Drawing for the curve removes the curve, not the nearness

A curved screen tilts a stereo pair both ways at once took a stereo pair drawn in the ordinary way — for a flat screen and two eyes level with each other — and showed it on a curved desk monitor, 700 mm wide on a one-metre radius, seen from 650 mm. With the head held level the curve alone gave the pair a vertical disparity, opposite in sign on either side of the middle, reaching 31.8 arcminutes at the corners for anything drawn at infinity. Rolling the head added the disparity a stereo picture is drawn for a level head had measured on flat screens, of one sign everywhere, so the two added in one pair of corners and cancelled in the other.

The essay ended by pointing out that the saddle was the curve acting on a pair drawn for a flat screen. Draw the pair for the curved screen instead — put each point’s two images where two level eyes’ sightlines to it actually cross the arc — and the curve’s own term should vanish by construction, leaving only the roll’s. The open question was whether the roll’s term on a curved surface is as even as it is on a flat one. If the pair drawn for the curve matches the flat screen at every roll, the curve’s whole cost is a rendering choice; if not, some part of it belongs to the surface itself.

Some part does, and it is a part with a simple name.

Drawn for the curve, the level head sees nothing #

The construction is direct. A point to be shown lies along a line from the middle of the viewer’s eyes through some place on the screen, at some depth. Its left image is where the line from the left eye to the point meets the screen’s arc; its right image, where the line from the right eye meets it. For a point at infinity both lines run parallel. Seen by the two level eyes the pair was drawn for, the two sightlines meet exactly at the point, and the vertical disparity is zero everywhere on the screen — the check every figure below makes before it measures anything else.

The hero figure puts the three cases side by side as the head rolls from level to ten degrees. The flat-drawn pair on the curve begins at 31.8′ with the head level and climbs to 90.6′ at ten degrees. The pair drawn for the curve begins at nothing, which is the construction working. At ten degrees it has reached 62.9′.

The comparison that answers the question is the third line: the same content, drawn for a flat screen of the same width and shown on one. At ten degrees it reaches 57.9′. The pair drawn for the curve is 8.6 per cent worse at every roll. The curve’s own term is gone completely, and something the flat screen does not have is left.

In the terms the earlier essay used: the 15-arcminute fusion limit, which a flat-drawn pair on the curve exceeds before the head moves at all, arrives at 2.58° of roll on the flat screen and at 2.38° on the curve drawn for it. Drawing for the curve buys back almost everything; it does not quite buy back the flat screen.

The excess sits at the sides #

Where on the screen does the extra disparity come from? The figure below maps the difference, place by place, at five degrees of roll.

At the middle of the picture the two screens agree to a fiftieth of an arcminute. The excess grows toward the left and right edges and reaches 2.52′ at mid-height on either side. Every cell with an excess visible at this scale shows the curve giving more; none shows it giving less. For content at infinity the curved screen is nowhere kinder than the flat one, and it is harshest exactly where the arc departs furthest from the flat.

The saddle was antisymmetric — one sign on one side of the middle, the opposite on the other. This excess is symmetric, the same on both sides and of the roll’s own sign everywhere. It is not a residue of the saddle that the rendering failed to remove. It is a different term, and the shape of the map — zero in the middle, largest where the arc has come forward — points at what it is.

A fixed separation seen from nearer #

The earlier measurement on flat screens stated the law for a rolled head: the vertical disparity is the image separation times the sine of the roll, over the viewing distance. For content at infinity the separation is the distance between the eyes, 63 mm. The figure below tests the law point by point across the middle of both screens.

Plotted against how far ahead of the eyes each point’s two images sit — measured straight ahead, not along the sightline — every point on both screens falls on one curve: the eyes’ 63 mm, times the sine of the roll, over that depth. The agreement is 0.19 per cent. Every point of the flat screen sits 650 mm ahead, so every point gets the same 29.0′. The curve’s middle also sits 650 mm ahead and gets the same. Its edges sit only 598 mm ahead, because the arc curls toward the viewer, and they get more.

