting them side by side.

The first is about the panel. Keystone correction spends the panel unevenly: a projector turned to its wall throws more of its panel’s pixels onto each millimetre of the near edge than of the far edge, and the even picture — the one that looks uniform on the wall — has to throw away the near edge’s surplus to match the far edge, which an even spend of the panel is an uneven picture priced. That surplus is the only thing any following layout has to give. It can go to the near edge, where it came from, or nowhere; and a layout that puts more on the near edge must put less on the far edge to stay within the budget.

The second is about the viewer. Detail per degree is samples per millimetre of wall times millimetres of wall per degree of the viewer’s sight, and a viewer sees the fewest millimetres per degree where the wall is nearest and most square to them. The screen sets the distance is the general form of it: how much a picture can show an eye depends on where the eye is. A viewer straight in front of the picture’s middle is nearest the wall there and sees it squarely, so a degree of their sight covers the least wall and carries the fewest samples: forty pixels a degree under the even picture, against seventy for a viewer 3.6 metres out to either side, who sees the whole picture obliquely and from further away.

Put together, these fix the ranking. The viewer nearest the centre is the worse-served, and can be helped only if their worst place is on the near half of the picture — that is, if they stand on the projector’s side of the centre — because only the near edge has samples to spare. Any other viewer can be charged for that help only if they have detail to spare at the far edge, which a viewer far out to the far side does and one near the centre does not. Every number in this essay is one of those two facts arriving at a particular pair of positions.

What a gallery’s projector should do with two visitors #

For a projector that can see who is in front of it, the measurements reduce to a short policy. With two visitors both on the projector’s side, follow the one nearer the centre: the other loses some of a gain it could barely see. With one visitor on each side, check which is worse-served; if it is the one on the far side, show the even picture, and if it is the one on the projector’s side, a following layout helps it at the far-side visitor’s expense, which is fair by the rule and visible only if the far-side visitor was near the threshold of what an eye resolves. With visitors at different distances from the wall, expect the nearer one to be served and the further one charged, sometimes below the even picture; a projector that cannot accept that should not follow at all.

The policy has one more condition that the single walker did not have. A layout is only as good as the projector’s knowledge of where the worse-served viewer is, and with two viewers that viewer changes: as they walk, the worse-served can switch from one to the other, and a layout drawn for the wrong one at the moment of the switch is the lagged layout’s problem twice over. A projector is a camera run backwards, and a camera that follows two subjects has to decide which to keep in focus; this one has to decide which to keep in detail, many times a second, and every decision it gets wrong it gets wrong for both.

What a shared layout can promise #

Put together, a following layout drawn for the worse-served of two walkers keeps nearly the whole gain for the one nearer the picture’s centre and a share for the other that falls from eighty-six per cent half a metre apart to forty-four at four. Walkers crossing keep nothing, because whenever they are on opposite sides some way out the one on the far side is the worse-served and nothing can be given to it. And the veto belongs to whichever walker is worse-served, not to whichever is on the far side.

One picture and three people found the conflict for a curved screen drawn for one seat at the expense of the others; the earlier essay found it for a single walker and a seated room. Here it is between two walkers, and it has a cleaner shape than either, because a rule — serve the worse-served — has been chosen. The rule is fair in the sense the seated room used, and the measurement shows what that fairness spends: the better-served walker’s gain, and at different depths more than its gain. A projector that wanted to give every walker something would need a different rule, one that weighs the walkers’ gains rather than their minimum, and would have to decide how much of the worse-served walker’s detail it is willing to give up for it.

What the walkers take for granted #

The projector knows where both walkers are, at once and without lag. The earlier essay found that a single walker’s layout lost a quarter of its gain at half a second of lag and turned harmful at a few seconds. Two walkers bring two trackers’ worth of lag, and the layout drawn for the worse-served is wrong for both when it is late.

Each walker’s worst place is what matters to them. The rule serves each viewer’s least-detailed part of the picture. A viewer looking at the middle of the picture, not its worst place, cares about a different number, and a following layout that knew where each was looking — not only where they stood — would be drawn differently.

Two walkers, not a crowd. A layout for the worse-served of a crowd is the layout for whoever is nearest the centre or nearest the wall, and every other walker is given that one’s layout. The more walkers, the more likely one of them is the worse-served on the far side, and the more often the layout is simply the even picture.

The projector is turned fifteen degrees. The earlier essay found the single walker’s gain levelling off past twenty degrees of turn and nearly vanishing at five. The pair’s shares are shares of that gain, and scale with it.

Still open: whether a layout that weighs the walkers’ gains does better than one that serves the worse #

The rule here gives the worse-served walker its best and lets the other take what falls out. A different rule would weigh the walkers: maximise the sum of their gains, or the sum of their gains over what each would get alone, or the smaller of their two shares of their own best — so that a walker far out on the projector’s side, who stands to gain most and needs it least, is neither starved nor favoured.

The measurement that settles what such a rule is worth draws, for the same pairs of walkers, the layout that maximises each of those three alternatives at every frame, and compares them with the rule here on three counts: the worse-served walker’s detail per degree, which the rule here maximises by construction and any other rule must give some of away; the better-served walker’s share of its own gain, which the rule here can leave below nothing; and how often either walker is pushed below sixty pixels a degree, the line a pixel is not a point draws for what an eye resolves — since a rule that gives up a little detail the worse-served cannot see in order to keep detail the other can see would be the better rule for the people in the room, whatever it does to the minimum.