The evenness a curve buys

Three rungs of this row have priced what a curved display costs: it names one seat, the seats it accepts are a few millimetres of room, and an audience of three is served by one of them. A trade needs both sides, and the other side has not been measured.

The argument for curving a screen is always the same sentence: the far edge of a flat panel is further away and more oblique than its middle, and curving it evens that out. Both halves of the sentence are true and measurable, and once measured they turn out not to be one quantity.

Three quantities, one word #

“Even” can mean at least three things about a screen and a seat, and each of them can be computed exactly.

How far the glass is, across the picture — the ratio of the furthest part to the nearest. This is the quantity the marketing sentence names.

How obliquely it is seen — the angle between the reader’s sightline and the glass’s own normal, at its worst point. This is what decides whether a panel’s brightness and colour hold up, since a display’s output falls away off-axis.

How large a pixel is at the eye, across the picture — the angular size of one panel pixel, largest over smallest. This is the one that decides whether detail is delivered evenly, and it is the composition of the other two: a pixel further away subtends less, and a pixel seen obliquely is foreshortened.

A reader would expect these three to agree about which screen is better. They do not, and the disagreement is the essay.

The reading the argument is made in gives up first #

Take the marketing quantity first, because it is the one that fails soonest.

From the middle seat, the curved television’s furthest glass is 1.0097 times its nearest and the flat panel’s is 1.0276. The curve wins, and by nearly three times.

Move along the sofa. At 400 mm off-axis the two are 1.0721 and 1.0735, which is a dead heat, and past about 450 millimetres the flat panel is the more even of the two. It stays that way for the rest of the room.

The reason is easy to see once it is stated. A cylinder is symmetric about its own axis, so it evens out the distance for a reader on that axis and for nobody else; a seat off to one side is nearer to the near edge of a curved panel than to the near edge of a flat one, because the curved panel’s edge is bent toward it. Half a metre is less than the distance between two people on a sofa.

The obliquity reading holds much longer #

The second quantity is kinder to the curve, and it is the one a panel engineer would actually care about.

From the middle seat the curved television’s worst obliquity is 4.7° and the flat panel’s is 13.3°. That is a large win, and it survives a long way: the two cross at 3.6 metres along the sofa, which is past the end of any real sofa and most of the way across a living room.

The mechanism is the same as before with the sign reversed. A cylinder points every part of its glass at its own axis, so an on-axis reader sees the whole screen nearly square-on; a far-off reader sees the far half turned further away than a plane’s far half would be, because the cylinder has already rotated it away from them.

So the curve buys a great deal of obliquity evenness and gives some of it back at the edge of the room.

The reading that matters most does not cross at all #

The third quantity — the angular size of a pixel — is the composition of the other two, and it behaves like neither.

The curved television delivers pixels whose largest is 1.0131 times its smallest from the middle seat, against the flat panel’s 1.0560. Move along the sofa and the curved screen stays ahead at every offset out to four metres, ending at 1.02 against 1.57.

That was not the expected result. The measurement was written to find a crossing — the shape of the first two readings makes one look inevitable — and there is none.

The reason the composition behaves better than either of its parts is worth stating, because it is the mechanism the whole essay turns on. Going off-axis makes a cylinder’s far half more distant and more oblique, and both of those make its pixels smaller at the eye; but they also make its near half closer and more square-on, and both of those make its pixels larger. The two errors are in the same direction as each other on a plane and in opposite directions on a cylinder, so on a cylinder they partly cancel and on a plane they compound.

Both readings from the middle seat are one line each #

The composed quantity is simple enough to write down, and writing it down turns the design question into a division.

A panel pixel of size $p$ at range $r$, seen at incidence $i$, subtends $p\cos i/r$. That is the whole model: distance shrinks it, obliquity foreshortens it, and the pixel-pitch reading is the ratio of that expression’s largest value across the picture to its smallest.

On a flat panel the ray to a point $\alpha$ off the axis has $r = d/\cos\alpha$ and $\cos i = \cos\alpha$, so the expression is $\cos^{2}\alpha/d$ and the ratio is

$$\sec^{2}\alpha_{\max} \;\approx\; 1 + \alpha_{\max}^{2}, \qquad \alpha_{\max} = \arctan\frac{w}{2d}.$$

For the 1.23 m panel at 2.6 m that is $\sec^{2}13.30° = 1.0559$, against the 1.0560 measured.