That is the whole of the surface’s share. The curve does not change the law; it changes the depth the law is evaluated at, and a fixed separation seen from nearer is a larger angle. The 8.6 per cent is the ratio of 650 to 598, a little diluted because the worst point is not quite at the edge. The rendering choice removed the curve’s shape from the vertical disparity. The curve’s nearness is a property of where the glass is, and no drawing can move the glass.

The same law explains why the saddle existed in the first place. A flat-drawn pair on a curve puts a point’s two images at slightly different depths ahead of the eyes, because they land on different parts of the arc, so even a level head sees them at different angles. Rendering for the curve moves the images so the level eyes’ sightlines meet; it cannot also make them the same depth ahead of a rolled pair of eyes, and it does not need to — the roll’s term depends on the images’ depth, not on the difference in their depths.

Sitting back shrinks both #

The law has a second consequence that a viewer controls without any help from the rendering. The roll’s term is inversely proportional to the depth ahead of the eyes, so every stereo screen punishes a rolled head less from further back; and the curve’s excess is the ratio of the middle’s depth to the edges’, which approaches one as both grow. Sitting back helps twice.

The numbers, for content at infinity and a head rolled five degrees, pairs drawn for their own screens: at the monitor’s design distance of 650 mm, 31.5′ on the curve against 29.0′ flat, an excess of 8.6 per cent. At 800 mm, 25.2′ against 23.6′, 6.9 per cent. At a metre, 19.9′ against 18.9′, 5.4 per cent. At 1.3 m, 15.1′ against 14.5′, 4.1 per cent. The arc brings its edges the same 52 mm nearer at every distance, and 52 mm is a smaller share of a longer way. The distance at which the eyes part found the curve’s other binocular effects falling steeply with distance too, and this one joins them, more gently: as one over the distance, where they fell as its cube and fourth power.

That is also a reminder of what a curved monitor is for. Its radius is chosen to wrap a wide picture round a viewer sitting close, and the price of sitting close is paid in every figure here. A viewer who pushes the chair back gains tolerance to a roll on any screen, and loses much of the reason for the curve at the same moment.

Near content prefers the curve #

For content at infinity the images are as far apart as the eyes, and the law makes the curve worse. Content drawn nearer has images closer together — content drawn exactly on the glass has both images at one place and no disparity to turn — and there the nearness helps.

For content drawn at 400 mm, well in front of the glass, the curve gives 22.26′ where the flat screen gives 23.93′. For content at 650 mm — the depth of the glass’s middle — the curve gives 2.32′ and the flat screen 4.14′. Beyond that the order reverses: at two metres the curve gives 20.83′ against 19.60′, and at ten metres 29.39′ against 27.15′.

The reason is the same nearness read the other way. A point’s two images are separated by an amount set by how far the content lies from the glass along the point’s line of sight. Content at 650 mm lies on the glass at the middle of both screens. Toward the sides, the flat screen’s glass recedes along the line of sight, so content at 650 mm is in front of it and has separated images; the curve’s glass stays closer to 650 mm along every line of sight, so the content is nearer to lying on it everywhere. For content behind the glass the curve’s nearness increases the separation, and the curve is worse; for content in front, it decreases it, and the curve is better.

So “is the curve kinder to a rolled head?” has no single answer. It depends on where the content is drawn. A monitor used for a stereo view of something at arm’s length — a molecule, a part on a workbench — is better curved; one used for a landscape is better flat.

The radius that suits the content #

The monitor’s one-metre radius is a compromise between two extremes: the flat screen, with an infinite radius, and a screen curved round the viewer’s eyes, with a radius equal to the viewing distance.

A screen curved round the eyes — a radius of 650 mm, its axis through the viewer’s head — is, for content drawn at 650 mm, the surface that content lies on. Along the middle row a point’s two images then coincide on the glass, and no roll turns anything vertical: the figure measures nothing at all there. Away from the middle row there is 1.17′ at worst, because a cylinder curves only across and the depth of 650 mm lies on a sphere. The flat screen gives 4.14′ for the same content. For content at infinity the order reverses: 33.02′ on the eye-centred cylinder against 29.04′ flat, and the monitor’s metre radius falls between at 31.5′.

