What the two eyes are sent

Everything the screen field has said about a reader’s seat applies twice, because a reader has two eyes and they are in different places.

Sixty-three millimetres apart, which on the scale of the acceptable region is a long way: a curved desk monitor’s region is about seven millimetres across, so the two eyes of a single reader are at seats nine acceptable regions apart. They are not receiving the same picture, and the difference is computable.

The pair a flat panel sends #

Start where the answer is exactly nothing.

A flat screen delivers a homography of the intended picture to each eye — that is the field’s founding result — so the map from what the left eye receives to what the right eye receives is a homography composed with the inverse of a homography, which is a homography. Fit one to four corners of the grid and every other point lands on its partner at the arithmetic floor.

That is worth pausing on, because it is the reason binocular vision does not complain about pictures. Two views related by a homography are two views of a plane, and a plane is exactly what is there. The reader’s stereopsis reports a flat surface at the screen’s distance with a picture painted on it, and it is right.

The pair a curved panel sends #

Curve the glass and the map between the two eyes’ pictures stops being a homography, because neither of the two maps composing it is one and the failures do not cancel.

The residual is measured the way everything in this row is measured — fit the homography on four corners, ask the rest of the grid — and it is reported in arcminutes at the eye, because that is the unit a comparison with a human limit has to be made in and a pixel count would hide the comparison.

Two numbers come out and they differ by a factor of seventy.

Horizontally, a curved desk monitor’s residual is 34 arcminutes; a curved television’s is 1.5; a cinema screen’s is 1.3.

Vertically, the same three are 0.50, 0.0025 and 0.0007 arcminutes.

What the horizontal part is #

The larger of the two is not an error, and calling it one would be the essay’s easiest mistake.

A horizontal difference between what two level eyes receive is a disparity, and a disparity is depth. Two pictures on one screen derives the relation: a point whose two sightlines cross is placed at $b \cdot D / (b - d)$, where $b$ is the separation of the reader’s eyes and $d$ the disparity on the screen. So the horizontal residual is a claim about where something is.

The claim is easy to test, because this collection insists on two routes wherever there are two, and the second route here costs nothing. Take the residual disparity, put it through the stereo field’s own depth relation, and compare the answer with the sag of the glass — how far the middle of the screen stands back from its edges, computed from the radius and the width and nothing else.

They agree. On a curved television the eyes report about 49 millimetres of depth and the glass sags 47. On a monitor, 74 against 61. On a cinema screen, 1,322 against 1,112.

So the horizontal residual is not a mystery quantity and not a defect: it is the shape of the screen, seen. Binocular vision is doing what it is for — measuring the surface in front of the reader — and it gets the answer right to a fifth on every screen tried. A reader of a curved display can see, binocularly, that the display is curved.

That single sentence explains a good deal about why curved screens are unobjectionable in a way an anamorph is not. An anamorph works by concealing its surface: the reader is meant to take marks on a pavement for an object in the air, and stereopsis is exactly the sense that gives the trick away, which is why an anamorph has one eye and why photographs of pavement paintings are more convincing than standing in front of them. A curved television conceals nothing. It shows a picture on a surface the reader can see is curved, and every reader knows they are looking at a screen.

What the vertical part is #

The vertical residual is the interesting one, and it is seventy times smaller.

A vertical difference between two level eyes’ pictures is the one difference that no arrangement of a plane can produce, and it is the quantity turning the cameras inwards uses to condemn toed-in stereo rigs: a pair with vertical disparity is a pair of pictures of no scene at all. Here it does have an explanation — the surface is curved, and a curved surface seen by two eyes produces vertical disparity honestly, the way every real object in a room does — but it is the component the visual system is most particular about, so its size is what decides whether the geometry has any perceptual consequence.

It is a fraction of an arcminute. On a curved television it is two and a half thousandths of one.

Every stated fusion limit is one to two orders of magnitude above that. The number is an argument here rather than a constant, because it is a fact about people rather than about projection and this collection does not assert facts about people — so it is drawn as a line the reader can move, and the finding is the same wherever the line goes within the plausible range: the effect exists, it is a fact about the glass, and it is far under the limit.

