Matching buys one seat

A reader who has followed this row this far is entitled to expect that feeding a curved screen its own surface fixes the audience problem. It does not, and it does not merely fail to.

The measurement #

Take a curved television. Feed it the picture drawn for its own surface, and separately feed it an ordinary flat picture. Sit at the matched distance and move along the sofa, and measure both.

On the axis the matched picture is exact and the flat one is 5.3 arcminutes out. At two hundred millimetres across the matched picture is 4.2 and the flat one is 7.2, so matching is still ahead.

At four hundred millimetres the matched picture is 10.8 and the flat one is 7.1. They have crossed.

Four hundred millimetres is well inside a sofa. Past it the matched picture is the worse of the two, and it stays worse for the rest of the sweep: at 1.5 metres across it is 80.8 against 75.7.

Why the better picture loses #

The shapes of the two curves are the whole explanation and neither of them is surprising once seen.

The flat picture is mediocre everywhere. It is 5.3 arcminutes out on the axis and 75.7 at a metre and a half, which is a factor of fourteen over the whole sofa. It was never right anywhere, so it has nothing to lose.

The matched picture is perfect at a point. It is zero on the axis and 80.8 at a metre and a half, which is a factor of infinity. It falls away from its minimum at a rate the flat picture’s shallow curve does not have to match.

So the two curves start far apart and converge, and because one of them starts at zero and the other does not, they cross. The crossing point is where the matched picture’s steeper rise has eaten its head start, and on this screen that is forty centimetres.

The same shape as the pre-warping result #

This collection has reached this conclusion before, from a completely different direction, and the agreement is worth naming because it turns two results into one.

One picture and three people applies a correction — a pre-warp that makes the picture exact for one chosen seat — and finds that the second seat is left worse than it was: correcting a curved screen for one viewer does not merely fail to help another, it takes away the one thing that other viewer had.

Here no correction is applied at all. The picture is simply drawn on the screen’s own surface, which is the arrangement a display engineer would call correct rather than corrected, and the second seat is left worse in exactly the same way.

So the effect is a property of the screen rather than of the fix. A curved screen has one seat — which the screen that names the seat recovers from the picture alone — and gives it to whichever picture is exact there; whatever is exact at one point is steeper elsewhere than something that was exact nowhere.

Whether a compromise is available #

The natural response is to author for the middle of the sofa rather than for the axis, and the row’s own machinery says what that buys.

The seats a screen will accept measures the solid of seats within a stated tolerance and finds that it is small and that widening it costs accuracy at every seat. The render is distorted on purpose is where the same trade is made deliberately in a rendering pipeline. The matched picture is the extreme member of that family — the picture whose solid is a single point and whose accuracy there is exact — and the flat picture is another member with a wide flat solid and no exact seat anywhere.

Everything between them is available and the trade is monotone. Nothing in the family is exact anywhere and good across the sofa, and the reason is not a limitation of the search: a curved screen delivers different angular relations to different seats, so a picture cannot satisfy two of them.

That is the same statement the curved screen has no shared audience makes, and this essay’s contribution is the number where the trade turns: on a curved television the matched picture stops being the better choice at forty centimetres, which is one seat over.

The two curves are one curve and a constant #

There is a number hiding in the sweep, and it is the same number twice.

The flat picture is 5.3 arcminutes out on the axis, where the matched picture is exact. At a metre and a half across the sofa the matched picture is 80.8 and the flat one is 75.7 — the matched picture is 5.1 arcminutes worse. The head start and the eventual penalty are the same quantity, to within four per cent, and it is not a coincidence.

The two pictures differ by a fixed map: the difference between drawing on a cylinder and drawing on the plane it approximates, which does not depend on where anybody is sitting. Call that difference $D$, about five arcminutes on this screen. A seat off the axis adds its own departure $S(x)$, which does not depend on which picture is being shown. What a viewer receives is the two together — and because they are angular displacements rather than magnitudes, they can add or cancel.

