Raise the gaze, and the line is gone

Both coordinates agree on a circle and a line took the horopter out of the plane of the eyes and found two pieces: the Vieth–Müller circle, and a vertical line through the fixation point. The circle came from the inscribed-angle theorem. The line came from somewhere more general — it is the axis of the rigid motion that carries one eye’s picture frame onto the other’s, and a point that motion leaves where it was is seen at the same place by both eyes.

That essay ended on an assumption it had been able to avoid. Every fixation it drew lay straight ahead, and for a fixation straight ahead it does not matter how an eye rolls about its own line of sight, because every sensible rule says it does not roll at all. This essay moves the fixation point: first aside, within the plane of the eyes, and then up and aside together. The first move tells where the line goes. The second tells whether there is a line.

Looking aside, in the plane of the eyes #

Turn the fixation point 20° to the right, keeping it 1.2 m away and in the plane of the eyes. Both eyes turn about vertical axes to face it — the right eye by a little less than the left, since it is nearer — and the pictures they make are no longer mirror images of each other.

The circle part of the answer barely needs checking. The inscribed-angle theorem does not care where on the circle the fixation point is: the horopter in the plane of the eyes is still the circle through both eyes and the point being fixated, simply a slightly different circle, because a circle through two fixed points is determined by any third.

The line is less obvious, and the obvious guess is wrong.

Seen from above, the vertical line is a dot, and the dot is not at the fixation point. It sits on the circle directly ahead of the midpoint between the eyes — in the median plane — at 1.277 m, while the point being looked at is at 1.2 m and forty-one centimetres to the right.

Why the line cannot leave the median plane #

The reason is short, and it holds for every fixation in the plane of the eyes, not only for this one.

The relative motion carries the left eye’s centre onto the right eye’s centre. For a fixation in the plane of the eyes it is a pure turn about a vertical axis — both eyes turn about vertical axes, so their difference does too — and the figure’s computation confirms it, with no slide along the axis at all.

A turn that moves one point onto another leaves both points at the same distance from its axis, because turning about an axis preserves distance from it. So the axis is equidistant from the two eyes. Every point equidistant from both eyes lies on the plane that perpendicularly bisects the segment between them, which is the median plane. The axis is a vertical line, so it lies in the median plane.

And it lies on the circle. The axis is a line of the horopter, so where it crosses the plane of the eyes is a point of the horopter in that plane, and the horopter in that plane is the circle. The foot of the line is therefore the one point where the circle crosses the median plane on the viewer’s side — the point of the circle straight ahead. For a fixation straight ahead that point is the fixation point itself, which is why both coordinates agree on a circle and a line found the line there. For any other fixation it is not.

Drawn obliquely, the separation is plain. The eyes’ lines of sight converge on a point to the right; the vertical line stands to their left, on the median plane. A point directly above the fixation point, which the straight-ahead result would have placed on the horopter, is not on it once the fixation has moved: its vertical disparity is no longer zero, because it is nearer one eye than the other. A point directly above the straight-ahead point of the circle is on it, whatever the eyes are looking at.

The roll an eye is free to choose #

Now raise the fixation point, and something the figures have not yet had to decide becomes decisive.

To look at a point, an eye’s optical axis must pass through it. That fixes two of the eye’s three rotational freedoms and leaves the third: a turn about the optical axis itself. A camera aimed at a target can still be rotated about its line of sight, and every such rotation aims it at the same target while rolling its picture. Fixation does not choose the roll. A rule does.

Three rules are standard in the description of eye movements, and the figures compute all three.

Helmholtz’s rule turns the eye up or down about the axis through both eyes first, then aside about the eye’s own turned vertical. A picture’s horizontal axis then stays parallel to the line joining the eyes, so the two eyes always share a single plane — the plane containing both eyes and the fixation point, tilted up with the gaze.

Fick’s rule turns the eye aside about the head’s vertical first, then up or down about the eye’s own horizontal. A picture’s horizontal axis stays level.

Listing’s law reaches every direction by a single turn from the straight-ahead position, about an axis perpendicular to both the straight-ahead direction and the new one. Human eyes with the head still are commonly described as following it closely; this essay treats all three as models and takes no position on which one an eye obeys.

For a fixation straight ahead and level, all three give no turn at all. For one aside in the plane of the eyes, all three give the same turn about a vertical axis. For one straight ahead and raised, all three still give a relative motion that is a pure turn, so all three keep a line — but not the same line: the axis leans back perpendicular to the plane of regard under Helmholtz’s rule, stays vertical under Fick’s, and leans halfway, by 10° at a 20° elevation, under Listing’s law. They part completely when the fixation point is both raised and aside, because only then does the order in which the two turns are made change the roll by different amounts in the two eyes.

