A curved screen tilts a stereo pair both ways at once

A stereo picture is drawn for a level head put a stereo pair on a flat screen and rolled the reader’s head. A pair is drawn for level eyes — each point’s two images share a row and differ only along it — so a rolled head compares them along a tilted line, and part of the separation drawn on the glass arrives as a vertical disparity, the one difference two eyes cannot fuse beyond a small limit. On a television 2.6 m away a 10° roll sends 14.46 arcminutes to anything drawn at infinity; at a desk monitor the 15-arcminute limit arrives at 2.58°.

That essay’s screens were flat, and it ended by pointing at the screens that are not. What the two eyes are sent and the distance at which the eyes part had found a curved screen producing a vertical disparity of its own, out of its shape. A curved screen’s disparity depends on where a mark is across the screen; a roll’s depends on the separation drawn. Being different in pattern, the two need not simply add. The question left was how they combine, what roll a curved monitor’s reader can take, and whether there is a roll at which a curved screen is kinder to a tilted head than a flat one.

Answering it turns up something the earlier essays did not measure, because they measured the picture itself rather than stereo content on it.

A curved screen tilts stereo content with the head level #

The earlier measurements sent both eyes the same flat picture on a curved screen and asked how the two views differed. Stereo content is different: each point is drawn twice, its two images a stated distance apart along a row of the displayed picture, and a curved screen lays that row round its arc. The two images of one point are then at two different depths in the room as well as two places across it.

Seen from a seat in the middle, a point above the middle row and to the right has its right image a little nearer the viewer than its left image, because the arc is curving toward the viewer there; a nearer point above the eye line is seen at a slightly larger angle of elevation. So the two eyes receive the point at different heights even though their own heads are level. Above and to the left the arc curves the other way and the sign flips; below the middle row both flip again. The result is a saddle across the screen, zero along the middle row and middle column and largest at the corners.

On the curved desk monitor — 700 mm wide, curved to a 1 m radius, seen from 650 mm — content drawn 63 mm apart, which is a point at infinity, reaches 31.8 arcminutes at the corners with the head perfectly level. That is twice the 15-arcminute limit the earlier essays used, before the head has moved at all. On the curved television, 1.23 m wide on a 4 m radius at 2.6 m, the same content reaches 1.5 arcminutes; its gentler curve and greater distance make the effect twenty times smaller.

The effect grows in proportion to the separation drawn, so it can be turned round into a statement about content.

The curved monitor stays within the limit, with the head level, only for content drawn less than 29.8 mm apart — anything depicted more than 58 cm behind the glass sends its corners past fifteen arcminutes. A flat monitor of the same size sends a level head nothing at any separation, since on a flat screen a point’s two images are at one depth and seen at one elevation by two level eyes. The curve’s own vertical disparity is a property of stereo content on a curved screen, and on a desk monitor it is a large one.

How large the saddle is, and why the monitor’s is so much larger #

The saddle’s size is worth understanding before it is combined with anything, because it explains why one curved screen is past the limit and the other barely notices.

A point’s two images are a drawn separation apart along the arc. Where the arc is turned toward the viewer by an angle, the two images differ in depth by the separation times the sine of that angle, and a depth difference seen at a height above the eye line becomes a difference in elevation: the height times the depth difference, over the square of the distance. So the saddle grows with the separation drawn, with how far toward the edge of the arc the point is — which sets how much the arc is turned there — with how high the point is, and falls as the square of the viewing distance.

The desk monitor has every one of those against it. Its arc is turned 20 degrees at the edges against the television’s 9; its corners are high relative to its viewing distance; and it is seen from 650 mm rather than 2.6 m, which on the square of the distance alone is a factor of sixteen. Together they make its saddle about twenty times the television’s for the same content, 31.8 against 1.5 arcminutes. The evenness a curve buys found the curve paying for itself on a desk monitor in how evenly it spends pixels across the reader’s view; the saddle is the stereo bill for the same curve.

