The seats a screen will accept

The screen that names the seat establishes that a curved display’s picture contains the reader’s position, exactly, and that a flat one’s contains nothing about it at all. That settles a question of principle and leaves the question a reader would actually ask.

How far may somebody move?

The answer is not a distance. It is a region, and the region has a shape, a volume, and two laws attached to it.

What is being collected #

A seat is acceptable if the picture it receives is within a stated tolerance of a projection of the intended one — and of a projection, rather than of the intended one, is the whole of the definition.

The reason is the one the screen field’s first rung turns on. A reader who moves gets two different things at once: a picture that is framed differently, which is a projective change and which any observer’s own head produces anyway, and a picture that has stopped being a projection at all. Only the second is a fact about the seat. So every measurement here fits a homography to four corners of the delivered grid and asks the rest where it landed, and the tolerance is on what is left.

Reported in pixels of a delivered picture 1,920 across, because a residual in focal lengths is a number nobody can place — and because pixels are the unit the chain from sensor to screen already carries this site’s viewing distances in.

Sampled rather than cast #

The region is measured on a grid of seats rather than by walking outward along rays from the design seat, and the choice is not arithmetic housekeeping.

A ray-cast measurement assumes the region is star-shaped about its centre — that once the tolerance has been exceeded in some direction it stays exceeded. Over a small tolerance that is true here. Over a large one it is not: the far side of the room re-enters, because a seat at twice the distance receives a picture whose departure from a projection happens to be small again, and a walk outward would report a radius through the gap and call it a region.

The same care is what the anamorph field’s eyeSolid takes, and for the same reason. It is worth stating in a place a reader can find it, because “measure the tolerance in each direction and multiply” is the natural way to write this and it is wrong wherever the answer is interesting.

The numbers #

A curved desk monitor — seven hundred millimetres of glass at a metre of radius, read from 650 mm — accepts about half a cubic centimetre of seats at one pixel. The solid is roughly twenty-two millimetres long, seven across and seven high.

A curved television — 1.23 m of glass at four metres of radius, from two and a half — accepts about a litre: 447 millimetres along the sight line by 66 across.

The two are not in the ratio of their radii, and the reason is worth a section of its own.

Three lengths, not one #

The radius sweep above has a slope of one, so it is tempting to read the allowance as proportional to the radius and to expect the television’s region to be four times the monitor’s. It is nine times. Both facts are correct and the second is the one that says what the sweep is a sweep of.

The previous rung puts the transverse allowance in closed form. The relief a cylinder carries is $w^{2}/8R$; a sideways step $e$ slides the eye across it, and what survives the corner fit, at a tolerance of $\tau$ pixels on a picture $N$ across, is

$$e_{\tau} \;\approx\; \frac{8\,\tau\,R\,d}{N\,w}.$$

Three lengths, and the radius is only one of them. Holding the width and the reading distance fixed — which is exactly what the sweep does, since it re-curves one screen in one room — leaves $e \propto R$ and a slope of one. Changing screens changes all three at once. Monitor to television, $R$ goes up by four, $d$ by 3.85 and $w$ by 1.76, so the allowance goes up by $4 \times 3.85 / 1.76 = 8.8$ — against the 9.4 the search returns from 3.5 mm to 33.

So the honest statement of the scale-freedom is the dimensionless one, and it is stricter than “measured in units of the screen”:

$$\frac{e_{\tau}}{w} \;\approx\; \frac{8\tau}{N}\cdot\frac{R}{w}\cdot\frac{d}{w}.$$

Scale the room and every ratio holds, which is the previous rung’s bit-identical picture. But two screens of different design are not each other scaled: a television is relatively flatter than a monitor — 3.3 widths of radius against 1.4 — and is watched from relatively further away, and both of those loosen it beyond what its size alone would buy. The four-times figure would be right only for a monitor photographically enlarged, which no manufacturer makes.

That also settles which of the three lengths a reader can do anything about. The radius and the width are the manufacturer’s, and the tolerance is the application’s. The distance is the reader’s, and it enters linearly: sitting twice as far back doubles the acceptable region in every direction and multiplies its volume by eight, at the cost of halving the angle the screen subtends. Which is the same trade a wide field on a small screen is about, arriving here as a tolerance rather than as a field of view.

A flat panel is the limit, not the exception #

That proportionality is the reason a flat screen has to be described carefully.

It would be easy to read the previous rung as saying there are two kinds of display — the ones that name a seat and the ones that do not — and that reading is wrong in a way the sweep above makes visible. Flatten the glass and the acceptable region grows in exact proportion to the radius. At a metre it is millimetres, at ten metres it is centimetres, at a hundred it is a metre, and at infinity it is the room. Nothing changes character anywhere along that curve.

So a flat panel’s total refusal to name a seat is a limit rather than a separate case, and the interesting quantity is not whether a display carries the information but at what scale. Which is a much more ordinary kind of statement, and it is the one a reader can act on: a tighter curve is a display that is more particular about where its reader is, in millimetres, and the constant of proportionality is its radius.

Ten times the tolerance is a thousand times the room #

The second law is about the tolerance rather than the screen, and it is the anamorph field’s law verbatim.

Every direction’s allowance is linear in the distance moved — the residual grows in proportion to the step, in every transverse direction and along the sight line alike — so three linear directions give a volume that goes as the cube. Measured over a factor of sixteen in tolerance, from a quarter of a pixel to eight, each doubling multiplies the volume by close to eight.

The room the eye may stand in reports exactly this for a pavement painting: thirty-one cubic centimetres at ten millimetres of tolerance, and a thousandfold for a tenfold loosening. The objects have nothing in common — a painting on a pavement and a television in a living room — and the exponent is the same, because the exponent is counting directions rather than describing a surface.

