One picture and three people

A television is watched by more than one person. That sentence is doing no work in the geometry of a flat panel, because a flat panel delivers a homography of its picture to every seat in the room and a homography of a picture is a correct picture of a scene. Three people on a sofa are three readers of three slightly different rooms, and nobody minds.

The screen that names the seat shows why a curved panel is a different object: what it delivers is not a projection of anything, and which nothing it is a picture of is a function of the seat. So the audience becomes a real problem with a real price, and the price can be computed.

What a correction is allowed to be #

A display can apply any fixed map to the content before showing it. It can move every pixel anywhere; the only constraint is that the map is the same for everybody, because there is one panel and one picture on it.

That is a strong power and it has one exact limitation, which the field’s earlier rung states and this essay prices. Choose the map that makes seat A see the intended picture exactly — a pre-warp — and ask what seat B is left with. On a flat panel the answer is a homography, because the correction for A is a homography and B’s own view is a homography and the composition of two is a third: the second viewer still has a picture of a scene, and at the arithmetic floor. On a curved panel neither map is a homography, the composition is not one, and the second viewer has a picture of nothing.

The point is not that the correction damages the second viewer. It is that on a curved screen the second viewer never had a picture to damage, and the correction cannot give them one.

Which seat to correct for #

If exactly one seat can be right, the natural question is which. The natural answer — the middle one — turns out to be right, and the interesting part is by how little.

The correction is defined by a virtual seat, which need not be a seat anybody is sitting in: the display warps the content as though for a viewer at some chosen place, and every real viewer then gets whatever that leaves them. Searching over the virtual seat and minimising the worst real seat is a minimax, and it can be run.

On a 1.8 m sofa in front of a curved television at two and a half metres, the minimax leaves the worst seat at 25.51 pixels of departure. Correcting for the middle seat instead leaves 25.53. That is a difference of one part in a thousand, and over sofas from 0.6 m to 3 m the gap never exceeds a quarter of a per cent.

So the honest report is a null result with a number in it: there is nothing to be clever about here. A designer who spends effort choosing the virtual seat is optimising a quantity that has already been optimised by symmetry, and a designer who simply corrects for the middle of the sofa has lost a fortieth of a pixel.

That is worth stating positively rather than as a disappointment. A search that returns the obvious answer is evidence that the obvious answer is the answer, and the collection has enough cases in it — the taught two-point cube reporting exactly 1.000, the by-eye depth method that beats the others — where the obvious answer is wrong, that it is worth checking each time rather than assuming either way.

What the correction is worth at all #

The second question is worth more than the first, and it is the one a manufacturer would rather not have asked. Correcting at all, for anybody: what does it buy?

Uncorrected, the worst seat on that 1.8 m sofa is 27.14 pixels out. Corrected by the best possible single map, it is 25.51. Six per cent.

The reason is visible in the per-seat figure above. The warp makes one seat nearly exact — the middle seat is left with 1.07 pixels, down from about thirteen — and the two ends of the sofa are hardly touched, because the change that would help them is one the middle seat’s correction has already spent. So the correction moves error around the sofa; it does not remove much of it.

The worst seat grows in a straight line #

Sweep the width of the audience and the worst seat’s departure grows almost exactly linearly: about fourteen pixels for every metre of sofa, from zero at one seat to forty-two at three metres.

Two things about that line are worth stating.

It goes through the origin. There is no width at which the worst seat is acceptable and beyond which it is not; the price starts the moment a second person sits down. A tolerance is a horizontal line drawn across the figure, and where it crosses is a matter of choosing the tolerance rather than of finding a threshold in the geometry.

It is straight. The departure is linear in how far a seat is from the one the content was warped for — which is the same linearity the tolerance region reports as a cube law on the volume — so the worst seat is linear in the half-width of the audience, and doubling the sofa doubles the price exactly.

