A quarter of a degree of roll keeps the line

A sliding pair keeps its line only near the middle measured what the slide costs. Two eyes obeying Listing’s law, looking up and to one side, are related by a motion that turns and also slides along its own axis, so the set of points both eyes see in one direction stops containing a straight line. Along the place the line would be, at a fixation 20° aside and 20° up at 1.2 m, the disparity reaches 3.2 px. The region of fixations where it stays under a pixel is a narrow cross, and it shrinks quickly with nearness. That essay also noted that Helmholtz’s rule has no slide at all, since it keeps both pictures in a common plane.

That makes a choice look starker than it is. Listing’s law is what human eyes do, and Helmholtz’s rule is what an instrument can be built to do, and they differ by several degrees of turn about the line of sight. If a viewer or a designer had to pick between them, the difference would be large. But Helmholtz’s rule does more than it needs to. It fixes the roll of each eye entirely, once its direction is chosen, and that takes care of a good deal more than the slide. The slide is one number: the component of the relative turn’s axis along the baseline. A much weaker condition should be enough to cancel one number.

One number, one knob #

Leave both eyes pointing where they point. Rotate each about its own line of sight by a small angle $\epsilon$, in opposite senses, the left eye one way and the right eye the other. This is a cyclovergence, and a pair can do it without changing where it is looking at all. Each picture turns about its own centre, the fixation point stays where it was, and the relative motion between the eyes changes a little, in a way that shifts the slide.

At a fixation 20° aside and 20° up, 1.2 m away, the slide under Listing’s law is 3.93 mm. A roll of 0.282° in each eye takes it to nothing. The search finds that to better than a billionth of a metre, and the slide’s response to the roll is almost linear over a much larger range than is needed.

Now the eyes have a motion with no slide, and a motion with no slide contains a line: its own axis. Every point on that axis is carried along itself, so both eyes see it in one direction. And so it is.

Along 60 cm of that axis, the disparity with no roll runs to 3.20 px. With the roll it is $2\times10^{-13}$ px. So a cyclovergence of 0.56°, and nothing else, has restored the straight line of the horopter at this fixation. That is exact, in the sense that the point-for-point check runs through the full camera model, with no small-angle assumption.

Much less than the whole rule #

The natural comparison is the turn Helmholtz’s rule would ask for at the same fixation.

At this fixation each eye has to turn by an average of 3.53° to reach Helmholtz’s rule. The roll that removes the slide is 0.282°, a twelfth of that. At 10° aside and 20° up the rule asks for 1.68°, against 0.276° for the roll. At 20° aside and 10° up, 1.85° against 0.140°. Closer in, the roll grows: at 40 cm, at 20° aside and 20° up, it is 0.845° against 3.52°, still about a quarter.

So the two rules’ large difference was almost all unnecessary, from the point of view of keeping the line. Helmholtz’s rule is one specific member of a large family of pairs that have no slide, and it happens to be a long way from Listing’s law. Another member is nearby. That member is what the question was asking for: a Listing pair is a fraction of a degree from one that keeps its line everywhere. The size of that fraction is a useful answer to the essay that asked, since a roll of that size could plausibly be invisible, or built into an instrument with no measurable cost, or even, as below, already happening.

The size has a short law #

The numbers above have a pattern. Going from 10° aside to 20° aside hardly changes the roll, and going from 10° up to 20° up doubles it. Tripling the nearness roughly triples it. That suggests something simple, and small-angle arithmetic supplies it. Under Listing’s law an eye looking at azimuth $\alpha$ and elevation $\beta$ has a false torsion of about $\alpha\beta/2$. The two eyes differ in azimuth by the vergence angle $\nu$, so their difference in torsion is about $\beta\nu/2$, shared between two eyes. That gives

$$\epsilon \approx \frac{\beta\,\nu}{4},$$

with the angles in radians, and the azimuth does not appear. The elevation and the vergence do, and the vergence depends on the distance.

