The distance at which the eyes part

What the two eyes are sent measures a real effect and reports a null result: a curved screen sends the two eyes pictures that differ vertically, a flat one does not, and at a television’s radius the difference is a fraction of an arcminute — well under any stated fusion limit.

That measurement was taken at each screen’s own sitting distance, and the essay said so. The sitting distance is a parameter, the chair moves, and the null result is therefore a null result about a distance rather than about a screen.

The sweep #

The same measurement, run from a quarter of each screen’s sitting distance out to four times it.

Two curves, because two quantities are worth having and they behave differently.

The raw difference is how far apart the two eyes’ pictures put the same mark, vertically, before anything is done about it. The residual is what is left after four corners have been used to fix a common frame — which is the earlier essay’s measure, and is the right one, because a difference the two eyes can absorb as a rigid alignment is not a difference they have to fuse.

The two exponents #

Both fall with distance and they fall at different rates, and the pair is the finding.

The raw difference falls like the cube of the distance. Over the far half of the sweep the fitted exponent is 3.11 on a curved monitor, 3.02 on a curved television and 3.09 on a cinema screen.

The residual falls like the fourth power: 4.05, 4.01 and 4.05 on the same three.

Three screens whose widths differ by a factor of ten, and both exponents agree to within a twentieth. So the exponents are the geometry rather than a property of any one display, which is what makes them worth quoting at all.

Why the residual falls faster #

The extra power is the common frame’s doing, and it is worth having because a reader told only the residual would conclude that the geometry itself is fourth order.

The raw disagreement is a picture-plane coordinate of a fixed offset — the interocular distance — seen across the screen’s own sag. Two of the three powers come from the offset being seen at a distance and the sag being traversed; the third comes from the coordinate being a tangent, which divides by the distance again.

The four corners that fix the common frame then absorb the leading behaviour: a homography can reproduce any difference that is projectively consistent, and the leading term of the two eyes’ difference is. What survives is the next order.

So the geometry is third order and the reading is fourth, and the difference between them is a fit rather than a fact about eyes. Reporting only the residual would attribute one power of the distance to the world instead of to the machinery.

Where the limit is crossed #

The earlier essay argued rather than assumed its fusion limit: fifteen arcminutes of vertical disparity is a fact about people, is quoted with its provenance, and every figure using it says which number it used.

Against that limit, a curved monitor’s viewer crosses at 286 millimetres. That is twenty-nine centimetres from a seven-hundred-millimetre screen with a one-metre radius, which is closer than anybody works and is not absurd — it is a person leaning in to read small type.

The seats a screen will accept measures the room a viewer has on the other battery, and the two agree about which screen is tightest. A curved television never crosses it: at a quarter of its sitting distance the residual is 0.67 arcminutes, and the screen’s own arc runs out before the limit does. A cinema screen likewise.

So the null result survives for the two large screens and fails for the small one, which is the direction one picture and three people would predict — a tighter radius at a nearer seat produces more disparity, and the curved monitor is both. The screen sets the distance is where the sitting distance enters this field in the first place.

What the third power is made of #

The raw curve can be written down, which is worth doing because it shows where each of the three powers comes from and makes the fourth one’s provenance obvious by contrast.

A point at world offsets $X$ across and $Y$ up, on glass at range $D$, is seen by the two eyes at ranges differing by about $bX/D$. Its vertical angle is $Y/r$, so the two eyes disagree by

$$\Delta \;\approx\; \frac{b\,X\,Y}{D^{3}} \;=\; \frac{b\,w\,h}{4\,D^{3}}$$

at the corner of a screen $w$ wide and $h$ tall. One power of $D$ is the vertical angle itself, one is the range difference the baseline produces, and one is that difference expressed as a fraction of the range — which is the account the section above gives in words, with the terms now separable.

Two things follow. The expression is linear in $b$, and it is precisely the part a common frame absorbs, which is why the residual is both a power steeper and a power of $b$ higher — the same pairing the two eyes’ two components show as slopes of one and two.