The constant is the screen, and it is not small #

Fourteen pixels per metre is a fact about a curved television at four metres of radius. The same sweep run on the other two curved surfaces this collection carries gives the constant its dependence, and it is the dependence everything else in this row has: the price is inversely proportional to the radius.

A curved desk monitor — a metre of radius, read from 650 mm — costs 126 pixels for every metre of audience, so two people sharing a curved monitor at arm’s length from each other are a worse-served audience than two people at the ends of a three-metre sofa. A cinema screen, at sixteen metres of radius, costs 6.5.

That ordering is the useful one for a reader deciding what to buy, and it is the opposite of the ordering the objects’ sizes suggest. The tightest curve is on the smallest screen, and the tightest curve is the one that most insists on a single reader — which is defensible for a monitor, where there is exactly one, and is the reason a curved monitor is a sensible object and a curved television is an argument.

Where the constant comes from #

Fourteen, a hundred and twenty-six and six and a half are three measurements, and the sentence that follows them — inversely proportional to the radius — accounts for a factor of four of the nine between the first two. The rest is worth recovering, because the missing part is the half that a buyer can act on.

The rung that measures the region gives the cost of a sideways step in closed form. A cylinder of width $w$ and radius $R$ carries a relief of $w^{2}/8R$; a step $e$ across the sight line at reading distance $d$ slides the eye over it, and what survives the best homography, on a picture $N$ pixels wide, is $New/8Rd$. The worst seat on a sofa of width $L$ is half a sofa from the middle, so

$$\text{worst departure} \;\approx\; \frac{N\,w}{16\,R\,d}\,L,$$

linear in $L$ with no constant term — which is the straight line through the origin the sweep reports, derived rather than fitted. Its slope, per metre of audience, comes out at 129 pixels for the desk monitor, 14.8 for the television and 6.4 for the cinema screen, against the 126, 14 and 6.5 measured.

So the constant is $w/Rd$ and not $1/R$. The radius is in it, and so are two lengths that move the other way: the television is 1.8 times wider than the monitor, which hurts it, and is watched from 3.8 times as far, which helps it more. Multiply the three and the television is 8.8 times the kinder object, of which the radius contributes four and the seating contributes the rest.

That matters because of which of the three a reader owns. The width and the radius are bought and cannot be changed; the distance can, and the price falls in proportion to it. Moving a curved monitor from 650 mm to a metre takes its cost from 129 pixels per metre of audience to 84, which is more than any choice of virtual seat in this essay was ever going to buy — the minimax saved a fortieth of a pixel, and pushing the monitor back saves forty-five.

The same expression explains the six per cent. On the 1.8 m sofa the offset term alone is $14.8 \times 0.9 = 13.3$ pixels for a seat at either end, and the two ends are 26.6 pixels apart in what they would each need — a difference between seats, which is precisely the quantity a single map applied to a single panel cannot address. Both the corrected worst seat and the uncorrected one sit within a pixel and a half of that figure. The correction is fighting the term that is common to every seat, and the term that is common to every seat was never the problem.

The control, which is the whole argument #

A flat panel of the same size, an audience three metres wide, and the same minimax: the worst seat comes back at the arithmetic floor.

Not small. Zero, to fourteen digits, at every width tried — because a homography corrected for one seat is a homography from every other, and there is no residual to minimise. The flat panel’s curve in the figure above is the horizontal axis.

That control is what makes the whole essay a statement about curvature rather than about audiences. Without it, “the worst seat grows with the audience” would be an unsurprising remark about people sitting off to one side, true of every display that has ever existed.

Why no fixed map can do better #

The minimax says the best is 25.51 pixels; the argument says no other approach does better either, and the argument is short.

What the display is being asked for is a single map that takes the intended picture to something whose delivery to seat A is a projection and whose delivery to seat B is a projection. Those are two conditions on one map. On a flat screen they are the same condition — both deliveries are homographies of whatever is shown, so making one right makes the other right — and one condition on one map has solutions.