On a cylinder the eye sits a distance $e = R - d$ from the axis, the normal is radial, and both $\cos i$ and $r$ follow from the same triangle: $\cos i \cdot r^{-1} = (R - e\cos\phi)/(R^{2}+e^{2}-2Re\cos\phi)$. Expanding to second order in the half-arc $\phi = w/2R$,

$$\text{ratio} - 1 \;\approx\; \frac{(R-d)(2R-d)\,w^{2}}{8\,d^{2}R^{2}},$$

which for the same panel at 4000R gives 0.01322 against the 0.0131 measured.

Two readings of that pair, and the second is the one worth having.

The advantage is $2R^{2}/(R-d)(2R-d)$, exactly, independent of the panel’s width — the widths cancel. For this television it is 4.23, which is the “factor of four” quoted at the design seat, and it is a property of the radius and the seat alone.

And it goes to infinity at $R = d$. The deficit carries a factor of $R - d$, so a screen curved at exactly the distance it is watched from puts the eye at the centre of curvature, where every pixel is the same distance away and square-on and the ratio is 1.0000. That is the ideal curve, it is not a compromise or a limit, and it is a single number: curve the screen at the seat.

Real displays are not curved that way and the shortfall is consistent. A 4000R television watched from 2.6 m has $R/d = 1.54$. A 1000R desk monitor read from 650 mm has $R/d = 1.54$. Two products of different sizes from different industries land on the same ratio, and it is half again flatter than the geometry’s answer — which is a manufacturing and shipping constraint rather than an optical one, since a screen curved at the seat has to know the seat, and a 650R monitor is a noticeably strange object on a desk.

It also explains the shape of the sweep in the next section. Every expression above is written for a reader on the axis, where $e$ is a distance and the geometry is one-dimensional; off the axis $e$ becomes a vector, the near half and the far half pick up opposite signs, and the cancellation those opposite signs produce is what makes the curved screen’s worst seat sit in the middle of the range rather than at the end of it.

The worst seat is in the middle of the range #

The cancellation has a consequence that shows up as a shape in the sweep, and it is the one genuinely surprising thing in this essay.

The curved screen’s pixel-pitch spread is not monotone in the seat’s offset. It rises from 1.013 at the middle seat to a worst case of 1.26 at about 1.5 metres, and then falls again, reaching 1.02 at 3.6 metres — as even, from far off to one side, as it is from the design seat.

A flat panel’s rises to 1.59 at 2.4 metres and flattens.

So a reader wanting the most even delivery from a curved television has two good places to sit and one bad one between them, which is not advice anybody would have offered and follows from the geometry rather than from a preference. The mechanism is the same cancellation: at moderate offsets the near half’s gain and the far half’s loss are unequal, and far enough round they balance again.

What the numbers are worth against the other side of the trade #

The three readings above are the whole of what curving a display buys, on the geometry this collection computes. Set them beside what it costs.

At the design seat the curve buys a factor of four in pixel-pitch evenness — 1.3 per cent of variation against 5.6 — and a factor of nearly three in obliquity.

It costs the ability to serve more than one seat at all: twenty-five pixels of departure on a 1.8 m sofa, growing at fourteen pixels a metre, against a flat panel’s exact zero at any width.

Whether that is a good trade is not a question geometry answers, and this essay will not pretend otherwise. What geometry can say is which quantities are being traded, and the answer is unusually clean: the curve buys uniformity for one reader and spends the projection for everybody. A monitor has one reader and should probably be curved. A television has several and the trade is a real one.

Where the evenness argument came from #

The uniformity argument is not a modern invention and it is not really about displays. It is the cone of vision in different clothes.

A picture drawn in perspective and read from its own station point stretches at the edges — that stretch is exact and correct, and a wide field on a small screen measures what happens when the reader is not at the station point. The traditional response was a rule of thumb limiting the field of view to sixty degrees, and this collection’s measurement of that rule is that it is a rule about readers rather than about pictures.