Raised and aside: two rules slide #

The figures fixate a point 20° to the right and 20° up, at 1.2 m, and compute the relative motion under each rule.

Under Helmholtz’s rule the relative motion is still a pure turn. Its axis has tilted back with the gaze, perpendicular to the plane that contains both eyes and the fixation point, and the in-plane argument above repeats inside that tilted plane: a circle through both eyes and the fixation point, and a line perpendicular to the plane through the circle’s point in the median plane — here 1.182 m ahead and 0.458 m up. The horopter has been tilted with the gaze and is otherwise the same object.

That is not a coincidence of this fixation. Helmholtz’s rule keeps both pictures’ horizontal axes parallel to the line joining the eyes, so both frames lie in one plane and differ only by a turn about that plane’s normal. A turn about an axis, with no slide, is exactly the kind of motion that fixes a line.

Under Listing’s law the relative motion turns by 2.99° and, as it turns, slides 3.93 mm along its axis. A motion that slides moves every point — points on the axis are carried along it, and points off it are carried round it — so nothing is left where it was, and there is no line of fixed points for the horopter to contain.

The horopter does not disappear. It is still the set of points both eyes see at the same place, and the figure’s computation still finds one such point for every value of its parameter and checks each by projecting it through both eyes. What it no longer does is split into a circle and a line. It is one curve, passing through both eyes, running up past the fixation region and on. In the classical analysis of the horopter that curve is a twisted cubic, of which the circle and line are the degenerate case; the figures sample it and verify every sampled point, and they do not test its degree.

Under Fick’s rule the two eyes’ rolls differ by more than they do under Listing’s law at this fixation — the measured consequence is the larger slide — and its relative motion slides 7.09 mm. The picture is the same kind of object as Listing’s: one curve, no line.

Why a roll becomes a slide #

It is worth seeing why a difference in roll, which is a turn, produces a slide, which is not.

Each eye’s roll depends on its own gaze direction, and the two eyes’ gaze directions differ — they converge on a point from 65 mm apart. Under Fick’s rule and Listing’s law, that small difference in direction produces a small difference in roll, and the two picture frames stop sharing a plane: each is rotated slightly about its own line of sight, by a different amount.

The motion carrying one frame onto the other must now do three things at once: move one centre onto the other, turn one line of sight onto the other, and undo the difference in roll. Chasles’s theorem says that any such motion can be written as a turn about some axis followed by a slide along it, and a slide of exactly zero is a special case that happens only when the three requirements are compatible with a pure turn. Helmholtz’s rule is built so that they always are. The other two are not built with binocular geometry in mind at all, and for them the compatibility fails as soon as the fixation is raised and aside.

The slide is the baseline, seen along the axis #

There is a shorter way to say which rules slide, and it turns the behaviour of all three into one line of arithmetic.

The relative motion carries the left eye’s centre onto the right eye’s. Written as a turn about an axis followed by a slide along it, the slide is the part of that displacement pointing along the axis. Put the two eyes either side of the midpoint between them, at −R and +R, and call the axis direction û and the turn Q. The turn leaves its own axis where it was, so the part of QR lying along û is the same as the part of R lying along û, and

$$s \;=\; \hat u\cdot(\mathbf R - Q\,(-\mathbf R)) \;=\; \hat u\cdot\mathbf R + \hat u\cdot Q\mathbf R \;=\; 2\,\hat u\cdot\mathbf R \;=\; \hat u\cdot(\mathbf R - (-\mathbf R)).$$

The last expression is the baseline — the 65 mm segment from one eye to the other — projected onto the axis. The slide is zero exactly when the axis of the eyes’ relative turn is perpendicular to the line joining them.

That settles Helmholtz’s rule without any further argument. Its two picture frames share the plane of regard, the axis of their relative turn is perpendicular to that plane, and the line joining the eyes lies in it. An axis perpendicular to a plane is perpendicular to every line in the plane, so the baseline has no component along it, and the slide is zero at every fixation there is.

It also reads the other two rules’ slides straight off their axes. Under Fick’s rule, 20° aside and 20° up, the axis of the relative turn has a component of 0.109 along the line joining the eyes; 0.109 of 65 mm is 7.09 mm, which is the slide the figure prints. Under Listing’s law at the same fixation the component is 0.0605, and 0.0605 of 65 mm is 3.93 mm. A rule slides by exactly as much as it tips the axis of the eyes’ relative turn toward the baseline — and a rule tips it only when the two eyes’ rolls differ, which is the geometric content of the paragraph above.