And a fourth-power fall makes the crossing distance remarkably insensitive to the limit it is crossed against. Tightening a fifteen-arcminute limit to two — a factor of 7.5, and more than the whole spread of figures anyone quotes — moves the crossing from 286 mm to 472, because $7.5^{1/4}$ is 1.65. The argument about where the limit is does not have to be settled to know roughly where the crossing is, which is an unusually comfortable position for a measurement that depends on a fact about people.

What the raw and residual readings are each for #

Two curves is one more than most measurements report, and the reason for carrying both is worth setting out because it decides what the answer means.

The residual is the reading that matters for comfort. The two eyes will absorb any difference between their images that is a projective transformation of one into the other — that is what a common frame is, and it is what happens when a viewer’s head tilts or when the two images are at slightly different scales. What they cannot absorb is what is left over, so the residual is the quantity a fusion limit is stated against.

The raw difference is the reading that says what the geometry did. It is the physical disagreement between the two images before any allowance, and it is the quantity whose exponent is a property of the arrangement rather than of the fitting.

Reporting only the first would attribute a power of the distance to the world; reporting only the second would compare a fusion limit against a quantity nobody’s eyes ever meet. The pair is honest and neither half is.

That is the same structure the screen field uses everywhere: the screen that names the seat reports a residual against a homography for exactly this reason, and here the homography is the two eyes’ own alignment rather than a display’s.

Where the disparity comes from #

One paragraph of mechanism, because a fourth-power law with no cause behind it is a curve rather than a finding.

The two eyes are level and a fixed distance apart. Each sees the screen’s edge at a slightly different angle, and because the screen is curved the edge is at a slightly different distance from each of them — the near eye is nearer the near edge and further from the far one.

That difference in distance changes the vertical angle each eye sees a mark at, because a mark’s height is fixed and its angular height is the height over the distance. So a curved screen sends the two eyes marks at different elevations, and a flat one does not, because on a flat screen both eyes are the same distance from every point of a vertical line through it. That the flat case is exactly zero rather than merely small is what makes the curved number a measurement, and it is the same control the screen is a picture surface too insists on.

The effect is therefore proportional to the screen’s sag, to the interocular distance, and inversely to the distance cubed — which is the raw law, and it comes out of that sentence rather than out of a fit.

Whether a tighter radius is worse or better #

The sweep is over distance and the other parameter is the radius, and the two interact in a way worth naming because it is not monotone in the obvious direction.

A tighter radius makes the sag larger, which increases the disparity. A tighter radius is also sold with a nearer sitting distance, which increases it again. Both push the same way, and that is why the curved monitor — the tightest radius and the nearest seat — is the only screen here that reaches the fusion limit.

But a tighter radius also makes the screen’s own arc larger for a given width, which at some point puts the screen’s edges behind the plane of the eyes and stops the measurement. So the disparity does not grow without bound as the radius tightens; the arrangement stops being a screen first.

That bound is the same one the evenness a curve buys meets from the other side, where a curve helps until it wraps too far and then stops helping. Both are consequences of a screen’s arc rather than of its radius, and both say that the interesting parameter is the angle the screen subtends rather than the number the manufacturer prints.

Where the sweep stops, and why #

The near end of the sweep is not a choice and it is worth saying what stops it.

A seat close enough to a wide screen has the screen’s own edges behind the plane through the viewer’s eyes. There is no flat picture of a point at ninety degrees, so the machinery that writes the two eyes’ views as ordinary pictures refuses, and it refuses correctly.

That is a limit of the coordinates the disparity is written in rather than of the eye. A viewer really can sit that close and really does receive something; what they do not receive is anything a single flat picture describes.

The refusal is reported rather than clamped. A sweep that silently dropped its near end would have the fall-off starting wherever the refusal did, and the exponents fitted from it would be fitted over a range chosen by the machinery.

What the null result was and was not #

Setting the two readings side by side, because the earlier one is not wrong and its scope was narrower than it looked.

At each screen’s own sitting distance, the curved screen’s two-eye disparity is a fraction of an arcminute and is under any stated fusion limit. That is what was measured and it holds.

At a quarter of that distance on a curved monitor it is two hundred and fifty arcminutes, which is four degrees, and the raw difference is over two hundred times the limit. Nothing about that contradicts the earlier reading; it is a different arrangement.