That is worth being careful about in one respect. A cube law is a statement about a product of three allowances, and it is only informative if the three allowances are genuinely independent. They are here, because the region’s cross-section is convex and its axes are the sight line and the two directions across it. A region shaped like a plate, with one direction unbounded, would show a square law and would be reported as a different animal.

The shape is a spindle, and it was expected to be a slab #

The prediction written into the first draft of this measurement was that a vertical cylinder would be forgiving of height.

The reasoning was geometric and sounded right: a cylinder’s generators are vertical, so moving the eye up and down does not change which part of the glass is nearer or further across the picture, and the region should therefore run away vertically — a slab rather than a solid, bounded in two directions and open in the third.

It is not a slab. The region is a spindle pointing along the sight line, and its two transverse half-widths are equal to twelve digits.

The mechanism the wrong prediction missed is that a vertical step does change the picture: it slides the eye against the screen’s relief exactly as a sideways step does, because the relief is a variation in depth across the picture and a sideways step and an upward step both move the eye across it. What the vertical generators buy is that the horizontal lines stay straight — which is a statement about a family of lines rather than about the map — and a map can fail to be a homography while keeping a family straight. The point to stand at is the collection’s first version of this distinction: a picture has one correct station and a family of things that stay true away from it.

The sight line is the direction that behaves differently, and it is cheaper rather than free: about a third of the cost per millimetre, because moving toward the screen scales the relief instead of sliding against it. Where the anamorph still works derives the same ordering for a pavement painting from the central collineation a wrong eye composes with, and finds the same two equal transverse directions and the same cheap one along the sight.

What a person actually does #

The measurement is now in a position to say something that reads as a criticism of curved displays and is not.

At one pixel, a curved desk monitor’s acceptable region is about seven millimetres across. A reader breathes, leans on an elbow, turns to a colleague, and reaches for a cup. Every one of those is several times the whole region.

So the picture on a curved monitor is, for practically the whole of its life, not a projection of anything. That is a strong statement in the geometry and it is very close to no statement at all about seeing, and the difference between those two is worth being explicit about, because the collection’s own habit is to compute the geometry and stop.

Two things make the geometry’s verdict harmless. The departure is a smooth distortion of a picture whose content the reader has never seen undistorted, so there is no reference to compare against; and the pictures displays carry are of scenes rather than of rulers, so a scene consistent with a slightly different room is a perfectly satisfactory scene. Standing in the wrong place makes the same point about ordinary pictures, where the wrong distance is the rule rather than the exception and nobody minds.

What the region is good for is not “how far may a reader move before it looks wrong”. It is how far may a reader move before a measurement made through the screen stops being trustworthy — and that question has customers: a camera watching a display, a calibration target shown on one, a photograph of a screen used as evidence. It also has one further consumer inside this row, because a reader has two eyes and they are at two seats a fixed distance apart, which what the two eyes are sent measures.

The tolerance is on the part that cannot be corrected #

One more distinction decides what the numbers mean, and it is the one that separates this measurement from a specification a manufacturer might print.

The residual here is what survives after the best homography has been removed. A reader at the wrong seat also receives a keystone — the picture is a trapezium rather than a rectangle — and that is a large, obvious, immediately visible change which this measurement scores at exactly zero.

Scoring it at zero is right for the question being asked, because a keystone is what every reader of every picture in every gallery has always had, and it is a correct picture of a tilted scene. It would be wrong for the question “does this look right”, which is a question about a person.

So the region is not a comfort zone. It is the set of seats from which the display is still doing the one thing a display is supposed to do — showing a projection of a scene — and it is small.

Where the region is not #

Two places, and both are the kind of thing a measurement of a region has to check rather than assume.

Inside the glass. A seat nearer the screen than the screen’s own bulge has part of the picture behind it, and the map is not merely large — it does not exist. Those seats are excluded rather than scored, which is the only honest thing to do with them, and a measurement that clamped them to a large value would have reported a region with a wall on one side.

Far off to the side. From a seat well round the curve, a corner of the screen falls behind the eye. Again the picture does not exist, and again the seat is outside the region rather than badly scored.

Both refusals come from the same place — the projection asking whether a point is in front of the eye — and it is the refusal every camera in this collection makes.

Against the four surfaces #

The four curved surfaces this collection carries give four regions of very different sizes, and the ordering is not the ordering of their curvatures.

A dome is the tightest, and it is tightest for a reason that is not about the seat at all: a sphere is not developable, so the picture is stretched on the surface before anybody sits down, and the residual has a term in it that no seat removes. The cylinder unrolls and the sphere does not is where that distinction is made, and it is the one place in this row where the surface rather than the seat is the subject.

A cinema screen is the loosest, and its allowance is a quarter of a metre sideways — which is why an audience of hundreds is a solved problem in a cinema and an audience of three is not solved in a living room, as one picture and three people prices out.

Two readers, two regions #

One consequence follows immediately and is worth stating because it is the bridge to the next rung.

The region is a region of seats for one design, and a display can be pre-warped for only one seat at a time. So two readers are inside the same region only if they are within one region’s width of each other — a few millimetres at a desk, a few centimetres in a living room — and two people never are.

That is the whole of the audience problem in one sentence, and it is why what one correction leaves three people is a separate measurement rather than a corollary: the region says two readers cannot share, and the audience measurement says what the one who is not served is left with.

The short version #

A curved display accepts a spindle of seats, a few millimetres across on a desk and a few centimetres in a living room, pointing along the sight line.

The spindle is proportional to the screen’s radius in every direction, so a flat panel accepts the room; its volume goes as the cube of the tolerance, so a tenfold loosening is a thousandfold region; and its two transverse directions are equal to twelve digits, which was predicted to be false and is the only part of the measurement that changed anybody’s mind.