The expression also explains why the sliding rules stop sliding straight ahead and in the plane of the eyes. In both places the two eyes’ gaze directions are mirror images of each other across the median plane, their rolls are mirror images too, and a relative turn between mirror-image frames has its axis in the median plane — perpendicular to the baseline, whatever rule produced the rolls.

How far the slide goes across the field #

The slide is the whole measurement: zero means a line survives, anything else means it does not. So the figures sweep it.

At 20° above the plane of the eyes, from straight ahead to 40° aside, Helmholtz’s rule never slides. Listing’s law starts from zero straight ahead and grows faster the further out it goes — 0.93 mm at 5°, 3.93 mm at 20°, and 9.50 mm at 40°, the last five degrees adding 1.67 mm. Fick’s rule grows faster, to 12.96 mm.

Halve the elevation to 10° and both sliding rules slide a little over half as far: Listing’s law 5.22 mm at 40° aside, Fick’s rule 7.07 mm. Helmholtz’s rule stays at zero, to the arithmetic floor, at every setting of both sweeps.

The numbers are millimetres against a fixation distance of more than a metre, and it would be easy to call them small. They are small in the sense that the curve replacing the line may pass near where the line was over a short stretch. They are not small in the sense that matters for the structure: a slide of one micrometre fixes no point, exactly as a slide of a centimetre fixes none. Whether the horopter contains a straight line is not a matter of degree, and the measurement that decides it is whether this number is zero.

What depends on the rule and what does not #

The pieces separate cleanly, and it is worth listing them.

Independent of the rule: the horopter’s trace in the plane of regard, for any fixation in the plane of the eyes, is the circle through both eyes and the fixated point; the relative motion for such a fixation is a pure turn about a vertical axis; that axis lies in the median plane, on the circle’s straight-ahead point, not at the point being looked at.

Dependent on the rule: whether a raised and turned fixation keeps any line. Helmholtz’s rule keeps one at every fixation; Listing’s law and Fick’s rule keep one only straight ahead, only in the plane of the eyes, or only in the vertical plane through the midpoint between them.

That division is worth stating for a stereo instrument, where the rule is not a fact about physiology but a choice somebody made. Turning the cameras inwards measured the vertical disparity a toed-in rig produces; a rig whose cameras are mounted on a tilting head that keeps their horizontal axes parallel to the baseline is a Helmholtz rig, and one whose cameras pan and tilt independently on gimbals is closer to Fick’s. The first keeps a circle-and-line horopter at every setting. The second does not, and nothing in either pair of pictures announces which was built.

It is also worth being clear what this does not touch. The pictures’ horizontal disparities and the depths depth is a reciprocal derives from them are unaffected by any of it, since the rule concerns only a roll about each line of sight. The relative pose that two pictures determine is the motion measured here; a pose-recovery routine run on a Listing pair would return a turn with a slide and be right to. And the eyes’ baseline and the fixation distance, which set every number in the straight-ahead result, set the scale of the slide and do not decide whether it is zero.

A degeneracy that is the usual case #

There is a habit, in recovering scenes from pictures, of calling an arrangement degenerate when some recovery has no answer there — a turn of the head, a flat scene, a pair far enough away to be one eye. The circle and line are a degenerate case in the other direction: a special configuration in which a general object, the curve, splits into simpler parts.

What is unusual about this one is how ordinary the special configuration is. Every fixation straight ahead, every fixation in the plane of the eyes, and every fixation of a Helmholtz pair lands on it. The general curve appears only when the gaze is raised and turned and the eyes roll by a rule that does not keep a shared plane. The textbook drawing of the horopter as a circle is not wrong; it is the picture of the most common special case, drawn without the one condition that makes it special.

Still open: how far the curve is from the line #

This essay establishes that a Listing or Fick pair loses the line and measures the slide that decides it. It does not measure how far the replacing curve lies from where the line would be, and that is the natural next quantity, because it is what a viewer or an instrument would actually notice. The measurement it wants takes the vertical line a Helmholtz pair would have — through the circle’s point in the median plane of the tilted plane of regard — and measures, for a Listing pair at the same fixation, the largest disparity in pixels anywhere along a stretch of it, across a grid of fixations up to 40° aside and 20° up. If that disparity stays below a pixel over the region where people and rigs usually look, the line survives as an excellent approximation and the slide is a structural fact without a practical one; if it does not, the circle-and-line drawing is wrong exactly where binocular work is hardest.