A hundred fixations, up to 30° up and 30° aside, from 30 cm to 2 m: the law holds to within 9 per cent at the worst of them, and the worst are the ones far from the axis. So the correction has one simple rule, and the rule depends on two numbers a pair of eyes always has available: how far up it is looking and how much it has converged.

That the azimuth is missing is interesting, because the slide itself depends on it strongly.

At 20° up, the roll is 0.274° at almost no azimuth and 0.315° at 43° aside, though the slide it removes goes from next to nothing to over 10 mm. This is not as paradoxical as it first sounds. The slide is a component of the turn’s axis, times a baseline, and the amount of axis component a given roll changes is itself larger when the azimuth is larger. Both ends of the balance grow together.

It has a practical consequence. A controller that wants to keep a horopter line, on an instrument or in a pair of eyes, does not need to know its azimuth. It can compute the roll from elevation and vergence alone. And an instrument builder designing a stereo head with gimbals can do this with a single small linkage or a line of software.

The sign, and two cases where it vanishes #

The law has a sign, and it is worth stating, since it says which way each eye should roll. It is set by the elevation. Look up and the roll goes one way, look down and it goes the other, with the same size at $-20°$ as at $+20°$. A fixation to the left or to the right does not change that sign. The two mirror-image fixations, 20° to the left and 20° to the right, have exactly the same roll, and the only thing to change is which eye turns which way when they are described in a head’s own terms. So a correction computed from elevation alone cannot go wrong in sign, as long as a pair knows which way is up.

There are also two arrangements where the law’s roll vanishes, and it does so for a clear reason each time. In the plane of the eyes, with no elevation, all three rules are the same rotation, as the earlier essay on raising the gaze already recorded, so no slide exists to remove. And as the vergence falls to nothing, with a fixation very far off, the two eyes point in the same direction and the two torsions are equal, so there is no difference to cancel. Between those limits the roll scales in the product of the two, as a term of this kind should. At a fixation five metres off and 20° up, with a vergence of 0.74°, the roll is about 0.06°, too small to matter for anything. At 25 cm, with a vergence of 14.8°, it is about 1.3°, and at that distance it is becoming a substantial movement.

So the correction matters most exactly where close work takes place. The distance at which the eyes part is another result that changes character at these short ranges, and vergence moves the shells describes what convergence does to depth quantisation there. A close-range instrument, with a vergence of several degrees, a stereo microscope for instance, or a pair of cameras on a surgical robot, sits in the range where a roll of a degree or so could be built in or computed, and is also the range where a slide has the largest effect on the picture.

What the roll costs each picture #

A roll is not free. It turns each picture about its centre, and a mark away from the centre moves.

At 1.2 m and 20° up, a mark at the edge of the picture moves by 2.18 px. At 40 cm, 6.55 px. At 30 cm and 30° up, 13.2 px. Those would not be noticed in a single picture: turning a whole scene by a third of a degree is far below what a viewer can detect by eye without a reference. And the relevant comparison is with the slide’s cost, 3.2 px across a long stretch of the very line the pair is using, in a place where matching actually depends on it.

But it does have a real effect on the rest of the field. Other than on the axis, the roll changes the disparity everywhere, and it adds a term that grows with distance from the fixation. Where the eyes are converged on a plane to be matched, the roll introduces a shear between the two pictures of about $2\epsilon$ times the distance from the centre: 4.4 px at the edge at 1.2 m. So a designer has a choice about which disparities to minimise: the ones along a straight vertical line, or the ones across a whole fronto-parallel plane. Those are not the same pair. The roll is the right answer to the question this essay asked, and it might not be the right answer to another one.

Turning the cameras inwards showed the plain case of this, where a verged pair with no elevation has a vertical disparity that grows toward the corners. Both coordinates agree on a circle and a line makes the point that all this is about one locus of perfect agreement. Most of what a stereo instrument does with its pictures happens away from that locus, for which the rectification in a pair’s lines is the usual treatment. So in an instrument, nothing above changes the established practice of rectifying the pictures after they are taken, and for that a slide costs little. It matters most in a system that does not rectify, and instead has to rely on correspondences being already nearly horizontal, as a pair of biological eyes does.

