A walker charged at its worst place gains where it looks

A rule that weighs two walkers spares the one serving the worse would starve put two visitors in front of a projector turned fifteen degrees to its wall and asked how one layout of its samples should be shared between them. Four rules picked from the same family of layouts: serve the worse-served, maximise the detail in all, maximise the sum of each walker’s share of their own best, or maximise the smaller of those shares. With the walkers two and five metres from the wall, serving the worse-served gave the near walker 11.7 per cent over the even picture and took the far one below it — to minus 84 per cent of its own gain, and below what an eye resolves in a fifth of the frames. Keeping the smaller share largest took nobody below the even picture.

Every one of those scores was taken at the walker’s worst place: the part of the picture they saw least finely, which a projector that tracks bodies and not eyes has to assume they care about. That essay ended on the obvious objection. A visitor walking past a picture looks at one part of it at a time — the part in front of them, or the part a little ahead — and the worst place for where they stand may be a part they are not looking at. Score each walker where they look, and do the rules still disagree, does the transfer survive, and how much of the near walker’s gain went to parts of the picture it was not watching?

Four places a walker might be watching #

The projector, the wall, the budget and the walk are the earlier essays’: the even picture’s budget of panel samples spread over the corrected picture’s columns, two walkers at a metre a second a metre apart, a frame every fifth of a second, and at each frame the four rules choosing from the even picture and a family running from the layout that would follow one walker alone to the layout that would follow the other. Only the scoring changes. A walker’s detail is now read at one of four places: their own worst place, as before; the corrected picture’s middle; the part of the picture straight in front of them, held to the picture’s edge when they have walked past it; and the part a stated angle ahead of straight across, along their walk.

Followed through the walk at the part straight in front of each, the case that motivated the question looks different. The near walker, two metres out, still gains under the worse-served rule — 12 per cent on average, as at its worst place. The far walker, five metres out, which lost 6 per cent on average at its worst place, gains 10 per cent where it looks. It drops below the even picture in 12 per cent of the walk rather than 45. The smaller-share rule gives the two 11 per cent each and takes neither below the even picture at any frame.

The reason is where the worse-served layout puts its samples and where it takes them from. The near walker’s worst place, for most of the walk, is the picture’s near edge, which it sees close and at a steep slant; the layout drawn for it moves samples there and takes them from the far edge. The far walker’s worst place is that far edge — but from five metres the far edge is off to one side, seen at a slant it does not have to look along, and what the far walker looks at is in front of it, where the samples were mostly not taken.

The rules nearly agree where walkers look #

At their worst places the four rules give the worse-served walker between 5 and 12 per cent over the even picture — the spread the earlier essay found and argued about. Straight in front, between 11 and 12 per cent; twenty degrees ahead, 9 per cent under all four. The disagreement between the rules was, in large part, a disagreement about the walkers’ worst places, and walkers do not look at their worst places. Scored where they look, a layout drawn for one walker’s worst place and a layout weighed between the two give both walkers nearly the same.

The middle of the picture is the exception that proves the arrangement. Scored at the picture’s middle, no rule gives either walker more than one per cent over the even picture, and the worse-served rule gives them a fifth of a per cent less. Every layout the projector can afford with the even picture’s budget moves samples from the middle of the picture towards whichever edge someone sees worst, because the middle is where the even picture already spends more than anyone’s worst place needs. A walker who watches the middle gains nothing from being followed, under any rule.

Detail an eye would have seen #

The earlier essay’s sharpest measure was the share of frames in which a rule held a walker below sixty pixels a degree — the line an eye resolves — where the even picture would have left them above it.

With the near walker leading, the worse-served rule takes visible detail in 21 per cent of the frames scored at worst places, 7 per cent scored straight in front, and none scored twenty degrees ahead. With the far walker leading, 31 per cent, 10 per cent — and 31 per cent again twenty degrees ahead, for a reason the last section gives. The smaller-share rule takes visible detail in none of these frames, scored anywhere. The earlier essay’s conclusion survives the change of scoring in the form that matters: the weighing rule never makes anybody worse off than the even picture where they look, and the worse-served rule sometimes does. What changes is how often. Most of the transfer the earlier essay measured fell on parts of the picture nobody was looking at.

Where the gain and the charge land #

The third question asked how much of the near walker’s gain went to parts of the picture it was not looking at, and the arithmetic answers it for both walkers at once.

The near walker’s gain under the worse-served layout is 12 per cent at its worst place and 12 per cent straight in front — the gain was real where it looks, not only where it sees worst — 9 per cent twenty degrees ahead, and nothing at the picture’s middle. The far walker’s charge is the other way round: 6 per cent lost at its worst place, 10 per cent gained straight in front, 8 per cent twenty degrees ahead. Very little of the near walker’s gain went astray. Nearly all of the far walker’s charge did.

That asymmetry has a geometric source. A walker two metres from the wall sees the picture’s near part steeply and close, and the part it sees worst is close to the part in front of it; feeding one feeds the other. A walker five metres back sees the whole picture more evenly, its worst place is the distant far edge, and the part in front of it is somewhere else entirely. A layout drawn for the near walker serves what that walker looks at, and charges the far walker at a place the far walker has little reason to look.

Why the middle gains nothing #

The picture’s middle deserves a sentence of geometry, because it is where most people look at most pictures and it is the one place no layout helps. A projector turned to its wall spends its panel unevenly before any layout is drawn: the corrected picture’s near side is drawn by panel pixels spread over less wall than its far side, so the panel’s own limit — how finely it can draw each column — falls from the near edge to the far. The even picture spends the budget so that every column of the wall gets the same density, which the far edge’s limit caps; everything a layout can then do is move density from columns that have more than some viewer’s worst place needs to columns that have less.

