A stereo picture is drawn for a level head

Two pictures on one screen placed a stereo point where two sightlines cross: a point drawn with a separation $d$ between its left and right images, on a screen $D$ away from eyes $b$ apart, is seen at depth $bD/(b-d)$, and the ceiling on depth is the width of a head. What the two eyes are sent and the distance at which the eyes part then measured the one disagreement between the two eyes that no scene can produce — a vertical one — and found a curved screen producing it out of its own shape.

All three held one thing fixed without saying much about it: the reader’s head was level. The two eyes lay on a horizontal line, parallel to the rows of the picture. A stereo pair is drawn for exactly that arrangement. Its two images of a point sit on the same row and differ only along it, because the cameras that made them, or the renderer that drew them, stood side by side.

A head is not held level. It tilts to read the side of a screen, rests on a hand, leans into a sofa cushion. So the question is what a level-head picture sends to a head that is not level.

The difference on the glass is fixed; the eyes turn under it #

The glass holds two images of each point, a horizontal distance $d$ apart. That is all it holds, and it does not change when the head moves.

What the two eyes receive is those two images seen along two sightlines, and the eyes compare them along the line that joins the eyes — the direction in which a scene’s disparities lie, because a scene seen from two places can only differ in the direction between the two places. Roll the head by $\theta$ and that line turns by $\theta$ against the glass. The glass’s horizontal separation $d$ is now partly along the line joining the eyes, $d\cos\theta$, and partly across it, $d\sin\theta$. The part across it is a vertical disparity, and for a screen $D$ away it is an angle of

$$\frac{d\,\sin\theta}{D}$$

to first order. The first figure computes it without the approximation, from the two actual sightlines, and the first-order form holds to a per cent out to 10° of roll.

The dependence is on the separation drawn on the glass, not on the depth the separation stands for. A point at infinity is drawn with a separation equal to the eyes’ own, 63 mm; a point half-way from the eyes to the glass is drawn with the same 63 mm crossed the other way. At 10° of roll in front of a television 2.60 m away both receive 14.46 arcminutes of vertical disparity. A point ten screen-distances back, drawn 56.7 mm apart, receives 13.02′; one two screen-distances back, drawn 31.5 mm apart, 7.23′.

The glass is the one depth a rolled head receives intact #

A point drawn with no separation at all — a point on the glass — has nothing to turn, and it receives no vertical disparity at any roll. The figure’s control is exactly that: zero at every angle to 30°.

So the arrangement has a single depth that is indifferent to how the head is held, and it is the screen’s own. Everything in front of the glass and everything behind it acquires a vertical disparity in proportion to its distance from that depth, measured in separation on the glass. It is also nearly uniform across the picture. At 10° of roll, content at infinity receives between 14.05′ and 14.46′ over the whole face of a 1230 mm television, so the effect is not something that happens at the edges; it is a property of the pair, delivered everywhere at once.

The search for a point’s second image, which a point is a line over there describes as a search along a line, runs for a level head along the row, because the line joining the two eyes is parallel to the rows. For a rolled head that line is tilted, and the pair on the glass does not lie along it. The second image is not where the eyes’ own geometry says it can be.

What comes out of the screen is the most fragile #

The dependence on the separation drawn has a consequence that runs against the way stereo content is usually made to impress.

Content behind the glass can never be drawn more than the eyes’ own 63 mm apart, because that separation already means infinity. Content in front of the glass has no such bound: the nearer to the eyes a point is meant to appear, the wider its crossed separation, without limit. A point a third of the way out from the glass toward the eyes is drawn with its two images 126 mm apart — twice the separation of anything at infinity — and at 10° of roll in front of the television it receives 28.93 arcminutes of vertical disparity, twice what infinity receives. It reaches fifteen arcminutes at 5.17° of roll. A point a quarter of the way out, drawn 189 mm apart, receives 43.39′ at 10° and reaches the limit at 3.44°.