Two laws, and why they differ by seventy #

The gap between the two residuals is not an accident of the arrangement. Sweep the reader’s own eye separation from thirty millimetres to a hundred and the two quantities come back with different exponents: the horizontal difference is linear in the separation and the vertical one is its square, both to two decimal places.

That is the explanation of the factor of seventy, and it is a better explanation than a ratio, because it says what would change it. The horizontal part is a disparity — the first-order effect of moving an eye sideways across a surface with relief — and first-order effects are proportional to the step. The vertical part is what is left after the first order has been accounted for, and second-order terms go as the square of a small number, which is a very much smaller number.

It also settles a question a reader might reasonably ask, which is whether the effect is worse for somebody with a wider head. It is, in both components; but the vertical one grows faster, so a reader with eyes eighty millimetres apart rather than sixty-three has 1.27 times the horizontal residual and 1.61 times the vertical. Both remain far under any limit, and neither ordering changes.

Where the residual lives on the picture is the other half of the same question. The vertical residual’s worst point is not at a corner: on both the monitor and the television it is at the middle of the top and bottom edges, where the glass is turning away from the reader in the direction the two eyes are not separated in. The horizontal residual’s is at the sides. Neither is where a reader looks.

The horizontal number is the row’s own expression again #

The agreement between the two routes is stronger than “they agree to a fifth”, because both of them reduce to arithmetic that can be written out.

The sag is $w^{2}/8R$ — 47.3 mm for the television, 61.3 for the monitor, 1,125 for the cinema screen, which is the second column exactly. And a depth offset $s$ behind a screen at $D$ puts a disparity $b\,s/D$ on the glass, subtending $b\,s/D^{2}$ at the eye. Composing them,

$$\text{horizontal residual} \;\approx\; \frac{b\,w^{2}}{8\,R\,D^{2}},$$

which gives 1.51 arcminutes for the television, 31 for the monitor and 1.24 for the cinema screen, against 1.5, 34 and 1.3 measured. It is the same $bw/8Rd$ the whole row is built on, with the reader’s interocular distance as the baseline and one more factor of $1/D$ because the answer is wanted as an angle rather than as a length on the glass.

Which is why the horizontal component is linear in the eye separation: $b$ appears once, and it appears because a disparity is what a baseline buys. The vertical component has no $b$ to first order at all — two level eyes separated horizontally cannot disagree about height on any surface whose relief is a function of horizontal position alone — so it is a second-order term and carries $b^{2}$, which is the pair of slopes the log sweep measures and the factor of seventy those slopes produce.

The expression also says what the dome’s 64 arcminutes is. A four-metre dome read from its centre has a relief comparable with its own radius rather than with $w^{2}/8R$, so the small-sag expansion above does not apply to it and the residual is not a hundredth of a screen’s but forty times one — which is the sense in which the dome is not a curved screen with a tighter radius but a different object.

A null result, priced #

A null result is worth reporting only if it is accompanied by the range over which it holds, which is the argument what a null result is worth in decades makes at length: an effect that is absent is not a finding until somebody says how absent, and over what.

Swept across radii from half a metre to sixteen, the vertical residual never reaches a fifteen-arcminute limit. Tightening the limit to two arcminutes — beyond any figure the literature offers for vertical fusion — puts the crossing at a radius of 285 millimetres, which is a screen bent almost into a half-cylinder around the reader’s head and is not a product.

So the honest statement is not “curved screens are fine”. It is: on this quantity, over the whole range of radii anybody builds, the effect is one to three orders of magnitude below the smallest limit worth arguing about, and it would take a screen four times more curved than the tightest monitor to bring it into contention.

That is a much more useful sentence than a reassurance, because it says what would have to change.

The dome, which is the exception again #

One surface in this collection breaks the pattern, and it is the same one that breaks the others.

A dome’s horizontal residual is 64 arcminutes, and at a four-metre sitting distance that is 75 millimetres of disparity on the surface. The reader’s eyes are 63 millimetres apart.

