Visual Plane
every picture states the point it is correct from
Concept

Relative pose — where it appears

The rotation and translation between two cameras, recovered from correspondences alone and only up to a scale. Four of them fit any set of correspondences and only one puts the scene in front of both cameras, which is what decides between them.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

0.30.50.7125×10⁻⁴0.0010.0020.0050.010.02how finely each point is read (px, log scale)worst camera-centre error, as a fraction of the track's mean radius (log)chained pairsadjusted togetheradjustment is 4.0–9.2× better6 views · identical observations

A chain and an adjustment

Composing pairwise poses along a sequence is supposed to drift. Measured over five links it wanders instead — one chain ends closer to the truth than its own worst link — and the real cost of chaining turns out to be something else entirely.

manyviews · Drift points in front of both camerasR₁, t0 of 44180° from the truthR₁, t reversed0 of 44180° from the truthR₂, t44 of 44chosenR₂, t reversed0 of 44180° from the truththe chosen pose is the true one to 0.0e+0°cheirality, not algebra, picks it

Four cameras fit, and one of them can see

The essential matrix does not determine a camera pair. It determines four, all of which reproject every correspondence exactly, and the thing that picks one is not more algebra — it is the assumption that the photographer could see what was photographed.

twoviews · Cheirality as recovered0.308 across× 3.4 — same two pictures1.046 acrossworst ratio deviation 6.3e-14size fixed only by a supplied 0.80 m

Two views give shape and no size

Every pairwise distance ratio in a courtyard recovered from two photographs matches the real one to fourteen digits. The courtyard's actual size is not merely uncertain — it is absent, and a reconstruction three and a half times larger fits the same two pictures exactly as well.

twoviews · Scale fixatedListing's law · 20° right and 20° uppitch 3.93 mm: one curve

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.

depth · Horopter -200204060050100150link along the streetscale error relative to the first link (%)average 28.6 %16 streets · ties by mean of ratios · marks read to 1 pxspread after 199 links ±7.3 %

A scale chain leans rather than wanders

A camera driven 200 m along a street, each pair's scale handed to the next through the points they share. Across sixteen streets the scale does not spread the way a random walk would — 7.3 per cent after 199 links, where independent ties would give 108 — and its average leans by an amount the choice of average decides, from +28.6 per cent to −12.2.

manyviews · Drift 0246-20020position along the line, cm from where it crosses the plane of regarddisparity there (px, f = 900)Listing's law, own axisListing's law, Helmholtz lineFick, own axisFick, Helmholtz line1.2 m, 20° aside, 20° upown axis: 3.20 / 6.01 px

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.

depth · Horopter

Named alongside it

The objects these essays reach for when they reach for this one.

BaselineBinocular disparityCorrespondenceDegeneracyDriftHoropterrandom walkreconstruction ambiguityscale ambiguityStereo pairTriangulationVergence

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