For a walker anywhere in front of the picture, the columns that have more than they need are the middle ones: seen squarely and from nearer than the edges, they give every viewer more pixels a degree than the edges do. So every candidate layout in the family takes from the middle — the even picture’s middle was already the richest part of the picture for everybody — and a walker watching the middle sees the even picture, give or take a per cent. Being followed is a way of improving the parts of a picture a viewer sees worst, and the middle is not one of them for anybody.

The same way, and crossing #

The two other arrangements the earlier essay walked tell the same story more quietly. Walking the same way at one depth, a metre apart, the worse-served rule gives the worse-served walker 9.4 per cent at its worst place and 9.4 straight in front; the weighing rules give 11 to 11.4 per cent straight in front, a little more than they gave at worst places, because a layout weighed between two walkers who want nearly the same thing serves both their fronts well. Crossing, nobody gains under any rule but one, as before: the walkers want opposite edges, the family’s layouts trade one for the other, and the rules that care for both settle on the even picture. The rule that maximises the sum of the gains still serves the worse walker 9 per cent below the even picture straight in front, and still takes visible detail in 16 per cent of the frames — summing the gains favours whoever is cheapest to help, wherever they are scored.

Looking ahead brings the charge back #

Walkers do not look straight across a picture as they pass it; they look ahead, towards the part they are walking towards, and sometimes back.

Walking from the projector’s side to the far side and looking ahead is looking towards the far edge — the edge the worse-served layout starves. Looking straight across, the far walker gains 10 per cent under the worse-served rule; looking 20 degrees ahead, 8; 30 degrees ahead, a loss of 1 per cent; 40 degrees ahead, a loss of 13. Looking behind, towards the near edge the layout feeds, it gains 9 per cent at every angle from ten to thirty degrees back. The smaller-share rule gives the far walker 11 per cent looking straight across, 5 looking 30 degrees ahead, and nothing worse than the even picture at 40.

So the gaze does not make the rules equivalent. It moves the question from where each walker sees worst to where each walker looks, and a walker who looks ahead along a walk towards the starved edge is charged again. The weighing rule is the one that does not need to know.

What a projector can know #

A projector that follows its viewers — a projector is a camera run backwards, and one that watches the room is a camera run both ways — knows where they are from a camera watching the room, and a camera can see a body far more reliably than it can see where the body’s eyes are pointed. That is why the earlier essays scored worst places: a viewer’s worst place follows from where they stand, and a viewer’s point of regard does not. This essay’s numbers say what that ignorance costs, and the answer depends on the rule. A projector using the worse-served rule, scored at worst places, looks as if it is robbing the far walker; scored where the walker looks straight ahead, it is mostly not, and scored where the walker looks well ahead along the walk, it is again. Which of those is true for a given visitor is exactly what the projector cannot see.

A projector using the weighing rule does not need to know. Under the smaller-share rule neither walker loses detail an eye would see where it looks, at any of the places tried, neither is below the even picture there on average, and the worse-served walker gets within a per cent of what the worse-served rule would give it straight ahead. The weighing rule was argued for in the earlier essay as fairer; here it is also the rule whose fairness does not depend on a guess about eyes.

What the measurement leans on #

The gaze is a point. Each walker is scored at one place. An eye takes in a few degrees sharply and the rest of the picture less so; a walker scored over a patch around the point of regard would sit between the scores at the point and at the patch’s worst part.

The layouts are the earlier family’s. The candidates are the even picture and blends of the two single-walker layouts, each drawn for a walker’s worst place. A projector that knew where each walker looked could draw layouts for the points of regard instead. The family here was drawn for the wrong place, and nothing guarantees that its best member is near the best layout for a point of regard.

Both walkers look the same way. Each looks a stated angle ahead along its own walk. Two walkers looking at different things — one at the picture, one at a companion — would be scored at different kinds of place.

The detail an eye resolves is sixty pixels a degree. As in the earlier essays. A pixel is not a point is why a number of pixels a degree, and not a number of pixels, is what a viewer sees.

Where the walkers’ worst places came from #

An even spend of the panel is an uneven picture and keystone correction spends the panel unevenly found where a turned projector’s samples go, and the even picture is fair to a centred room found which arrangement of them serves a seated audience’s worst seat. A walker is followed only on the projector’s side and two walkers share a layout drawn for the worse carried the worst-seat rule onto people who move. Every one of those measured a viewer at their worst place, because a projector that cannot see eyes has to; and every one of them, read here, was measuring something about the picture’s edges. Walkers look at its middle and at what is in front of them, and the picture’s middle is the one place no layout improves.

Still open: whether a projector that knows where each walker looks should draw for it #

Every layout here was drawn for some walker’s worst place and only scored at the place they look. A projector with a camera that follows eyes as well as bodies could draw for the points of regard themselves: move samples to the part of the picture each walker is looking at, from the parts nobody is. Such layouts can only do better at the walkers’ points of regard than the earlier family, since that family is among the layouts they would be chosen from; the question is by how much, and at whose expense.

The measurement that settles what that is worth draws, for each frame, the layout that maximises the detail at a walker’s point of regard within the even picture’s budget and the panel’s limit — and the shared layouts the four rules pick between two such — and asks how much more a walker gains where it looks than under the worst-place layouts; whether the four rules then disagree again, now that the layouts can concentrate rather than tilt; and what a gaze-drawn layout does to the parts of the picture the walkers are not looking at, which are the parts a third visitor, arriving, would see first.