The screen is a picture surface too

Everything the screen field has said so far treats a display as a rectangle of a stated width at a stated distance. That is all the arithmetic needed: the chain from sensor to screen cares how wide the picture is and not what shape it is, and the answer — that almost nobody is standing where the picture says to — comes out of two lengths and a division.

A screen that curves breaks that, and it breaks it in a way this site already has a whole field for. A curved display is a picture surface: the pixels are equally spaced along the glass, so the arc length across the screen is the picture’s width, and a viewer on the axis receives azimuth in proportion to the picture coordinate. That is the cylindrical picture surface, in a living room.

From the axis it is exact, and only from there #

Put the eye at the centre of curvature and every screen point is at the same distance and at an azimuth proportional to its position along the glass. That is the cylindrical map’s definition, and the agreement is at the arithmetic floor — not a good approximation, the same function.

Move the eye off that point, along the axis, and it stops. The azimuth is no longer proportional to anything; at six tenths of the radius the departure is degrees rather than parts per million.

So there is one exact arrangement and it has two conditions: the eye is on the axis, and the picture was drawn for a cylinder. Neither is what happens. A television at four metres of radius is watched from two and a half, and it is shown a rectilinear picture, which is what every camera and every renderer produces.

Two families stay straight, and it is the cylinder’s own two #

A drawn line on a curved screen is seen straight when the eye and the line are coplanar. Two families manage it and they are not chosen for convenience.

A horizontal line lies in a horizontal plane, and a viewer at the same height is in that plane. It is seen straight, exactly.

A vertical line is one of the cylinder’s own generators, which is a straight line in space. It is seen straight, exactly.

Everything between them is a curve on the cylinder and is not coplanar with anything.

Those are the same two families the curved field measures on a cylindrical picture, arriving here without being sent for. A curved television is not similar to a cylindrical picture surface in the way a metaphor is similar; it is one, so its straight family is the cylinder’s straight family, and a reader who knows the field already knows which lines will bend.

The maximum is not in the middle either, which is worth a moment. A drawn line at fifteen degrees off horizontal bends more than one at forty-five, because the bend depends on how much of the line’s length is running round the curve and on how far the two ends are from the eye’s own plane, and those two peak at different angles.

The two families are not equally straight #

The two exact families are stated side by side above, and they are not on the same footing. One holds from anywhere on the axis and the other holds at exactly one height, and the asymmetry is the reason curved screens draw the complaint they do.

A vertical is a generator of the cylinder — a straight line in space — so the eye and the line are coplanar wherever the eye is. Seen from any height, from any distance, a drawn vertical is straight.

A horizontal is a circle on the cylinder, at constant height. The eye is coplanar with it only if the eye is at that height. From the centre of curvature at eye level, the line at eye level is seen straight and every other horizontal is a small circle of the visual sphere — which is not a great circle, and therefore not straight.

So the honest statement is: verticals always, and exactly one horizontal — the reader’s own.

The bow on the others has a closed form. A horizontal at height $h$ on a cylinder of radius $R$ sits at a constant elevation $\varepsilon = \arctan(h/R)$, while the great circle through its two ends rises to $\arctan(\tan\varepsilon/\cos\varphi_{\max})$ in the middle. The departure is therefore

$$\varepsilon\,(\sec\varphi_{\max} - 1)$$

for small angles, with $\varphi_{\max}$ the screen’s own half-angle.

Put a monitor in it. A metre of radius and 600 mm of width give $\varphi_{\max} = 17.2°$ and $\sec\varphi_{\max} - 1 = 0.047$. A horizontal 150 mm above eye level sits at $\varepsilon = 8.5°$, so it bows by 0.40° — which at a metre is about 7 millimetres of departure from straight, on a line 600 mm long.

That is not below anybody’s threshold, and it is the effect a viewer describes as the top and bottom edges of the screen looking bowed. It is also exactly what the geometry requires: the edges are the horizontals furthest from eye level, so they carry the largest $\varepsilon$ and therefore the largest bow.

Three consequences follow, and the third is the one that decides whether a curved screen suits a subject.

The bow is proportional to the height above eye level. So it vanishes for the line the viewer is level with and grows linearly outward — which means a screen sat squarely in front of a viewer has its worst bowing at its top and bottom edges and none in its middle, and a screen viewed from below has it all at the top.

And it grows as the square of the screen’s half-angle, through $\sec\varphi_{\max} - 1 \approx \varphi_{\max}^{2}/2$. A screen twice as wide at the same radius bows four times as much. Curvature and width are not independent choices; what matters is how much of a turn the glass covers.

So the arrangement suits pictures with no long horizontals in them. A landscape with a level horizon running the width of the frame is the worst case, and a picture of a room seen obliquely, whose long lines are neither horizontal nor vertical on the screen, is barely affected — because those lines were never in the straight family and the reader has no expectation about them.

Which is the same reading the curved field gives of every picture surface: the surface keeps one family straight, and whether that matters depends entirely on whether the subject has lines in the family it does not keep.

Where to sit, and the answer is not in the room #

For a picture drawn for a cylinder there is an exact seat. For an ordinary picture there is a best one, found by search rather than by formula, and the search is over how far back the viewer sits.

The search is over the seat’s distance and the cost it minimises is the residual against a homography, which is the measurement the next section is about. It is worth noticing that the search returns a minimum rather than a zero, and bestSeatFor asserts exactly that: the best cost has to be positive. A search that reported zero would mean the map had become projective somewhere, and it never does.

The best seat is nearly seven metres from a television a metre and a quarter wide. That is not a recommendation; it is the plot’s way of saying the obvious thing, which is that the further back a viewer sits the less the curve does, and the limit of the best seat is the seat at which the screen might as well be flat.