The correction is to the scope rather than to the number. A null result whose parameter was not swept is a null result about a point in a parameter space, and stating it as a property of a screen is the ordinary way such a reading goes wrong. This collection’s own habit — that a quantity held fixed should be swept before it is concluded from — is what the shortfall recorded and what this discharges.

What a viewer would actually notice #

Two things worth separating, because “disparity” covers both and only one of them is about fusion.

Vertical disparity is the one measured here and it is the one that matters for comfort: the two eyes cannot rotate independently about the horizontal axis, so a vertical difference between the two images has to be absorbed by the fusion machinery and there is a limit past which it cannot be.

Horizontal disparity is depth, and the eyes are built for it — two pictures on one screen is where this collection uses it deliberately. The curved screen produces plenty — 34 arcminutes at a curved monitor’s sitting distance, against half an arcminute of vertical — and a viewer interprets it as the screen being slightly curved, which it is.

So a curved screen’s honest description is that it tells the two eyes the truth about its own shape, in the horizontal channel where they expect it, and adds a small vertical error where they do not. The vertical error is the whole of the complaint and it is small everywhere except very close in.

The interocular distance, which is also a parameter #

The sweep holds one number that varies between people, and it is worth saying what it is worth.

Sixty-three millimetres is the ordinary adult figure and it is what every number here uses. The disparity is proportional to it, so a viewer at fifty-five sees twelve per cent less and one at seventy-one twelve per cent more, and the crossing distance moves by about three per cent either way — the third root of twelve per cent, because the residual falls like the fourth power and the crossing is where it meets a fixed limit.

A child’s is smaller and a child sits closer, and those push opposite ways. At fifty millimetres and half the sitting distance the residual is up by a factor of eight from the reduction in distance and down by a fifth from the narrower eyes, so a child leaning into a curved monitor is comfortably past the limit.

That is a real consequence and it is the one place in this row where a held parameter’s variation between people changes the conclusion rather than the number. It is also the one this collection’s own machinery is least able to say anything about, because a fusion limit is a fact about people and the fifteen arcminutes here is a stated figure rather than a measured one.

What this does not measure #

Three things the sweep leaves out, each worth naming because a reader could reasonably think they are in it.

Accommodation. The eyes focus as well as converge, and a screen at a fixed distance asks for one focus and a range of convergences. That is the whole of the vergence-accommodation problem in stereo displays and it is nothing to do with the curvature.

Head rotation. A viewer looking at the screen’s edge turns their head, which changes both eyes’ geometry and absorbs part of the disparity. The measurement here is of a viewer facing the middle, which is the worst case and is stated as such.

And the screen’s own pixel grid. A disparity smaller than a pixel is not delivered at all, and on a curved monitor at its sitting distance a pixel subtends about a third of an arcminute. The residual at that distance is half an arcminute, which is a pixel and a half — so the effect is real and it is close to the display’s own resolution, which a curve in flat pieces is the other place this collection meets that bound.

The general lesson about held parameters #

Worth stating outside the screen field, because the correction here is procedural rather than geometric.

A null result is a statement that a quantity is below a threshold. It has as many caveats as the measurement had held parameters, and a held parameter that a reader can vary — a distance, a temperature, a rate — turns a general claim into a local one.

The remedy is not more caution in the prose. It is to sweep, and to report the exponent, because an exponent says what the held parameter was worth: a quantity falling like the fourth power of a distance is one whose null result is very safe going back and very unsafe coming forward, and a quantity that is flat in it needs no sweep at all.

This collection’s habit of recording what it could not finish is what surfaced this one. The earlier round wrote in its own shortfall queue that the sitting distance was a parameter and had not been swept, and the sweep found a crossing.

The short version #

The two eyes’ vertical disagreement on a curved screen falls like the cube of the sitting distance, and what survives a common frame falls like the fourth power. Both exponents are the same on three screens ten times apart in width, so they are the geometry rather than a display.

The earlier null result holds at every screen’s own sitting distance and fails on a curved monitor at 286 millimetres, where the fifteen-arcminute fusion limit is crossed. A null result taken at one point in a parameter the reader can move is a null result about that point.