Two pictures on one screen establishes what happens then, and it establishes it as a property of the reader rather than of the display: the disparity that puts a point at infinity is exactly the separation of the eyes, and beyond that the eyes are being asked to diverge, which they cannot do. So a dome’s own shape, read binocularly at its own sitting distance, is past the ceiling.

The reading is not that domes are unwatchable — a dome’s audience sits much further from most of the surface than the nominal radius, and the residual is a worst case over the whole field rather than a value at the middle. The reading is that a dome is the only surface here whose shape, not its content, reaches a limit that is a fact about human eyes, and it does so on a quantity that has nothing to do with the picture being shown.

What a reader’s own head is worth as a ruler #

One more consequence follows from the linearity, and it is the inverse of everything above.

The horizontal residual is proportional to the separation of the reader’s eyes and depends otherwise on the screen. Turn that round: a reader who knows their screen — its width and its radius, both printed on the box — can read their own interocular distance off the disparity, because there is exactly one unknown left in a relation that is linear in it.

That is not a practical instrument; a ruler is easier, and the measurement needs the reader to judge a disparity of a millimetre on glass. It is a good illustration of what the field’s habit of counting unknowns is for. The arrangement has three quantities in it — the screen’s shape, the reader’s seat, the reader’s eye separation — and the observations available are the picture each eye receives. Two eyes give one difference; the difference names the surface if the separation is known and the separation if the surface is known; and no single observation gives both, which is the same shape as what one picture of a plane determines.

What this does to stereo content #

The row’s practical consequence is for displays showing stereo material, and it is an addition rather than an interference.

Stereoscopic content arrives as a pair with a designed disparity field, which the display shows on its glass. The glass then contributes its own residual disparity — 1.5 arcminutes on a television — and the two add. So the depicted depth of every point in a stereo scene shown on a curved screen is displaced by the screen’s own sag, about 49 millimetres at a television’s radius, in the direction that makes the scene shallower at the edges.

Against a depth budget measured in metres that is nothing, and it is worth having the number rather than the impression. It also has a shape: the contribution is largest at the edges of the picture, where the glass is furthest from the chord, and vanishes in the middle. A stereo scene on a curved screen is therefore slightly bowed, and the bow is the screen.

Why this is a different measurement from the seat #

Two rungs of this row have now measured a departure from a homography, and it is worth saying plainly what makes them different questions.

The seat is recovered from the difference between the picture that arrived and the picture that was intended. That needs the content: the reader has to know what was supposed to be shown.

The two eyes’ residual needs no such thing. It is the difference between two simultaneous observations of the same content, so whatever was shown cancels, and what is left is a property of the surface and of the reader’s own separation. That is why it measures the glass rather than the seat, and it is the same division of labour a stereo pair always has: two views give shape and no size, and the size here is supplied by the sixty-three millimetres between the reader’s eyes.

The seat where it would matter #

Everything above is measured at each screen’s own sitting distance, and the one arrangement that would bring the effect into contention is worth naming rather than leaving to be inferred.

The residual grows as the reader approaches the glass, because the relief the two eyes are separated against is a larger fraction of the viewing distance. A curved monitor read from 300 millimetres rather than 650 — which is closer than anybody sits and is where a reader leaning in to inspect something would be — is the direction in which the numbers move, and it moves both components together.

It is still not enough to reach a fusion limit, for the same reason the radius sweep is not: the vertical component is second order in a quantity that stays small. What would reach it is a screen whose radius is comparable to the sitting distance, which is a display wrapped around the reader’s head — and at that point the object is a headset rather than a screen, and its own geometry is a different subject with a different failure.

The short version #

A curved screen sends a reader’s two eyes a pair that no plane could have produced. The part of the difference that carries depth is not an error at all — it is the shape of the glass, measured correctly, to within a fifth of the sag the radius says it has. The part that no scene could produce is a fraction of an arcminute, one to three orders of magnitude under any limit worth quoting, and it takes a radius no manufacturer builds to bring it into contention.

The one surface where the reader’s own eyes reach a hard limit on the shape of the display is the dome, and what it reaches is the divergence ceiling — which is a fact about the width of a head and not about the picture.