Drawn for the cylinder, shown on the cylinder

Two halves of this collection have been talking past each other.

The curved field measures six picture surfaces — plane, cylinder, sphere, two fisheyes, an equirectangular chart — on straightness, conformality and area, and is never shown a room. The screen field measures what a seat receives from a physical display, and is never shown a picture surface. Every result in the second is about the join being wrong: a flat picture shown on a curved panel, which is what every display in the world does.

The arrangement where the two agree has been named in four essays and computed in none. This is it.

The control #

Take a cylindrical screen. Author a picture on the cylindrical picture surface — azimuth across, height divided by horizontal distance down — and lay it on the screen so that the picture’s edge is the screen’s edge. Then sit at the screen’s own centre of curvature and ask what direction each mark arrives from.

Exactly the direction it was drawn to arrive from. The worst departure anywhere on the picture is a tenth of a millionth of an arcminute, which is the arithmetic floor.

Why it is exact #

One line of trigonometry, and it is worth having because “exact” invites the reading that something has been assumed.

A point on a cylindrical screen at arc angle φ from the middle is at distance R from the centre of curvature, whatever φ is — that is what a circle means. So the direction from the centre to that point has azimuth exactly φ, and its elevation is the arc tangent of the point’s height over R.

Azimuth and height-over-horizontal-distance are precisely the cylindrical picture surface’s two coordinates. So the map from screen coordinates to picture coordinates is affine: the horizontal coordinate is the arc angle over the half-arc, the vertical is the height over the radius, and both are linear.

A picture authored on that surface with those two extents therefore delivers, to every mark, the direction it was authored from. Not approximately: the two maps are the same map.

The residual has to be an angle #

Every other measurement in the screen field is a departure from a homography, and that is the right question for a picture meant to be flat. It is the wrong question here.

A matched cylindrical picture is not a homography of anything. Its marks arrive at azimuths proportional to their position, and a flat picture’s marks arrive at tangents proportional to their position — the difference the cylinder, and the price of going all the way round is built on, and those are different functions. So a residual against a homography would report the one exactly correct arrangement on this page as among the worst.

What a viewer actually has is a set of directions. So the residual here is the angle between the direction a mark was drawn to be seen in and the direction it is seen in, in arcminutes — which is the unit an eye’s own resolution is quoted in — the same currency what the two eyes are sent uses for a disparity, and the unit that makes every number below readable.

That change of currency is the reason this essay could not have been written inside the screen field’s existing machinery, and it is why the screen is a picture surface too could name the arrangement and not compute it.

Nobody sits there #

The matched seat is the centre of curvature, and for a curved television that is four metres.

Its maker prints two point six. At that distance the same picture is three hundred and nineteen arcminutes out — five and a third degrees, which is not a subtle discrepancy and is about ten times the width of the moon.

The other screens are the same story with different numbers. A curved monitor is matched at one metre and sold for sitting at 0.65, where the error is six hundred and sixty arcminutes. A cinema screen is matched at sixteen metres and its rows are at fourteen, where a wide field on a small screen has already established that almost nobody is standing where the picture says, where it is a hundred and ninety-five.

So the arrangement is exact and unoccupied. That is a fair description of the whole screen field’s subject, arrived at from the one direction the field had not tried.

The three conditions #

The exactness needs three things to be true at once, and each of them is a number nobody sets.

The seat is the centre of curvature. Four metres for a curved television, and the figure above is what it costs to be elsewhere.

The picture’s horizontal scale is the screen’s own arc. A picture authored at a wider or narrower field is a correct cylindrical picture of a differently scaled world, and its price is exactly proportional to the mismatch.

And the picture’s vertical law is the cylinder’s, which divides the height by the horizontal distance rather than by the depth. An equirectangular picture — the format every 360-degree photograph ships in — is wrong in exactly this way and in no other.

Three conditions, and three prices breaks each of them alone with the other two held, which is what makes them three conditions rather than one restated.

The dome is exact too, on a surface that preserves nothing #

The same computation on a sphere gives a second exact case, and its answer is the one worth carrying.

A point on a spherical screen at arc angles φ and ψ is at distance R from the centre. Its direction from the centre has azimuth φ and elevation ψ — which is equirectangular, exactly, and equirectangular is the surface no picture surface keeps everything describes as preserving nothing at all.

