Both coordinates agree on a circle and a line

The depth a pair calls zero found, by bisection along 121 azimuths, that two eyes verged on a point agree — the same image coordinate in both pictures — on a circle through both eyes and that point. It was a plan: every search ran in the plane of the eyes, and in that plane the answer is the Vieth–Müller circle to the last digit the fit could print.

Near the end the essay allowed itself one sentence about the third dimension. In space, it said, the full zero-disparity locus is a surface, and for a verged pair with no roll it is the circle crossed with a vertical line: a cylinder.

That sentence joins two different sets under one name, and taking them apart is the whole of this essay. A picture has two coordinates. The set of points at which the eyes agree about the horizontal one is indeed a cylinder. The set at which they agree about both is not a surface at all. It is the circle and a single vertical line, and the line has a reason for being there that the plan could never show.

Two coordinates, two questions #

A world point seen by two eyes lands at a position in each picture, and a position is two numbers. The difference between the two pictures’ horizontal numbers is the horizontal disparity, the quantity every stereo depth measurement is built from. The difference between their vertical numbers is the vertical disparity, and in most of what has been written here it has been zero — by construction, because a rectified pair of cameras looking the same way cannot produce any.

A verged pair is not rectified. Each eye is turned inward to face the fixation point, so their two picture planes are not parallel, and a point above or below the plane of the eyes can land at different heights in the two pictures. Turning the cameras inwards met this from the display side, where a stereo pair built by rotating two cameras toward a common point puts the same world point up to thirty pixels apart vertically, and concluded that such a pair is a pair of pictures of no scene.

So “the eyes agree” is two conditions, and the horopter proper — the set of points both eyes see at the same place — needs both.

Where the horizontal coordinates agree #

The first search repeats the search in the plane of the eyes exactly, raised off it. Along each of 27 azimuths from 26° left to 26° right, and at heights of zero, ten, twenty and thirty centimetres above the plane of the eyes, the distance at which the two horizontal image coordinates agree is found by bisection.

The distances do not depend on the height. At every azimuth the four searches return the same distance, and they return it not to a tolerance but exactly: the spread across heights is zero metres. That is a cylinder standing vertically on the Vieth–Müller circle, which is what the earlier essay’s sentence described.

The exactness has a reason, and it is worth having because it would otherwise look like luck. Each eye here turns about a vertical axis to face a point in the plane of the eyes, so each picture’s horizontal axis stays horizontal and its vertical axis stays vertical. A picture’s horizontal coordinate is the ratio of a point’s offset along the picture’s horizontal axis to its distance along the line of sight, and both of those are measured in the horizontal plane. Raising the point changes neither. The horizontal disparity of a point is therefore the same as that of its foot on the plane of the eyes, and the set where it vanishes is every point directly above or below the circle.

Where the vertical coordinates do not #

The same searches read the vertical disparity at every point they found, which is the quantity a search confined to the plane never looks at.

In the plane of the eyes it is zero at every azimuth, as it must be: a point at the height of the eyes projects onto the horizontal line of both pictures. Straight ahead it is zero at every height. Everywhere else on the cylinder it is not zero, and it grows with both the height and the angle aside: 0.74 px at ten degrees and ten centimetres up, 1.67 px at twenty degrees and the same height, 5.02 px at twenty degrees and thirty centimetres up, and 7.35 px at the edge of the searched field.

The mechanism is size. A picture’s vertical coordinate is a point’s height divided by its distance along the line of sight. A point in the median plane — straight ahead of the midpoint between the eyes — is equally far from both eyes, so its height divided by either distance is the same number and the vertical coordinates agree. A point off to the right is nearer the right eye than the left one. Its height is the same in both, its distances are not, and the nearer eye sees it higher up in the picture. The difference is the ratio of the two distances, which is one only on the median plane.

So the cylinder is the set where horizontal disparity vanishes, and the vertical disparity on it vanishes along exactly two curves: the circle, where the height is zero, and the vertical line through the fixation point, where the distances are equal.

The set where both agree, drawn #

Put the two conditions together and the horopter in space is what is left.

The figure draws it obliquely, over a patch of the plane of the eyes. The circle runs from one eye round through the fixation point to the other, and a vertical line rises and falls through the fixation point itself. Nothing else in space is seen at the same place by both eyes.