On the axis $S = 0$, so the flat picture carries $D$ alone and the matched picture carries nothing. Far off the axis $S$ is large and points the other way, so the flat picture carries $S - D$ and the matched picture carries $S$. The flat picture is winning by exactly the amount it was losing by, and in between there is a crossing, which is what the forty centimetres is.

So matching does not buy a better picture in general; it removes a constant. Removing a constant is worth everything where the rest of the error is nothing and worth less than nothing where the rest of the error is large and happens to point the other way. That reading also says what the crossing distance depends on — the ratio of $D$ to the rate $S$ grows at, both of which the row’s closed form supplies — so a flatter screen, whose $D$ is smaller and whose $S$ grows more slowly, keeps its crossing at a comparable fraction of the sofa rather than at a comparable number of millimetres.

What matching does buy #

Two things, and both are real.

A reference. An exactly correct arrangement is what everything else is measured against, and before this row the screen field had no such thing — every residual was a departure from a homography, which is a departure from a picture that is itself wrong on a curved screen. Having a zero makes the other numbers mean something.

And the middle of the room. Within about a quarter of a metre of the axis at the matched distance, the matched picture is better than a flat one by a factor of several. If there is one viewer and they sit where the screen says, matching is the right answer and it is exactly right.

What it does not buy is a wider room, and the phrase is worth keeping because it is what the row set out to test. Matching the surface is not the fix for the audience problem; it is the best possible answer to a different question, which is what one viewer at one place should be sent.

The monitor is the interesting case #

A curved monitor has one viewer, sitting on the axis, at a distance the manufacturer states — which is the arrangement matching is exactly right for.

Except that the stated distance is 0.65 metres and the matched seat is one metre, so the viewer is at 65 per cent of the matched distance where the matched picture is 320 arcminutes out. The one display in the world with a single viewer on the axis is a display whose viewer is in the wrong place.

That is not a criticism of the design. A metre is further than a desk allows, and the radius is chosen for how the screen looks rather than for where its centre of curvature lands. What it means is that the arrangement matching is right for does not occur, even in the one place it should.

The two curves’ arithmetic #

The shapes are worth deriving, because “one starts at zero and is therefore steeper” is a description rather than an explanation.

A seat displaced across the axis by a receives the screen’s marks at azimuths that are no longer proportional to their arc. To first order in a, the azimuth of the mark at arc φ is shifted by (a/R)·cos φ — a shift that is not constant across the picture and therefore not absorbable by turning the head. So the residual after the best rigid alignment grows linearly in a, with a coefficient set by how much cos φ varies over the screen’s arc.

The flat picture’s residual has that same linear term, plus a constant: the amount by which a flat picture is wrong on a curved screen from any seat at all. So the two curves are a line through the origin and a line through 5.3, with nearly the same slope, and two lines with nearly the same slope and different intercepts do not cross.

They do cross, so the slopes are not the same, and the measurement says which is steeper: the matched picture’s. The reason is that the flat picture’s own error partly cancels the seat’s — a flat picture is wrong in the direction of the tangent law, and an off-axis seat shifts the delivered azimuths toward the tangent law on one side of the picture. The cancellation is partial and it is enough to give the flat curve the shallower slope.

That is an unlovely mechanism and it is the honest one. The matched picture is not beaten by anything elegant; it is beaten by a wrong picture whose wrongness happens to point the same way as the seat’s.

What a second matched picture would need #

Since one picture cannot serve two seats, the obvious question is whether two pictures could — one per viewer, somehow separated.

They could, and the machinery for it is already in this field. Two pictures on one screen is a stereo display, which sends two different pictures to two places by polarisation or by shutter, and the same machinery pointed at two seats rather than two eyes would give each viewer their own matched picture.

What stops it is not geometry. A screen that could send different content to different seats would solve the audience problem completely, and such screens exist as lenticular and light-field displays; what they cost is resolution, divided by the number of views, and that division is why they are not televisions.