Curving the surface is the other response to the same complaint, and it is the response the curved field has been describing since six flat pictures of everything: a cylindrical picture surface delivers azimuth in proportion to the picture, so a panorama on a cylinder has no edge stretch at all. The catch, which is this row’s whole subject, is that a display is not shown a cylindrical picture. It is shown a rectilinear one computed for a plane.

The dome, where the surface is uneven before anybody sits down #

Everything above is a statement about a seat, and one of the four surfaces this collection carries has an unevenness that has nothing to do with where anybody is.

A cylinder can be unrolled. Cut it along a generator and it lies flat with nothing stretched, so an image laid on it keeps every pixel’s area and shape, and the whole of what goes wrong is the reader’s position. A sphere cannot. Laying an image on a dome stretches it by a factor of 1.18 between the middle and the rim before a reader exists.

So the dome’s answer to all three questions above is contaminated by a fourth term that the other surfaces do not have, and the honest way to report a dome’s evenness is as two numbers rather than one: what the surface did to the picture, and what the seat did.

That distinction is the curved field’s own, and it is the property that decides which floors unroll. It arrives here as the reason a dome is a different kind of object from a curved television rather than a more curved one.

What would actually make it even #

The measurement above prices the compromise. It also says what the uncompromised version would be, and the answer is not a shape at all.

A curved screen delivers uniform pixels to a reader on its axis if the content is drawn for a cylinder. It is not: the content is rectilinear, so the map from the intended picture to the delivered one is not a projection from any seat, and the uniformity measured here is a property of the glass rather than of the picture on it.

Drawing the content for the cylinder would fix that exactly — and would require every source of pictures in the world to know the shape of the screen it will be shown on, which is a fact about the industry and not about geometry. That is the honest end of the argument: the curve is a fix applied at the last surface for a decision made at the first, and the reason it can only be partial is that the two ends of the chain do not talk to each other.

The screen sets the distance makes the same point about the viewing distance, which is the other quantity the chain decides and nobody controls.

Two things this does not measure #

The three quantities above are geometric, and a display has properties that are not.

Brightness and colour off-axis are the reason obliquity is worth measuring at all, and how much either falls with angle is a fact about a panel’s optical stack rather than about a cylinder. The geometry says what angle each part of the glass is seen at; the panel decides what that costs. A screen whose output were perfectly Lambertian would make the second reading above irrelevant and leave the other two untouched.

Reflections are the other half of the room and cut the other way: a curved panel gathers light from a wider arc of the room into the reader’s eye than a flat one does, which is why a curved screen in a bright room is a worse object and not a better one. That is a real effect with a real geometry — the mirror field’s, rather than this one’s — and a curved mirror has no eye is where the collection measures the reason: a curved reflector has no single centre of projection, so what it shows of the room is not a picture from anywhere.

Both are named here rather than measured because the site’s boundary is the geometry of pictures, and a claim about brightness would be borrowing authority from arithmetic that has none in it.

What a reader can check for themselves #

The three quantities are computable from two numbers printed on the box and one measured in the room, which makes this the rare rung a reader can apply without any of the machinery.

The screen’s width and radius give the arc it subtends. The sitting distance gives the rest. From those three, the distance to each part of the glass and the angle it is turned through follow by trigonometry, and the pixel pitch is their product.

Two shortcuts are worth having. A curved screen’s evenness advantage is largest when the sitting distance is close to the radius, because that is where the glass is most nearly equidistant — a television with a four-metre radius watched from two and a half is well short of its own best case, and one watched from four metres would be better served by its curve than by anything else it could do. And the advantage collapses toward nothing as the sitting distance grows past the radius, because a distant reader sees any screen as nearly flat.

That second point is the one worth carrying out of the essay: the curve is bought for a reader at about a radius, and most people sit further away than that. A cinema screen at sixteen metres of radius watched from fourteen is the arrangement closest to its own design intent, which is why the cinema’s numbers are the best in the set on every reading.

The short version #

Curving a display buys evenness, and evenness is three things.

The plain distance from eye to glass is evened out for one seat and made worse for every seat past half a metre. The obliquity is evened out substantially and holds until three and a half metres. The angular size of a pixel — the composition of the two, and the one that decides delivered detail — is evened out from every seat in the room, by a factor of four at the middle one, with a worst case in the middle of the range rather than at its end.

None of the three is the quantity the argument is usually made in, and the one the argument is made in is the one that gives up first.