The figure does not draw what a formula says the horopter should be. It computes the locus from the eyes’ own geometry, then projects every sampled point back through both eyes and checks that both coordinates agree — to two millionths of a pixel, the arithmetic floor of the projection. A drawn point that failed that check would be a failure of the figure rather than part of the picture.

Seen from directly above, the result looks exactly like the search in the plane: a circle, with the line collapsed to a single dot at the fixation point. That is why the plan could not distinguish between a cylinder and a circle with a line. Both project onto the same circle.

Why there is a line at all #

The circle has a classical explanation, the inscribed-angle theorem, and the depth a pair calls zero gave it in a paragraph. The line needs a different one, and it is the more useful of the two, because it survives when the circle’s explanation does not.

Each eye’s picture is a camera frame: a centre, a direction it faces, and a roll about that direction. There is a rigid motion that carries the left eye’s frame exactly onto the right eye’s — its centre onto the right eye’s centre, its axes onto the right eye’s axes. A point that this motion leaves where it was is, by construction, at the same position relative to both frames, and so appears at the same place in both pictures. Points fixed by the relative motion are on the horopter.

By a theorem of Chasles, every rigid motion is a turn about some axis followed by a slide along that axis. If the slide is zero the motion is a pure turn, and every point on its axis stays put: the axis is a line of the horopter. For two eyes that turn about vertical axes to fixate a point in their own plane, the relative motion is a pure turn about a vertical axis — the vergence angle, 3.10° here — and the figure’s computation finds that axis passing through the fixation point with a slide of exactly zero.

That is the line. It is not a feature of the circle and it does not come from the inscribed-angle theorem. It is the axis about which one eye’s view of the world is turned into the other’s.

The two explanations then fit together. Every point on the circle is seen at the same place for the inscribed-angle reason, and every point on the axis is seen at the same place for the fixed-point reason, and the general account — which the figure’s computation uses throughout — contains both: for each positive number μ it gives one point seen in the same direction by both eyes, and as μ runs over its range those points trace the circle, except at one value of μ where the whole axis is fixed at once.

A nearer fixation, and a further one #

The vertical disparity that separates the cylinder from the horopter is not a constant of human eyes. It depends on how near the fixation is, and it depends on it strongly.

Bring the fixation point in from 1.2 m to 60 cm and the vertical disparity at the corner of the field rises from 7.35 px to 29.26 px, very nearly four times as large for half the distance. The reason is the same ratio of distances: at arm’s length the difference between a point’s distance from the left eye and from the right eye is a larger fraction of either, so the heights they see differ by more. A point thirty centimetres above a book held at reading distance is nowhere near the horopter even when it is directly above a word being read.

Push the fixation out and the effect falls away.

At 2.4 m the horopter is the same two pieces on a larger circle, and the vertical disparity on the surrounding cylinder is a quarter of its 1.2 m value. The horopter’s shape does not change with distance; the penalty for being off it, in the vertical coordinate, does. That is the vertical-disparity counterpart of the bow that essay measured in the horizontal one, where a wall at the fixation distance departs from the circle by the fixation distance times sin² of the angle aside: both effects are largest for near work and wide fields, which is where stereo vision and stereo instruments are most used.

The size of it, in one expression #

The three searches give three numbers — 7.35 px at 1.2 m, 29.26 px at 60 cm, 1.84 px at 2.4 m, all at the corner of the field — and they are worth turning into a formula, because a formula says which of the arrangement’s quantities the effect depends on and how.

Put the eyes a baseline b apart, fixating a point D straight ahead, and take a point on the cylinder at an angle φ aside and a height h above the plane of the eyes. Its horizontal distance from the midpoint between the eyes is D cos φ, which is what the searches found — 1.128 m at 20° aside for a 1.2 m fixation. Its distances along the two eyes’ lines of sight differ by about b sin φ times that distance over D, because each eye’s line of sight is turned inward by about b/2D. A vertical image coordinate is f h divided by the distance along the line of sight, so the difference between the two eyes’ vertical coordinates is

$$v \approx \frac{f\,h\,b\,\sin\varphi}{D^{2}\cos^{3}\varphi}.$$

At the corner of the searched field — 26° aside, 30 cm up, 65 mm of baseline, 900 px of focal length, a 1.2 m fixation — the expression gives 7.36 px, against the 7.35 px the search measured. Halving D to 60 cm multiplies it by four, to 29.4 px, against 29.26 measured; the small shortfall is the approximation beginning to strain at arm’s length, where b/2D is no longer small. Doubling D to 2.4 m divides it by four, to 1.84 px, which is what the search found.