So the part of a stereo picture that comes out of the screen toward the reader — the part chosen for effect — is the part a tilted head loses first, and there is no roll small enough to be safe for all of it: draw a point close enough to the eyes and even a degree or two sends it past the limit.

How much roll a screen allows #

The size of a vertical disparity a pair of eyes will still fuse is a fact about people rather than about geometry. What the two eyes are sent argued for fifteen arcminutes and said where the figure comes from, and the measurements here use the same number and say so. Its exact value matters less than it seems, because the roll that reaches it enters through a sine.

Setting $d\sin\theta/D$ equal to the limit gives the tolerated roll: its sine is the limit times the screen’s distance over the separation drawn. For content at infinity, where the separation is the full 63 mm, a desk monitor at 650 mm tolerates 2.58°; a television at 2.6 m, 10.4°; a cinema screen at 14 m, 75.8°. Content two screen-distances back, drawn at half the separation, tolerates twice the sine: 5.17° at the monitor, 21.1° at the television, and in the cinema no roll short of lying down reaches the limit.

The cinema’s 75.8° is close to a distance beyond which no roll matters at all. Content drawn at infinity has the widest separation anything behind the glass can have, the eyes’ own 63 mm, and a head rolled a full 90° turns all of it vertical. So the largest vertical disparity content behind the glass can ever send is 63 mm over the screen’s distance, and it falls to fifteen arcminutes at a distance of 63 mm divided by fifteen arcminutes in radians: 14.44 m. A stereo screen further away than that cannot push anything drawn at or behind the glass past the limit, however the head is held. Only what is drawn in front of the glass, with separations wider than the eyes, still can.

That ordering is the one the screen sets the distance would predict and it is not the one screen size suggests. A stereo monitor is the least tolerant of a tilted head because it is the nearest, not because it is the smallest. At arm’s length a tilt of a few degrees — well inside what a head does without its owner noticing — is enough to send content drawn at infinity past the limit.

And the uncertainty in the fusion limit passes straight through. Because the tolerated roll enters through a sine, and a small angle’s sine is nearly the angle, a limit twice as generous allows about twice the roll: thirty arcminutes instead of fifteen moves the television’s crossing for content at infinity from 10.37° to 21.11°. That is the opposite of what the distance at which the eyes part found for a curved screen’s viewing distance, where a fourth-power fall made the crossing distance nearly indifferent to which limit was chosen. Here the roll a reader may take is exactly as uncertain as the fusion limit itself, so the fifteen-arcminute figure is better read as setting the scale of the tolerance than its value.

The depth that is left at a given roll #

Turned round, the same arithmetic says which depths a head held at a given roll can still fuse.

With the head level, no depth has any vertical disparity, and the only ceiling is the one two pictures on one screen found: a separation wider than the eyes asks them to diverge. As the head rolls, a far limit appears as soon as content at infinity crosses the fusion limit — at 10.4° for the television — and it closes in from infinity toward the glass. At 15° the depths that still fuse run from 1.53 m to 8.55 m around a screen at 2.60 m. At 30° they run from 1.91 m to 4.06 m. The band narrows on both sides and is centred, in separation, on the glass.

At the desk monitor the band is already closing at 2.58°, and at 15° only 554 mm to 787 mm is left around a screen at 650 mm: a volume a quarter of a metre deep, where the level head had everything to infinity. This is a comfort budget of the same shape as the vergence one two pictures on one screen set out for the split between where the eyes focus and where they converge, but for a different reason, and it multiplies with it rather than replacing it: content has to stay near the glass for a tilted head, whatever the eyes’ focus is doing.

A roll, in pixels #

Arcminutes are the unit a fusion limit is stated in; pixels are the unit a picture is made and checked in, and the conversion is worth doing because it shows how large the effect is against the errors a stereo pair is normally checked for.

