A sliding pair keeps its line only near the middle

Raise the gaze, and the line is gone found that two eyes looking up and aside keep the horopter’s straight line only if they roll by Helmholtz’s rule. Under Listing’s law or Fick’s rule the motion that carries one eye’s view onto the other’s does not only turn; it slides along its own axis, by 3.93 mm and 7.09 mm at a fixation 20° aside and 20° up, and a motion that slides leaves no point where it was. The horopter becomes one curve with no straight part.

That settled the structure and left the size. A slide of a few millimetres at a metre and more might leave a curve that hugs the line so closely that nothing a viewer or an instrument does could tell them apart — or it might not. The essay proposed a measurement: take the vertical line a Helmholtz pair would keep at the same fixation, and ask how large a disparity a Listing pair shows anywhere along a stretch of it.

The proposal turns out to measure two things at once, and only one of them is the slide. This essay separates them, measures the one that matters, and finds where in the field the line survives.

The line to measure against is the pair’s own #

Start where no rule slides: straight ahead, 20° up, 1.2 m away. Every pair keeps a line there, since all three rules give a relative motion that is a pure turn.

Along the Helmholtz pair’s line a Listing pair nevertheless shows a disparity: zero where the line crosses the plane of regard, rising steadily to 2.15 px thirty centimetres above and below. A Fick pair shows 4.43 px. Neither slides here, so none of that is the slide. It is the fact the previous essay recorded in passing — straight ahead and raised, the three rules keep three different lines, perpendicular to the plane of regard under Helmholtz’s rule, vertical under Fick’s, and leaning halfway under Listing’s law. Along its own line, each pair shows $5 \times 10^{-13}$ px.

The Helmholtz line was the natural thing to measure against, because it is the line the textbook drawing of the horopter shows: a circle in the plane of regard and a line standing perpendicular to it. That drawing is the Helmholtz pair’s horopter exactly, and for the other two rules it stands the line the wrong way before any slide has entered — by 10° under Listing’s law and 20° under Fick’s rule at this elevation.

So a disparity along the Helmholtz line answers two questions at once: whether a rule keeps a line, and which way that line points. The second question has an answer even where the first does not arise. For the size of the slide to be read by itself, the stretch has to be the pair’s own axis — the line the curve would contain if the slide were switched off.

Along its own axis, the disparity is the slide #

The own axis has a property that makes it the right instrument, and it is exact rather than approximate.

The relative motion carries the left eye’s picture onto the right eye’s: the left eye sees a point $X$ exactly where the right eye sees the point the motion carries $X$ to. For a point on the axis, the motion is a slide of $s$ along the axis direction $\hat u$, so the left eye sees $X$ where the right eye sees $X + s\hat u$. The disparity at $X$ — the difference between where the two eyes see it — is therefore

$$\text{right}(X + s\hat u) - \text{right}(X),$$

the slide carried into one eye’s picture. That identity holds at every sampled point to $10^{-8}$ px, and it is what licenses reading the disparity along the own axis as a measurement of the slide and of nothing else.

The figure draws both lines at the fixation the previous essay used, 20° aside and 20° up at 1.2 m. Along its own axis a Listing pair shows at most 3.20 px and a Fick pair 6.01 px. Those numbers change along the 60 cm stretch only as its points come nearer the eyes or go further — the same slide seen from a different distance. Along the Helmholtz line the same pairs show 3.73 px and 7.02 px, rising toward both ends of the stretch where the two lines’ directions pull apart. At this fixation the tilt adds only half a pixel to the slide’s own three; straight ahead, the tilt was the whole of it. The slider moves the fixation aside: at 0° the own axis shows nothing, and at 40° it shows 7.95 px under Listing’s law.

The slide stays; its picture shrinks with distance #

A slide is a length. How large it looks depends on how far away it is seen from, and the slide itself turns out not to care.

From 30 cm to 6 m the slide changes by less than a per cent: 3.903 to 3.935 mm under Listing’s law, 7.033 to 7.096 mm under Fick’s rule. The previous essay derived why: the slide is the 65 mm baseline projected onto the axis of the eyes’ relative turn, and that axis’s direction is set by where the eyes point far more than by how far. Moving the fixation point out along its direction hardly changes the direction either eye is turned to, so it hardly changes the slide.

