Divergence — the series
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The picture whose lines spread
In a Byzantine icon the sides of a table diverge with depth. The standard account says the vanishing point is behind the viewer. It is not — it sits below the near edge, in front of the eye, and it is the vanishing point of a direction running down and away.
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An inverse perspective is a leaning plane
Ask a divergent construction what solid it depicts and it answers: a rectangle, four right angles, near edge equal to far. What the splay encodes is not the shape but the plane's tilt — and a real square on a plane leaning toward the camera really does photograph with its far edge wider.
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Four surfaces, and no one camera that draws them
Read under one assumed camera, the floor, footstool, table and book of a constructed divergent picture imply tilts of 36.7°, 65.1°, 54.2° and 78.2°, where one camera photographing four parallel surfaces gives each of them 35.0°. But the spread between the tilts is 46.8° under a 260 px lens and 6.7° under a 5,000 px one, so it measures the lens as much as the picture. The measure that owes nothing to a lens is on the page: the nearest drawing one camera could make moves the far corners by 17.6 px.
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A camera count needs a tolerance
Asked how few cameras could have drawn a splayed picture, a constructed stack answers four, three, two or one depending on how many pixels of redrawing a reader will allow — one camera at 17.6 px, two at 11.1, three at 4.6. The count is real: a picture built with two groups hidden in it gives back exactly those two, anywhere between nothing and 47.0 px. What it is not is a property of the picture alone, and the floor under it belongs to the hand — a one-camera drawing made by a hand that scatters its far corners by σ splits below about 0.57σ, where the 0.57 belongs to that hand and runs from 0.17 to 1.34 across a hundred and twenty of them.
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One camera means one horizon, not one point
The test this field has been using asks whether a picture's surfaces share a meeting point. One camera photographing four parallel surfaces turned by different angles in their own planes gives them meeting points 1,065 px apart in column and identical in height to 3 × 10⁻¹² px — so the shared-point test charges 28.7 px to a picture one camera really took, and the charge grows with the turn. What one camera imposes is a shared vanishing line. The earlier verdicts survive intact, and for a narrower reason than they looked to have.
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The rows under a splay measure the bays, not the lean
A splayed strip drawn with its rows evenly spaced is still an exact picture of one flat plane leaning toward the eye — the straight sides fix the plane, whatever the rows do. What the rows decide is how deep each bay is, and evenly spaced ones make the near bay deeper than the far by nearly the square of the splay. That, and the six-per-cent gap between even rows and a camera's, depends on the splay alone — not on the lean, the distance or the lens.
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The rows count hands, not cameras
Rows drawn between two straight sides charge a camera nothing — every strip of a four-strip divergent picture reads back as a flat plane, whatever placed them, so a single viewpoint redraws the whole picture exactly. What they do fix is one number per strip, and that number survives the lens. Read the same drawing at focal lengths twenty to one apart and the lean runs from 36.4° to 86.0° while the habit stays at 1.000000000.
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A vanishing line with a slope in it
Turn the plane about the view direction and no family of any surface's edges is level; each vanishing line acquires a slope, and a group must agree about two numbers rather than one. The count does not change character — a hand of four pixels costs the test 3.00 px at no slope and 3.27 at thirty-eight degrees of it. What the slope does expose is the redraw: holding each far edge at its drawn height charges 0.95 px to a picture one camera really took.