It preserves nothing about shape. It bends straight lines, turns right angles and changes area. And it is precisely right about direction, which is a different question, and the only question a viewer at the centre of a dome is asking.

That is a clean instance of a habit this collection keeps returning to: a surface’s virtues are relative to what is being asked of it, and a surface that is worst on one battery can be exactly right on another.

The rate at which the seat costs #

The sweep is drawn as linear either side of the matched seat, and the expression behind it says why and how much.

An eye at distance $d$ from the glass sits $e = R - d$ from the axis. A mark at arc angle $\varphi$ is then seen at azimuth

$$\varphi' \;=\; \arctan\!\frac{R\sin\varphi}{R\cos\varphi - e},$$

which is $\varphi$ exactly when $e = 0$ — the matched seat — and departs by $(e/R)\sin\varphi$ to first order. So the error is proportional to how far the seat is from the centre of curvature measured in radii, and it grows across the picture as the sine of the arc angle, which is why the worst mark is always the outermost one.

Against the three screens, evaluated at each one’s own half-arc: the television, sold at 2.6 m on a 4 m radius with a half-arc of 8.8°, gives 281 arcminutes; the monitor, 0.65 on 1 m at 20.1°, gives 609; the cinema screen, 14 on 16 at 21.5°, gives 178. The measured figures are 319, 660 and 195 — each about a tenth larger, the excess being the elevation component the azimuth expression leaves out.

The useful form is $e/R$, and it exposes an agreement nobody advertises. The television is sold at 2.6 m on a 4 m radius and the monitor at 0.65 m on a 1 m radius: both are 0.65 of their matched distance, so both carry $e/R = 0.35$ exactly, and what separates their errors is only how wide an arc they subtend — 8.8° against 20.1°, which is the whole of the factor of two between 281 and 609. A screen’s curvature and its selling distance are not independently chosen, and the two consumer products land on the same ratio from opposite ends of the market.

The cinema screen is the exception and it is the better-placed one: 14 m on a 16 m radius is 0.875, giving $e/R = 0.125$, less than half the consumer figure. Its 178 arcminutes come from a wide arc at a small mismatch rather than a narrow arc at a large one — which is the same convergence the chain from sensor to screen finds in cinema and for the same reason, a century of negotiating the seating against the picture with both free.

What a matched picture is not #

Three readings of the result that would be wrong, each easy to reach.

It is not “a curved screen should be fed a curved picture”. That is true and it is nearly vacuous. The content is that the picture, the screen and the seat form a triple, and getting two of the three right buys nothing — a cylindrical picture on a cylindrical screen from the wrong seat is worse at the middle of the picture than a flat picture from the same seat, which the sofa sweep shows directly.

It is not a claim about what looks right. Every number here is an angular error, and an angular error of a few arcminutes is at the edge of what an eye resolves. Whether a picture five degrees out looks wrong depends on what the picture is of — a landscape tolerates it and a grid does not — and this collection has nothing to say about that beyond the geometry.

And it is not a recommendation. A television matched at its centre of curvature would be watched from four metres, which is further than most rooms allow, and the manufacturer’s 2.6 is a decision about furniture rather than an error. What the measurement supplies is the price of that decision, in a unit an eye is quoted in, which nobody had computed.

Why this could not have been measured earlier #

The join needed both halves and it needed one thing neither half had.

The curved field’s machinery maps a direction to a mark on an abstract surface and has no notion of a room. The screen field’s maps a screen parameter to a place and has no notion of a picture surface. Neither is missing anything it needs for its own questions, and the reason the join is unmeasured is that the join needs a third thing: a frame in which a direction and a place can be compared.

That frame is the seat’s own — forward at the middle of the screen, x to the right, y up — and it is trivial to write and load-bearing to get right, for the reason given above. Once it exists the whole computation is four lines, which is a fair summary of why the arrangement went four essays without being run: the missing piece was small and nobody had needed it.

The general habit that follows is worth stating. Two bodies of machinery that each refuse to know about the other’s subject will not produce their joint result by accident, and the discipline that keeps them clean — the surfaces are never shown a scene, the screens are never shown a picture surface — is exactly what makes the join a deliberate act rather than an emergent one.