Whether eyes do this #

The arithmetic does not mention people. But the roll’s law predicts something about real eyes that is checkable, and researchers have checked closely related things. Binocular eye movements are known not to obey Listing’s law exactly in convergence. Measurements dating back at least to work by Van Rijn and Van den Berg in the early 1990s show a cyclovergence that rises with elevation and with vergence, so that the eyes’ torsion changes with where they are looking in depth. The direction and size of that term have been discussed since as a possible way to keep the two eyes’ epipolar geometry useful, as in work by Schreiber and colleagues in the 2000s. This essay has done nothing to establish that claim. The law here is geometric and exact, and measured eye movements carry scatter and individual differences well beyond it.

Yet the size is suggestive. A cyclovergence of half a degree, at this distance and elevation, is well within what those studies report. A system that needs to keep its corresponding points in a useful arrangement has this very cheap way of doing so, a quarter of a degree in each eye, with no need for the several degrees that Helmholtz’s rule would cost in the eyes’ mechanical range.

That is about as far as the geometry can carry this. It says that the cost is small and what its law would be. Whether a real visual system pays that cost, and for this reason, is a matter for people measuring eye movements.

What a pair with no slide has given up #

A pair corrected in this way keeps a straight line. It has not become a Helmholtz pair. The line it keeps stands at a slant different from the line Helmholtz’s rule would keep, and that was the other half of the earlier result: each rule has its own line, and they differ by 10° and 20° at this elevation. A small roll does nothing substantial to which way the line points. So a person drawing the textbook figure of the horopter, with the vertical line standing square to the plane of regard, would still be drawing it wrongly for this pair, in the same way as for an uncorrected Listing pair. That figure belongs to Helmholtz’s rule alone.

The size of the circle is also unaffected to any useful precision. The motion’s turn angle changes in the third or fourth decimal place. Only the slide is removed, which is what was asked for, and there is a nice economy in that. It is the only property of Listing’s law that this correction needed to touch.

A real eye translates as it turns, and its torsion lags #

The eyes are points with perfect rotation about their centres. A real eye translates a little as it turns, and that adds a tiny baseline change that can also contribute to a slide. Whether that effect is comparable with the roll’s would need an eye model with its centre of rotation offset from its optical centre, and none is used here.

The fixation point is treated as known exactly, and the roll is exact for it. Points other than the fixation are considered only along the motion’s axis. How large a region of three-dimensional space stays within a pixel of agreement under the roll, compared with under Listing’s law alone or Helmholtz’s rule, would be a separate sweep.

Everything is in terms of a roll computed for the fixation. A real eye’s torsion takes time, and in a fast movement it lags. Over a saccade of a few tens of milliseconds, the roll would not be keeping up, and it is not obvious that the slide would matter at those moments.

Still open: what the roll does to the whole plane of regard #

A pair that keeps a line has been the question of this and the preceding essays, because the horopter’s line is a clean, exact object. A viewer does not spend most of the time looking at a vertical line, however. Most of what stereo vision is used for involves a surface near fixation, and the amount of vertical disparity across such a surface is what decides how easily it can be fused. A roll changes that shear directly.

The measurement that settles what the roll costs on a surface takes a fronto-parallel plane through the fixation, at each of many fixations, and maps the vertical disparity across it under three rules: Listing’s law alone, Listing’s law with the roll, and Helmholtz’s rule. It asks which of the three keeps the largest area under a stated threshold of vertical disparity; whether the roll helps or hurts on that measure; and whether a different roll, chosen for the surface and not for the line, would differ from the one found here in size or even in sign.

Two eyes obeying Listing’s law are a quarter of a degree of roll away from a pair that keeps its line. Between Listing’s law and Helmholtz’s rule, the large difference was not the one that mattered.