Three candidate seats are worth naming because a reader will have heard the first two argued for.

The centre of curvature is the one the marketing describes: every part of the screen at the same distance. It is the exact seat for a cylindrical picture and it is four metres back from a domestic television, which is further than most rooms.

The station point is the one the picture asks for: the distance at which the screen subtends the field of view the picture was rendered at. It is the shortest of the three and the worst of the three, because the curve does most at close range.

And the sofa, which is where the viewer is, and is in between.

The difference that is not a matter of degree #

Everything above is a quantity, and a reader could reasonably conclude that a curved screen is a flat screen with a small extra error on it. It is not, and the distinction is this essay’s headline.

Sit off to the side of a flat screen and the picture reaches the eye through a projective transformation — a plane seen from a different centre is a homography of itself, which is the keystone and is exactly the same statement as the picture plane being a choice. Four picture points fix that homography and every other point lands where it predicts, at the arithmetic floor, from every seat in the room including one a metre and a half off to the side where the distortion is violent.

So the off-axis viewer of a flat screen is seeing a correct picture of a differently-shaped scene. That is this site’s recurring answer and it holds here without qualification.

Sit anywhere in front of a curved screen and the map is not a homography. A cylinder is not a plane, and no centre of projection turns one into the other; the fifth point misses by pixels, from every seat, including the axis.

That is not a worse version of the flat case. It is the other kind of thing: the flat screen’s error is a change of scene, and the curved screen’s is not a scene at all.

And no correction fixes it for two seats #

The obvious repair is to pre-warp: apply a fixed map to the content so that one chosen seat sees the intended picture. Projectors do this and panels can.

On a flat screen the correction is a homography, and composing two homographies gives a homography — so correcting for the near seat leaves the far seat seeing a projective transformation of the picture, which is still a correct picture of a scene. The second viewer loses nothing that a viewer of an uncorrected flat screen had.

On a curved screen the correction is not a homography and neither is what the second seat is left with. Correcting for one seat more than doubles the residual at the other.

The finding is not that the correction damages the second viewer, and stating it that way would be a small mistake worth avoiding. assertTheCurvedScreenHasNoSharedAudience checks the third thing explicitly: the curved screen’s uncorrected picture is not a homography for the second seat either. There was nothing there to damage. What the correction does is take the picture from wrong-for-everybody to right-for-one-and-worse-for-the-rest, and calling that a loss requires forgetting what the starting point was.

The measurement, and why it is four points and not a fit #

The instrument used throughout is one this site keeps reaching for and it is worth stating plainly, because a different one would have measured a different thing.

Four correspondences determine a homography — eight degrees of freedom, two equations each — so four picture points and their perceived positions fix the map exactly, with nothing left over. Then the other forty-five points of the grid are asked where they landed. On a flat screen they land where the four predict, at the arithmetic floor. On a curved one they do not.

The alternative is a least-squares fit over all forty-nine, and it is worse for a reason that has nothing to do with accuracy. A fit returns a compromise, and its residual is a mixture of two things — that the map is not a homography, and that the fit split the difference between the points. Four-and-predict separates them: the residual at the four fitted points is zero by construction, and everything else is the map’s own departure.

That is exactly the instrument the shadow on a curved floor uses, and it was written there for the same reason. A reader who has met it once will recognise the shape of the answer: fitted points exact, predicted points out by the surface.

The dome, where the surface does something a cylinder does not #

A cylinder unrolls. Cut it along a generator and lay it flat and nothing stretches — which is the developable surfaces result and is why the pixels can be equally spaced along the glass in the first place, with no distortion introduced before anybody looks.

A sphere does not. An image laid on a dome is stretched before a viewer exists, by an amount that is a property of the surface, and the stretch between the middle and the rim of a hemispherical dome is a fifth again.

So a dome has two problems where a cylinder has one. It is not a homography from any seat, like the cylinder; and its content has been stretched non-uniformly to get onto it, which no cylinder does. Reporting both as “distortion” hides that one of them is a viewing problem and the other is a printing problem.

What the curve is actually for #

None of this says a curved screen is a bad object, and it is worth saying what the geometry does support.

The curve does one thing exactly: at the centre of curvature every part of the screen is the same distance from the eye. For a display whose focus matters — a monitor at arm’s length, where the eye refocuses between the middle and the edge of a wide panel — that is a real ergonomic property and it has nothing to do with projection at all.

And a curved screen delivers more angular width for the same viewing distance, because the edges wrap toward the viewer. On a cinema screen that is most of the argument — and it is the same argument the cylinder makes against the plane about how much of the world a surface can hold before it runs away.

What the geometry does not support is the claim that a curved screen is more correct, or more immersive in any sense that has a number attached. It is correct for a cylindrical picture from the axis, and it is being sent a rectilinear picture and watched from a sofa.

What this does not say #

It does not say the numbers are large. A domestic television’s residual at a normal seat is around ten pixels of a nineteen-hundred-pixel picture, which is under a per cent and is nothing anybody would notice on a moving image.

It does not say what a viewer perceives. This site computes the geometry of pictures and says nothing about seeing, and a reader at the wrong distance in front of any screen usually does not notice and does not care.

It does not treat a screen wrapped far enough round to be a room. The arithmetic here refuses a screen past a quarter turn, because at that point the viewer is inside the surface and the questions change — which is where a set built for one eye takes over.

And it does not say a rectilinear picture is the right thing to send. A picture rendered for the cylinder, watched from the axis, is exact — and the one place that arrangement is routine is a planetarium, which uses a dome and gets the printing problem instead.

A display is a picture surface with a viewer in front of it. Ask which surface, and the question of where to sit stops being a matter of taste and becomes the same question the field has been asking of pictures all along.