Stereo pair — where it appears
Named by 23 essays across 6 fields — each of them below, with the objects they name alongside it.
Far enough away, a pair is one eye
Hold the baseline and walk the scene away, and the parallax a single homography cannot explain falls as the distance to the power −0.968 — one over the distance, which says the ratio of baseline to depth is the whole of it. The recovered translation direction follows it down, from 3.3° at four metres to 74.5° at two hundred and fifty-six.
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.
A longer lens buys parallax and not the direction
Walk a pair away from a scene and the parallax a homography cannot absorb falls. Lengthen the lens to keep the scene filling the frame, and one power of that loss comes back, exactly. The recovered direction between the cameras does not come back with it. At the same parallax in pixels, a long lens is three times further from the answer, because what fixes a direction is the angle the marks span and no lens can widen that.
One shutter, two views
A photograph with a mirror in it is a stereo pair, and a peculiarly well-behaved one. Its fundamental matrix is skew-symmetric, so both epipoles are the same point; that point is where the camera would see its own lens; and every line joining a mark to its reflection passes through it, to 1.4 × 10⁻¹² px. The baseline is twice the distance to the glass, which is the one number a single view cannot supply and a tape measure can.
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.
A scroll through two slits ranges in a straight line
Draw a scroll twice, through a slit leaning 10° forward along the track and one leaning 10° back, and every point appears in both drawings on the same row, separated by 9.169 px for every metre of its depth — at four metres and at fifty-two. Depth is proportional to that separation rather than reciprocal to it, so a pixel of error costs 10.9 cm at every distance, averaging leaves no bias, and there is no range past which the depth runs off to infinity. The price is paid in roll: a 100 m scroll ranges nothing past 283.6 m.
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.
A point under water has two depths
The apparent depth of a submerged point is not one number even along one line of sight. The rays it sends to an eye pass through two focal lines, and at 60° from the vertical a point 1.50 m down has an image 0.740 m down and another 0.320 m down. Two eyes side by side read the first, a head moving up and down reads the second, and a pair of eyes tilted between them reads neither — their two rays miss each other by as much as 6.39 mm.
The stick a stereo pair puts back
Two eyes side by side reconstruct a submerged stick exactly as the sagittal image — kinked 14.96° — and two eyes one above the other exactly as the tangential one, kinked 9.59°. Roll the baseline between them and the two rays to a point miss each other by up to 3.13 millimetres, past the 2.85 a pixel covers at that range, and the reconstruction is a third stick that is neither — 551 millimetres of a one-metre stick, with its tip at 0.405 metres against a true 0.866.
Two mirrors are three cameras
A photograph with two mirrors in it holds three views of the scene from three places, at baselines of 2.90, 3.10 and 2.26 metres. Two of the three pairwise geometries are mirror pairs and are skew; the third is a rotation by twice the angle between the mirrors, and it is not new evidence — five numbers read off the print rebuild it to 9.4 × 10⁻¹⁴ pixels, where a general three-view arrangement needs eighteen.
The second disparity cuts cells
A point off the plane of the eyes has a vertical disparity as well as a horizontal one, and quantising both, on an 86,400-point lattice of a room, gives 7,663 labels where one coordinate gives 179 — a count that belongs to the lattice rather than the room, as the essay after this one found. The gain is entirely vergence's — two eyes looking straight ahead have no vertical disparity at all, exactly — and it is largest where the first reading is already finest: 60.8 in the near metre and 3.7 in the far band.
The residual does not warn
Fit a straight line to the stick a stereo pair puts back and the fit looks best exactly where the reconstruction is least supported: the residual is 0.535 mm at a level baseline, where the two rays meet perfectly, and 0.038 mm at sixty degrees of roll, inside the band where they miss by more than a pixel covers. Over the same sweep the fitted line is 285 to 584 mm short of the stick's metre — fourteen thousand times its own residual at worst.
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°.
A symmetric object is its own stereo pair
A building with a plane of symmetry photographed once gives fourteen correspondences whose joining lines meet at one point to 1.9 × 10⁻¹² pixels, a skew-symmetric matrix, and the object's whole shape to fifteen digits — with no mirror anywhere and no second exposure. What it does not give is the size, and the instrument that decides whether any of it applies is the same meeting point, which opens to 12.7 pixels when the symmetry is half a per cent out.
A fit weighted by the miss trusts only the surface
Weight each point of a reconstructed underwater stick by how well its two rays meet, and the fit hands all but a ten-billionth of its trust to the one point where the stick enters the water — and reports a residual of nothing at every rolled baseline. Floored at the reading error, the weighting changes the answer by a few thousandths of a millimetre. And the miss itself, the one honest number, is exactly zero at a level baseline where the stick comes back 285 mm short.
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.
Drawing for the curve removes the curve, not the nearness
A stereo pair drawn for a flat screen and shown on a curved one sends a level head 31.8 arcminutes of vertical disparity before anyone tilts anything. Draw the pair for the curve and that vanishes exactly. What remains when the head rolls is the roll's own disparity, and on the curve it is 8.6 per cent worse than on a flat screen — not because of the curve's shape but because the arc brings its edges nearer the eyes, and a fixed separation seen from nearer is a larger angle.
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.
Two wide pairs are most of a symmetry
A facade with a symmetric frame and an asymmetric middle is still its own stereo pair. Two pairs whose halves stand wide apart find the symmetry plane to 0.40° with marks read to 0.4 px; twelve pairs find it to 0.26°; two narrow pairs only to 1.35°. The middle that pairs with nothing comes back too, to 0.8 per cent, carried along its rays to the wall the pairs have fixed. The risk is not too little symmetry but a false pair — and a window set wider than its partner is one the standard test cannot see.
The wall convicts a pair set too wide
A window set wider than its partner's reflection keeps every joining line on the mirror's point, so the one test a symmetric photograph was known to carry cannot see it. The wall the other pairs fix can: it says where the partner should be seen, and a window set ten centimetres too wide misses by 6.0 pixels. With marks read to 0.4 px it convicts a 3-centimetre offset eight or nine times in ten and a 5-centimetre one every time. What no single photograph can see is a move along the camera's own ray to the window.
A false pair is named by its neighbours, not by itself
Read against the rest of a symmetric facade, a pair whose right half has moved six centimetres is explained exactly by two stories: the right half moved, or the left half moved the mirror way from a pair standing six centimetres over. Predicted from left to right and from right to left, the pair gives one residual twice. What names the half is a mark that shares a line with it — the window's upper corner names a move across ninety-three times in a hundred at two centimetres — and what the readings recover is the move as the camera sees it, blind along its ray.
Named alongside it
The objects these essays reach for when they reach for this one.
BaselineBinocular disparityVergenceDisparityEpipoleFundamental matrixHoropterMirror planeTriangulationCorrespondenceDegeneracyReconstruction