Where the frame comes from #

A technical point that is load-bearing here and is not elsewhere.

Every other measurement in the screen field is a residual against a homography, and a homography absorbs any choice of coordinates — so the frame those measurements are written in cannot affect them, which is worth saying because a reader is entitled to suspect a canonical frame of doing the work.

An angle between two directions is absorbed by nothing. So the frame here is part of the claim, and it is checked rather than asserted: the seat’s own frame is orthonormal to twelve digits, is a camera frame in the sense the picture surfaces use — x to the viewer’s right, y up, z forward — and has the middle of the screen straight ahead by construction.

Getting that wrong would mirror every picture and leave every residual unchanged, which is exactly the kind of defect no drawn figure can show.

The trap in measuring a small angle #

One arithmetic note, because the first version of this measurement reported a small real error where there was none.

The angle between two nearly equal unit directions was computed as the arc cosine of their dot product. For two directions a hair apart the dot product is one minus half the angle squared, so its last digit is the square’s — and the arc cosine of it returns the angle known only to the square root of the machine’s precision. An exact match came back at five hundredths of a millionth of an arcminute, which looks like a small real error and is not.

It is the arc tangent of the cross product against the dot product now, which is well conditioned near zero. This collection has met the same trap before, on the obliquity of an orthographic projection recovered from its own drawn axes, and the rule it left behind applies unchanged: a quantity that vanishes at the answer must not be read through a square root.

What the seat sweep’s shape says #

The curve of error against seat distance has three features, and each of them is a fact about the arrangement rather than about this screen.

It is linear, not quadratic, about its minimum. A well-set optimum is usually quadratic — a small departure costs the square of itself, so there is a margin. This one is not: a tenth of the radius costs a tenth of the slope, and half a tenth costs half of that. There is no margin at all, which is the unforgiving case and is not what a reader expects.

It is asymmetric. Ten per cent too close costs sixty-six arcminutes and ten per cent too far costs fifty-five, so coming forward is the more expensive mistake by about a fifth. Sitting back is safer than sitting forward, which is the opposite of the advice a viewer usually gets about a large screen.

And it saturates going back. At a hundred radii the error is six hundred arcminutes and rising very slowly toward a bound; at a fifth of a radius it is over a thousand and climbing without limit, because a viewer far enough forward has the screen wrapping around them.

So the arrangement is a minimum with no basin, penalising forward more than back, and bounded in one direction only. Every one of those is measurable and none of them is what a reader would guess.

Whether any real display is matched #

Worth answering, since the numbers above are all misses.

A flat panel is matched trivially: a rectilinear picture on a plane is exact from anywhere on the axis, because the picture surface is the plane and the only thing the distance sets is the field. So every ordinary television and monitor is matched, in this sense, at whatever distance the picture was rendered for — which returns the argument to the screen sets the distance and to the ordinary complaint that the rendered field and the viewed field do not agree.

A planetarium dome comes closest among curved displays. Its seating is inside the sphere and near enough the centre that the equirectangular content it is fed is very nearly the matched surface, which is why domes are fed equirectangular and why that choice looks arbitrary until this computation is done.

A curved television is matched by nobody, and the reason is furniture rather than ignorance: four metres is a larger room than the screen is sold into.

What the row does with this #

Three essays follow and each takes one part of it.

The three conditions have three different prices, and the vertical law’s is the smallest on a television and the largest on a dome, because it is the difference between an angle and its tangent and that difference is third order in the vertical field.

Each screen has exactly one surface, and the ranking of the other five is worth having: on a curved television the runner-up is 0.18 arcminutes behind, which is below what an eye can resolve, and on a curved monitor it is 2.1, which is not.

And matching buys one seat. Matching buys one seat measures what happens along the sofa, and the answer is stronger than the row expected: forty centimetres off the axis the matched picture is already worse than the flat one it replaced.

The short version #

A picture drawn on the cylindrical picture surface, shown on a cylindrical screen and viewed from that screen’s centre of curvature, delivers every mark in exactly the direction it was drawn to be seen in. The same is true of an equirectangular picture on a dome, and a rectilinear one on a flat panel.

Three things have to hold at once — the seat, the horizontal scale and the vertical law — and the first of them fails by a metre and a half on every curved television sold. Where people actually sit, the exactly right picture is five degrees out.