So the audience problem on a curved screen has a known solution and its price is stated in the wrong currency for this row: it is a resolution budget rather than an angular error. That is worth naming because it stops the row’s conclusion sounding like an impossibility when it is a trade.

What the numbers are not sensitive to #

Three things that could have been objections and are not, checked rather than argued.

The picture’s aspect. The residual is dominated by the horizontal law on a cylindrical screen, and the sweep along the sofa is horizontal, so a taller or shorter picture moves the numbers by under a per cent.

The grid’s density. The residual is a worst case over a thirteen-by-thirteen grid, and the error is smooth in both directions, so a finer grid moves the crossing by less than the last digit quoted.

And the choice of the matched distance for the sweep. The sofa sweep is taken at the matched distance, which is where the matched picture is best; taking it at the sold distance moves both curves up by hundreds of arcminutes and removes the crossing entirely, because at that distance the matched picture’s head start is already gone.

That third one is the interesting non-sensitivity, and it sharpens the conclusion rather than softening it. The crossing exists only in the one arrangement where matching is worth anything at all, and at every real sitting distance the matched picture is simply worse than the flat one everywhere.

Reading the crossing correctly #

One caution, because the crossing is the essay’s headline and it is easy to over-read.

The crossing is a comparison of two worst-case angular errors over the picture, and at four hundred millimetres both are around ten arcminutes. Ten arcminutes is a sixth of a degree — visible, but not a picture anybody would call broken.

So the honest statement is that past forty centimetres the matched picture is the worse of two acceptable pictures, and it does not become the worse of two unacceptable ones until much further out, where both are bad. The crossing is a real feature of the curves and it is not the point at which anything fails.

What makes it worth reporting is the direction. A reader would predict that the better picture stays better, and it does not; the reason it does not is structural rather than incidental, and it applies to any arrangement where one option is exact at a point.

What a display engineer should take from it #

Three statements, ordered by how much effort they are worth.

Fix the distance first. The seat’s term dominates everything else by an order of magnitude on every curved screen measured here, and it is the one term nothing in the signal can change. A curved television watched from its own centre of curvature is a good arrangement; one watched from two-thirds of it is not, whatever it is fed.

Feed the arc’s field, not the chord’s. Content authored for a flat panel of the same width uses a slightly narrower field than the arc subtends, and the mismatch is worth tens of arcminutes. It is a number in a pipeline and it is free to get right.

And do not bother with the surface. On a curved television the whole surface question is worth 1.79 arcminutes between the best choice and the worst, and 0.18 between the best and the second — which is below what an eye resolves. A curved monitor and a cinema screen are worth a little more, and a dome is worth all of it.

The ordering is the point and it inverts the usual attention. The surface is the interesting question and the seating is the one that matters, which is a fair summary of the whole screen field.

The general shape #

Worth stating outside the screen field, because it is not a fact about displays.

An estimator that is exact at a point is steeper about that point than one that is uniformly mediocre. There is nothing special about screens in that; it is a statement about how a function that touches zero behaves compared with one that does not, and it holds wherever an arrangement is optimised for one condition and then run at another.

This collection has the same shape in three other places. A camera calibrated at one distance and used at another, which how well the floor has to be known prices. A shadow construction exact on a plane and run on a floor that is not one, which the floor that is not a plane measures. A stereo rig converged for one depth and used across a range, which turning the cameras inwards measures. In every case the exactly-right answer degrades faster than the never-right one, and in every case the choice depends on how tightly the condition is actually held.

Which is to say the finding here is not that matching is a mistake. It is that exactness at a point is a design decision with a cost, and the cost is measurable, and on a sofa it is paid one seat over.

The short version #

Feeding a curved screen its own picture surface makes it exact on the axis and worse than a flat picture at forty centimetres across — inside the sofa, on a screen sold for three people.

That is the same conclusion this field reached about pre-warping, arrived at with no correction applied, so it is a property of the curved screen rather than of any fix. What matching buys is a reference and the middle of the room; what it cannot buy is more of the room, because a curved screen sends different angular relations to different seats and one picture cannot satisfy two of them.