On a curved screen the two deliveries differ by something that is not a projective map, so the two conditions are genuinely different, and a single map cannot satisfy both unless the difference between them is itself absorbed by the freedom left over. It is not: the freedom left over in a fixed map, once seat A is exact, is nothing at all.

This is the same counting that decides what a second view buys a reconstruction and why three parallel views do not fix a solid. Conditions and freedoms, counted; and when the count is short, no cleverness recovers it.

What a display could do instead #

Three routes, and this collection can price the first two and only describe the third.

Track the viewer. If the display knows where one reader is, it can warp for that seat continuously and deliver an exact projection to one person. The measurement that makes it possible is the previous rung: the seat is in the picture, so a camera on the display can find it. This is a real answer for a single viewer and no answer at all for three.

Flatten the glass. A flat panel serves everybody exactly, which is the control above, and everything a curved panel is bought for is what is being given up. What the curve is worth measures that side of the trade.

Or accept it, which is what every curved display does and which is defensible. The 25 pixels of departure on a 1.8 m sofa is a smooth distortion of a picture whose undistorted form nobody in the room has seen, on a scene rather than on a ruler. It is a large number in the geometry and it is not a claim about what anybody notices, which is the boundary this collection keeps and does not cross.

The cinema, where the audience is enormous and the problem is not #

A cinema screen is curved and holds hundreds of people, which looks like a counter-example and is a scaling result.

The allowance scales with the screen: a cinema screen’s acceptable region is about a quarter of a metre across against a curved television’s few centimetres, because the arrangement is measured in units of the screen and a cinema screen is ten times the size. That is a real improvement and it is nothing like enough — an auditorium is twenty metres across, so nearly every seat in it is outside the region by two orders of magnitude.

What actually rescues the cinema is not geometry. It is that the audience has no reference: a scene shown to a viewer who has never seen it undistorted, on a surface whose departure is smooth, is a scene. The same argument works for the sofa, which is why curved televisions sell.

The one case where an audience really is served #

There is an arrangement in which a curved surface serves several viewers exactly, and it is worth naming because it shows what the obstruction actually is.

Put the projector at the viewer’s eye. Then the wall’s shape stops mattering entirely — the picture is painted along the very rays the viewer reads it back along — and a projector in the viewer’s eye measures that this is exact rather than nearly right, on a cylinder, on a dome, and on a plane alike.

That does not serve an audience either: it serves one eye, and it is the same one-seat result wearing different clothes. The obstruction is not the screen’s shape and not the correction’s power. It is that a projection has one centre.

What the pixels are, and what they are not #

Every number here is in pixels of a delivered picture 1,920 across, measured as the largest departure over a grid of forty-nine points after the best homography between the intended picture and the delivered one has been removed.

Three consequences of that definition are worth carrying, because the number is easy to over-read.

It is not a displacement of the content. A reader at the end of a sofa also sees a keystone — the rectangle is a trapezium — and that is a large, obvious change which this measurement scores at exactly zero, because it is a projective one and a projective change of a picture is a picture of a scene.

It is not spread evenly over the picture. Twenty-five pixels is the worst point of a grid of forty-nine, and the grid’s points are not all alike: the departure grows toward the edges, where the glass turns furthest from the reader, and the middle of the picture is the part a homography fits best. A single worst-case number is the right thing to gate a design on and the wrong thing to imagine as a uniform blur.

And it is not a measure of anybody’s discomfort. The tolerance region makes the same distinction from the other side, and it is the boundary this collection draws around itself: what is computed here is the geometry of the picture, and how the picture is read is a fact about seeing.

The short version #

A curved display serves one seat. It can choose which, and the choice is the middle of the sofa to a quarter of a per cent; it can correct, and the correction buys the audience six per cent; and the price of every extra metre of audience is fourteen pixels, in a straight line through the origin.

A flat display serves all of them, exactly, at every width — which is not a virtue anybody advertises, because it has never been a problem anybody had.