A television 1230 mm wide showing 1920 pixels has a pixel 0.64 mm across, and from 2.6 m that pixel subtends 0.847 arcminutes. So the 14.46′ of vertical disparity a 10° roll gives content at infinity is the angle of 17.1 pixels of vertical misregistration between the left and right pictures. Turning the cameras inwards measured the vertical disparity a toed-in camera pair puts into its own pictures in pixels, up to thirty on a frame four hundred high, and treated it as a defect of the pair. A pair delivered seventeen pixels out of vertical register would be rejected by anyone who checked it. The same disagreement produced by a reader’s head passes every check the pair was ever given, because every such check is made with the pair level.

The two sightlines do not meet #

The vertical disparity has a geometric meaning beyond a number of arcminutes, and it is the reason a rolled head is a different case from a moving one.

With the head level, the sightline from the left eye through the left image and the sightline from the right eye through the right image lie in one plane and cross, and where they cross is the depicted point. Roll the head and they are skew lines: they pass each other without meeting. At 10° of roll, the two sightlines to a point drawn two screen-distances back pass 10.6 mm apart; to a point ten screen-distances back, 50.9 mm apart; to a point half-way out, 5.5 mm. There is no point in the room that both eyes are looking at.

That is the property turning the cameras inwards found in a toed-in camera pair, which puts the same point at different heights in the two pictures: a pair with vertical disparity is a pair of pictures of no scene at all. A level stereo pair read by a rolled head is exactly such a pair, produced by the reader instead of by the cameras.

It is worth setting against what a head that moves sideways receives. The second eye is a shear found that a picture seen from somewhere other than its own centre is a picture of a sheared room — a different room, but a room: every pair of sightlines still meets, and the depicted scene is still a scene. A stereo reader who steps sideways sees the depicted room shear by the same law, the step over the distance to the glass, and the scene survives as a scene. A reader who rolls their head has no such consolation. No room, sheared or otherwise, would send those two pictures to those two eyes.

What a display could do about it #

Three things follow, and the geometry prices the first two.

Keep the content near the glass. Every tolerance above scales with the separation drawn, so a stereo picture whose depth stays within a screen-distance or so of the glass has a tolerance several times that of one using its whole budget out to infinity. That is already how stereo content is composed for comfort; the roll gives a second reason to do it.

Sit further back. The tolerated roll’s sine grows in proportion to the screen’s distance. The same content that fails at 2.58° on a monitor holds to 10.4° on a television and beyond 75° in a cinema, and the size of the screen plays no part: a monitor-sized stereo picture viewed from 2.6 m, with the same separations drawn on it, would tolerate exactly the television’s 10.4°.

Draw for the head that is there. A display that knows the reader’s head roll can draw the pair for eyes on that tilted line, separating the two images along it instead of along the rows. That turns the vertical disparity back into zero at every depth, for one reader, which is the same single-seat answer one picture and three people found a curved screen allows: exact for the head it is drawn for and for nobody else in the room.

What this does not settle #

The limit is stated, not measured. Fifteen arcminutes is the figure the earlier measurement argued for. Vertical fusion limits vary between people and with the size and duration of what is shown, and every crossing roll here moves with the figure chosen, through its sine.

The eyes turn a little against the head. When a head rolls, the eyes counter-rotate slightly in their sockets, which reduces the roll the retinas actually receive. That is a fact about eyes rather than about the picture, it varies, and it was not modelled; the rolls here are head rolls.

A flat screen only. The pair is on a flat screen. A curved screen already sends a level head a small vertical disparity of its own, and how that combines with a roll was not computed.

Still open: a rolled head in front of a curved screen #

A curved screen’s own vertical disparity depends on where a mark is across the screen, and a roll’s depends on the separation drawn; the two are different in pattern, and they need not add.

The question that leaves is how they combine: whether a curved monitor’s vertical disparity and a head’s roll add at one side of the screen and partly cancel at the other, what roll a curved monitor’s reader can take before the worst point on the screen crosses the fusion limit, and whether there is a roll at which a curved screen is kinder to a tilted head than a flat one.