The ideal surface for a rolled head, then, is not a property of the screen but of the content: a sphere round the eyes at the depth the content is drawn. What the two eyes are sent and the distance at which the eyes part found that a curved screen’s binocular effects are a matter of its radius against the viewer’s distance; the roll’s term adds a third length, the depth of the content, and says the screen is best matched when all three agree. The screen is a picture surface set out why a screen’s shape is part of the picture it shows; the evenness a curve buys found that a curve evens out how much of the picture each degree of the view spends; it does not, by the same token, even out what a rolled head receives from content that is far behind the glass.

How far a head may roll #

The practical question the earlier essays asked was how far a head may roll before the worst point on the screen reaches the 15-arcminute limit at which two images stop fusing. The last figure answers it for content at every depth, on the three screens.

For content at infinity, a head may roll 2.58° before a flat screen reaches the limit and 2.38° before the curve drawn for it does; a flat-drawn pair on the curve has exceeded the limit before the head moves. For content at a metre, the flat screen and the curve drawn for it both allow 7.39°, while the flat-drawn pair on the curve allows 1.79°. Content drawn near the glass tolerates a large roll on any screen, because it has little separation for the roll to turn vertical, and the flat-drawn pair on the curve is the one that fails everywhere, because its saddle does not depend on content depth at all.

The comparison that matters for anyone making stereo content for a curved monitor is between the second and third lines. Drawing for the curve lifts the tolerance for content at a metre from 1.79° to 7.39°, and for content at infinity from nothing to 2.38°. That is the rendering choice, and it is worth nearly everything. What it cannot do is lift the curved screen to the flat one for distant content, and the reason is the nearness of its edges, which no rendering reaches.

Rendering and the surface, separated #

The question left open was whether the curve’s cost belongs to the rendering or to the surface. It belongs to both, in separable parts. The saddle — the curve’s own vertical disparity, present with the head level — is entirely a rendering choice: a pair drawn for the curve has none of it. The roll’s term is governed everywhere by one law, the eyes’ separation times the sine of the roll over the depth of the images ahead of the eyes. That law belongs to the surface only through the depth, and the curve’s contribution to it is that its edges are nearer.

Two pictures on one screen found that the disparity reaching infinity is exactly the distance between the eyes, at any screen distance whatever. That fact is what makes the roll’s term depend on nothing but depth for distant content: every stereo display sends distant content with the eyes’ own separation, and a rolled head turns that separation partly vertical by an angle set by how far away it is drawn. A curved screen draws its edges nearer, and so it pays more at the edges for distant content, whatever it is drawing.

What the figures assume #

The eyes are 63 mm apart and the pair is drawn for exactly those eyes. A viewer whose eyes are wider or narrower than the pair assumes adds a horizontal mismatch that a roll also turns partly vertical.

The head rolls about its own line of sight, at the design distance. A head that rolls and also moves sideways or forward changes the depth of every image, and so, by the law above, every number.

The curved screen is a cylinder. Every radius measured is a curve across and none down. A screen curved both ways — a sphere, or a dome — is the case in which content at one depth could be matched everywhere rather than along one row.

Still open: a pair that follows the head #

Everything here draws the pair once, for level eyes, and asks what a rolled head receives. A display that tracks the head can do better: it knows the roll and can draw each point’s two images where the rolled eyes’ sightlines cross the screen, removing the roll’s term the way rendering for the curve removed the saddle. The law says what is left when it does, which is nothing, for a head whose roll is known exactly.

A tracker is never exact. Its estimate of the roll lags and scatters, and the error in the estimate is what the viewer then receives, turned vertical by the same law: the eyes’ separation times the sine of the roll error, over the images’ depth ahead. The measurement that settles what tracking is worth takes a head rolling at a stated rate, a tracker with a stated latency and noise, draws each frame for the roll the tracker reports, and asks how the worst vertical disparity compares with drawing once for a level head — and whether, for content near the glass, where the untracked tolerance was already seven degrees, tracking buys anything at all.