Vergence — where it appears
Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.
The depth a pair calls zero
Two eyes verged on a point agree — the same coordinate in both pictures — not on a plane at the fixation distance but on a circle through both eyes and that point. Found by bisection along 121 azimuths and fitted rather than assumed, it is a circle to 0.0000 cm; at 26° off centre it lies 23 cm nearer than a flat wall does.
Both coordinates agree on a circle and a line
Two eyes fixating a point straight ahead see their horizontal image coordinates agree on a whole vertical cylinder over the Vieth–Müller circle, the same radius at every height to the last bit. Their vertical coordinates agree on almost none of it — 7.35 px apart at 26° aside and 30 cm up, 29.26 px when the fixation is brought to 60 cm. The points where both agree are the circle and one vertical line, and the line is the axis of the motion that carries one eye onto the other.
Raise the gaze, and the line is gone
Turn two eyes 20° aside in the plane they share and the horopter keeps its vertical line — but the line stays in the median plane, 1.277 m ahead, not at the point being looked at. Raise the gaze as well and the rule by which each eye rolls decides the rest: Helmholtz's rule keeps a line; Listing's law and Fick's rule make the eyes' relative motion slide 3.93 mm and 7.09 mm along its axis, no point stays put, and the horopter becomes one curve.
Vergence moves the shells and does not respace them
Turn two eyes inward and the depths a whole-pixel reading can report stop being planes and become a family of near-circles through both eyes — the twenty-pixel shell standing at 0.74 m forty degrees aside where a parallel pair puts it at 3.82. The spacing between consecutive shells is the same to 0.07 per cent across the whole field, so vergence relabels the rays and does not sharpen them, and the resolution argument for turning the eyes in does not exist.
A stereo picture is drawn for a level head
Every stereo pair is drawn for two eyes level with each other — a point's two images share a row and differ only across it. Tilt the head 10° in front of a television and that difference turns partly vertical, 14.46 arcminutes for anything drawn at infinity, and the two sightlines to a point stop meeting. At a desk monitor the same fifteen-arcminute limit arrives at 2.58°.
Rectifying a pair spends what its epipolar lines lean
Counted from the two disparities alone, rectifying a verged pair looks as if it throws away at least 99.7 per cent of what the pair can tell apart. That cannot be true of a warp that loses no ray, and it is not: once the place in the picture is counted, the vertical disparity adds 4.1 per cent for eyes verged at 1.2 m, and rectifying at the same focal length gives back all but 4.0 of it.
A curved screen tilts a stereo pair both ways at once
A stereo pair drawn for a flat screen and shown on a curved one gets a vertical disparity from the curve alone, with the head level — opposite in sign either side of the middle, and 31.8 arcminutes at the corners of a curved desk monitor for anything drawn at infinity. A rolled head adds its own, of one sign everywhere. The two add in one pair of corners and cancel in the other, so the curved screen is kinder than a flat one in two corners and harsher in the two that decide.
A sliding pair keeps its line only near the middle
Two eyes that roll by Listing's law lose the horopter's straight line as soon as they look up and aside, and the question worth a number is whether they lose it by much. Measured along the line they nearly keep, the disparity is exactly the slide seen by one eye — 3.20 px at 20° aside and 20° up, a metre and a bit away — and the slide does not shrink with distance. At arm's length the line survives to within a pixel only in a narrow cross through the middle of the field.
A quarter of a degree of roll keeps the line
A Listing pair's horopter loses its straight line because the motion between the eyes slides along its own axis. Helmholtz's rule cures that by turning each eye several degrees. A roll a tenth that size, in opposite senses, cures it just as exactly, and its size is a short law: a quarter of the elevation times the vergence, with hardly any dependence on the azimuth.
Named alongside it
The objects these essays reach for when they reach for this one.
Binocular disparityStereo pairDisparityHoropterBaselineDegeneracyRelative poseViewing positionDepth uncertaintyinstrument limitInterocular distanceQuantisation