The roll adds one sign everywhere #

A rolled head adds the disparity the earlier essay measured: the separation drawn, turned partly across the line joining the eyes. On a flat screen it is nearly even over the whole face — 14.05 to 14.46 arcminutes on the television at 10° of roll — and of one sign, set by which way the head leans.

A curved screen now carries both. At each place on the picture its own saddle term and the roll’s nearly even term arrive together, and because the saddle changes sign from quadrant to quadrant and the roll’s does not, the two add in one diagonal pair of quadrants and cancel in the other.

Before the curved television, a head rolled 8° receives 11.6 arcminutes in the middle of the picture, where the curve contributes nothing. Toward one pair of opposite corners the curve’s term adds and the corners reach 13.2; toward the other pair it cancels and they fall to 10.5. On the curved monitor the saddle is much larger than the roll’s term at small rolls, and the picture changes character.

At a roll of 3° before the curved monitor the middle of the picture receives 17.4 arcminutes. Along the cancelling diagonal the saddle overtakes the roll near the corners, the disparity passes through zero and the corners end at 12.6 — below the middle, and within the limit. Along the adding diagonal the corners reach 50.3. The same screen is kinder than a flat one in two corners and more than three times harsher in the other two.

No roll makes the whole screen kinder #

That answers the question the earlier essay asked about a kinder roll, and the answer depends on what “kinder” is taken to mean.

Locally, yes. At any roll there is a pair of corners where the curve’s term opposes the roll’s, and there a curved screen sends less vertical disparity than a flat one would. On the curved monitor at 3° of roll those corners are the best-fused places on the screen.

Over the picture, no. A picture fuses only if every part of it does, so what matters is the worst place on the screen, and the worst place on a curved screen is always one of the adding corners.

For content at infinity, the flat monitor and the flat television start at zero and reach fifteen arcminutes at 2.58° and 10.4° of roll. The curved television starts at 1.5 arcminutes and reaches the limit at 9.3°, a degree sooner than the flat one. The curved monitor starts at 31.8 — already past the limit — and only gets worse. At no roll is a curved screen’s worst point better than the flat screen’s, because whichever way the head leans, one pair of corners has the two terms adding.

The one way a curved screen could be kinder over the whole picture would be a roll in the direction that cancels at every place, and the saddle rules that out: its sign flips between quadrants, so no single roll can oppose it everywhere.

That has a consequence for the way a reader actually holds a head. Two pictures on one screen placed a stereo point where the two sightlines cross, and a reader who finds one corner uncomfortable tends to tilt the head toward it. On a flat screen that makes no difference to which corner is worst — the roll’s term is even. On a curved screen, tilting toward a corner changes which pair of corners is adding, so the discomfort follows the head round the screen rather than going away.

What stereo content can do about it #

The earlier essay found the depth budget for a rolled head: content near the glass is robust, content far behind or far in front is fragile. The curve changes the budget’s size and not its shape.

For content drawn 20 mm apart — a point about 30 cm behind the glass on the monitor — the flat monitor tolerates 8.2° of roll and the curved one 2.5°. The flat television tolerates 34.6° and the curved 33.7°, nearly the same, because its curve is gentle. For content at infinity the curved monitor tolerates no roll at all: its own saddle has used the whole limit before the head moves. Past about 30 mm of separation, any roll is too much.

So for a curved desk monitor the practical rule is sharper than for a flat one. Stereo content has to be kept shallow — within a few tens of millimetres of separation — not to make it survive a tilted head, which is what the earlier essay recommended for flat screens, but to make it survive a level one.

Why the earlier measurement did not see it #

It is worth saying why this was not already in the two essays about the curved screen’s vertical disparity, since both measured a vertical difference on the same screens.

They measured what a curved screen does to one picture seen by two eyes, and they took the difference after fitting a common frame to the four corners of the picture — the part of the two views’ disagreement that no single projective correction removes. That part is small, because most of what the curve does to two views of one picture is a smooth distortion the corners absorb. The distance at which the eyes part found the residual falling faster with distance than the raw disagreement, for exactly that reason.