The disparity the slide makes does change. It falls from 17.3 px at 30 cm to 3.20 px at 1.2 m and 0.59 px at 6 m under Listing’s law — a little faster than one over the distance, because at close range the 60 cm stretch reaches well off the line of sight. Under Listing’s law it falls below a pixel at about 4 m; under Fick’s rule it is still above a pixel at 6 m. So the question “is the line a good approximation?” has no single answer. It depends on how near the thing being looked at is, and for a slide that does not shrink, near is where the answer is worst.

The axis holds still while the turn shrinks #

The reason the slide ignores distance is visible in the rest of the relative motion, and it changes how the motion should be pictured.

Move the fixation point out along its direction, from 30 cm to 20 m, and take the relative motion apart each time into its turn, its slide and its axis. The axis direction does not move: its components are (0.0600, 0.9804, 0.1878) at 30 cm and (0.0605, 0.9802, 0.1884) from 1.2 m out to 20 m. The turn shrinks as one over the distance — 11.94° at 30 cm, 2.99° at 1.2 m, 0.60° at 6 m and 0.18° at 20 m — because the turn is the eyes’ vergence, the angle between two lines of sight that converge on a point, and that angle falls as the point recedes. The slide stays at 3.90 to 3.94 mm throughout.

So the character of the motion changes with distance even though none of the numbers the previous essay tracked does. At 30 cm the motion is mostly a turn: 3.9 mm of slide for every 12° of turn, 18.7 mm of slide per radian. At 20 m it is mostly a slide: the same 3.9 mm for a fifth of a degree, 1,256 mm per radian. And the place where the axis crosses the plane of regard runs out with the fixation point, from 29 cm ahead at a 30 cm fixation to 19.6 m ahead at a 20 m one, while keeping its direction.

The axis direction depends only on where the eyes point because each eye’s roll does. Under Listing’s law, an eye’s orientation is fixed by its gaze direction alone, and the two eyes’ gaze directions differ by the vergence and in no other way. The component of the relative turn’s axis along the baseline — the component that makes the slide — is then set by the direction of gaze, and multiplying it by the fixed 65 mm baseline gives a slide that the distance never enters. The depth a pair calls zero found the circle of the horopter through the fixation point and both eyes, a circle that grows with the distance; the slide is the part of the geometry that does not grow with it.

Where in the field the line survives #

A pixel is a threshold with a meaning for an instrument — at a focal length of 900 px it is 3.8 minutes of arc — a stereo matcher reading disparity to whole pixels, which whole pixels cut space into shells counted in, cannot see a disparity below it. The region of fixations over which the own axis shows less than a pixel is the region over which the circle-and-line drawing of the horopter is as good as exact for that instrument.

At 1.2 m a Listing pair keeps its line to under a pixel nearly all the way to 40° aside only when the fixation is within about 2.5° of the plane of the eyes. At 10° up the boundary is 12.8° aside, and at 20° up it is 6.6°. A Fick pair does worse by about a factor of two at every elevation: 6.4° aside at 10° up and 3.3° at 20° up. The product of the two angles along each boundary stays within a narrow range — 99 to 133 square degrees for Listing’s law and 64 to 69 for Fick’s rule — which is what a slide growing roughly with both angles at once predicts.

At 40 cm — a book held in the hands, a workbench, a close-range rig — the region shrinks by more than a factor of three. Under Listing’s law the boundary runs from 16.6° aside at 2.5° up to 1.9° aside at 20° up; under Fick’s rule to 0.7°. What is left is a cross: the band close to the plane of the eyes, and the band close to the vertical plane through straight ahead, where the rules’ rolls are mirror images of each other and the slide vanishes. Everywhere off that cross, the horopter of a near-looking Listing pair is a curve for any instrument that reads a pixel.

That answers the question the previous essay closed on, and it answers it in the less comfortable direction. The circle-and-line drawing is exact on a cross and good near it. It is wrong by more than a pixel over most of the field a pair looks at closely, and the error is not a matter of fine structure: it is several pixels a few tens of degrees off the cross.