Three things follow from the form, and one of them runs opposite to the result in the plane.

It is proportional to the height, so the cylinder and the horopter meet only in the plane of the eyes, and the vertical disparity grows linearly as a point rises off that plane.

It is proportional to the baseline. The bow that the earlier essay measured — the depth by which the horopter departs from a flat wall — does not depend on the baseline at all, because it comes from the circle’s shape and the circle’s diameter is the fixation distance. The vertical disparity does, because it comes from the two eyes being at different distances from the point, and that difference is the baseline’s doing. Eyes twice as far apart see twice the vertical disparity on the same cylinder.

And it falls as the square of the fixation distance, twice as fast as the bow. The horopter’s three-dimensional structure is therefore a near-field phenomenon in a stronger sense than its plan: at reading distance it is a matter of tens of pixels, and across a room it is below the precision most instruments read to.

What the vertical difference is worth #

A reading of this that stops at “the horopter is smaller than it looked” misses the constructive half.

Vertical disparity is a quantity a pair of pictures contains, and on the cylinder it is determined by two things: how far aside the point is and how near the fixation is. Given the angle aside — which a single picture already supplies — a measured vertical disparity says something about viewing distance that horizontal disparity, which is zero on the cylinder by definition, does not. The geometry makes that information available in the two pictures without any knowledge of the scene. Whether a visual system uses it is a question about vision rather than projection, and this site leaves it where it belongs.

What belongs here is the instrument side. A rectified camera pair throws this information away before it starts, because rectification makes every vertical disparity zero; a point is a line over there is written for pictures in which a match’s search is along a horizontal line. A verged pair keeps it, and pays for keeping it with a search that is no longer along a row. Neither is wrong. They are two different decisions about which of a picture’s two coordinates to spend on depth, and what the two eyes are sent is where the display that has to satisfy both was taken apart.

It also sharpens what depth is a reciprocal and the range a pair cannot see past mean by disparity. Both were written for a rectified pair, where the word has one meaning. For a verged pair it has two, and a stereo measurement that reports only the horizontal one is reporting half of what the pictures differ by.

What the circle and line depend on #

The result is exact for a stated model, and the model has one assumption that the figures have so far been careful never to test.

Each eye is a pinhole whose centre of projection is its centre of rotation, turned to face the fixation point. Fixating a point fixes where the eye looks and does not fix how far it is rolled about that direction; a camera aimed at a point can still be turned about its own line of sight. For every figure above the fixation lies in the plane of the eyes, and there every reasonable rule for that roll gives the same answer — the eye turns about a vertical axis and does not roll at all — so the choice was never made because it never mattered.

It matters as soon as the gaze leaves the plane. An eye looking up and to the side has a roll that depends on which rule it follows, and different rules give relative motions that are not pure turns: they slide along their axis. A motion that slides fixes no point, so it has no axis line on the horopter, and the circle-and-line structure has nothing left to hold it together.

Two further limits stand as they did for the circle alone. This is the geometric horopter, not the empirical one a person reports; and it is computed for pinhole eyes 65 mm apart with a 900 px focal length, so the pixel values scale with that focal length while the shape of the locus does not.

Still open: whether the line survives a raised gaze #

The one assumption above is the thing to vary. With the fixation point turned 20° aside in the plane of the eyes, the line survives under every rule but moves: it stays in the median plane, at the point of the circle straight ahead of the eyes, rather than passing through the point being fixated. Raise the fixation 20° as well and the rules part. An eye that rolls by Helmholtz’s rule keeps a shared plane of regard and a line; an eye that follows Listing’s law, or Fick’s rule, produces a relative motion that slides — 3.93 mm and 7.09 mm at 20° aside and 20° up — and the horopter loses its line and bends into a single curve through both eyes. Raise the gaze, and the line is gone measures that slide across the field for all three rules, and draws the curve that replaces a circle and a line.