The saddle here is a different quantity. It is the curve acting on a stereo pair’s separation, which is different at every point of the picture according to the depth the point depicts, and no correction fitted to the picture’s corners knows about it. It is also the quantity the rolled-head essay measured on flat screens, raw, because a pair’s vertical disparity is compared point by point by the eyes. On the same footing, a curved monitor’s stereo corners are past the limit before a roll, and the combination with a roll is the sum measured here.

Where this reading stops #

Raw vertical disparity, compared to one limit. As in the essay before it, the vertical disparity is taken point by point and compared with fifteen arcminutes. The visual system does adapt to some kinds of vertical disparity across a whole field — a vertical size difference between the eyes, a small cyclotorsion — and a saddle is close to a pattern the eyes can partly correct by rotating about their lines of sight. How much of the curve’s saddle a real reader absorbs is a question about people, not measured here.

A pair drawn at the eyes’ own separation. Every figure uses eyes 63 mm apart, the value the pair is drawn for. Turning the cameras inwards is about pairs whose own vertical disparity comes from how they were shot; a pair from toed-in cameras would bring a keystone term of its own, which would add to both of these.

One seat in the middle. Every figure places the reader on the screen’s axis at its designed distance. The seats a screen will accept is about readers elsewhere; off the axis, the saddle’s centre moves across the picture and the cancelling and adding corners move with it.

The pair is drawn for a flat screen. Stereo content could be rendered for the curved screen it will be shown on, laying each point’s two images round the arc so that the curve’s own term is removed. The render is distorted on purpose is about pre-warping a picture for its screen; a stereo pair pre-warped for a curved screen would have no saddle, and only the roll’s term would remain.

And a monitor and a television, measured. The two curved screens are the ones the earlier essays on curved screens measured. The saddle grows as the square of how sharply the screen curves relative to the viewing distance, roughly, which is why the monitor’s is twenty times the television’s; a cinema screen, gently curved and far away, sits closer to the television.

The two tilts, combined #

A stereo pair drawn for a flat screen and shown on a curved one receives a vertical disparity from the curve alone, with the head level: a saddle across the picture, zero along the middle row and column, opposite in sign from quadrant to quadrant, and proportional to the separation drawn. On a curved desk monitor, content at infinity reaches 31.8 arcminutes at the corners — twice the fifteen-arcminute limit — and only content drawn within 29.8 mm, depicting less than 58 cm behind the glass, stays within it. On a curved television it is 1.5 arcminutes.

A rolled head adds a disparity of one sign everywhere, so the two add in one pair of opposite corners and cancel in the other: 13.2 against 10.5 arcminutes on the television at 8° of roll, 50.3 against 12.6 on the monitor at 3°. The curved screen is kinder than a flat one in the cancelling corners, and a picture fuses only if its worst corner does. The curved television reaches the limit at 9.3° of roll against 10.4° flat, and at no roll is a curved screen’s worst point better than a flat screen’s.

Still open: whether a pair rendered for its curve survives a roll better than a flat one #

The saddle is the curve acting on a pair drawn for a flat screen. Draw the pair for the curved screen instead — lay each point’s two images at the places on the arc that two level eyes would see it through — and the curve’s own term vanishes by construction, leaving only the roll’s.

That leaves a question with a number in it. A pair rendered for its curve lies on the arc rather than on a plane, so a rolled head’s comparison runs across a surface whose depth varies from middle to edge, and the roll’s own term need not be as even as it is on a flat screen. The measurement renders the pair for the curved monitor, rolls the head, and compares the worst vertical disparity on the screen with the flat monitor’s at the same roll. If the rendered pair on the curved screen matches or beats the flat one, the curve’s cost is entirely a rendering choice; if it does not, some part of it belongs to the surface itself.