The slide shows up and down #

The disparity along the own axis has two components, and they do not share it equally.

At 20° up and 20° aside, a Listing pair’s own axis shows 3.17 px of vertical disparity and 0.40 px of horizontal. At 40° aside the two are 7.53 and 2.56 px. The axis of the relative turn is close to vertical, so a slide along it moves the right eye’s picture of each point mostly up or down.

That matters for how the error is met. Both coordinates agree on a circle and a line found that the horopter needs both disparities to vanish, and it was the vertical one that confined it to a line. A pair of cameras that is rectified before matching — its pictures turned so that corresponding points share a row, which rectifying a pair spends what its lines lean priced — reads horizontal disparity and discards vertical by construction. For such a pair the part of the slide that shows is the smaller part, and a matcher searching along rows sees at 40° aside a horizontal disparity of 2.56 px where the whole disparity is 7.95. The vertical part is not lost information about the scene, either; the second disparity cuts cells and its correction found that a verged pair’s vertical disparity is almost entirely fixed by the other readings, so what a sliding pair adds to it is, to that extent, a statement about the rig rather than about the world. Depth is a reciprocal turns a horizontal disparity into depth; the vertical part of the slide has no such conversion, and a reader who takes it for depth information is reading the eyes’ roll.

Instruments choose their rule #

For eyes, the rule an eye rolls by is a fact to be measured, and this essay takes no position on which rule a particular eye obeys. For a rig it is a choice, and the numbers above are a price list for it.

Turning the cameras inwards measured the vertical disparity a toed-in rig produces. A rig whose two cameras sit on one head that tilts about the baseline and then pans each camera about its own turned vertical is a Helmholtz rig: its horopter keeps its line at every setting, and the price table has zeros in it everywhere. A rig whose cameras pan about a fixed vertical and then tilt about their own horizontal, each on its own gimbal, is a Fick rig, and at 40 cm it leaves the pixel less than a degree off the vertical plane when looking 20° up. A rig that turns each camera by the single shortest rotation to its target, as a motion controller computing orientation from a direction naturally does, is a Listing rig and sits between the two.

None of that shows in either picture alone. Four cameras fit, and one can see recovered relative pose from a pair of pictures; a pose recovered from a Listing or Fick pair is a turn with a slide, correctly, and the slide is exactly what this essay has been measuring. A reader with the recovered pose can compute the disparity along its axis and so know, for that rig and that fixation, how far its horopter is from a line.

What the stretch leaves out #

The stretch’s length. Every number above is the largest disparity over 60 cm of the axis, centred where it crosses the plane of regard. A longer stretch reaches further from the eyes’ lines of sight and shows more; a shorter one shows less. The slide itself is independent of the stretch, and at the fixations drawn the disparity at the crossing point alone is within a fifth of the stretch’s worst. At 20° aside and 20° up under Listing’s law, a stretch of 10 cm shows 2.95 px, 30 cm 3.04, 60 cm 3.20 and 1.2 m 3.54: the length moves the answer by a tenth, and the slide supplies the rest.

Other pixel sizes. The focal length is 900 px throughout. A finer picture sees the same slide as proportionally more pixels, and every boundary above moves toward the middle of the field.

Whether eyes follow any of these rules. The three rules are models. Measurements of real eye movements depart from each of them in ways that depend on vergence, and a departure in roll is exactly what changes the slide.

Still open: a rule that keeps the slide at zero #

Helmholtz’s rule keeps the slide at zero by keeping both pictures in one plane, and that is a strong condition: it fixes the roll of each eye completely once its direction is chosen. The slide depends on only one number, the component of the relative turn’s axis along the baseline, so a much weaker condition should be enough to cancel it.

The measurement that settles it lets each eye roll by Listing’s law plus a small correction that depends on its direction, chooses the correction to make the slide zero at every fixation, and asks two things: how large the correction is — how far, in degrees of roll, a Listing pair is from one that keeps its line everywhere — and whether a correction that small leaves the two eyes’ pictures nearly as they were. If a roll of a fraction of a degree removes the slide, then the choice between keeping the horopter’s line and following Listing’s law costs almost nothing in either direction.