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

Distortion — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. principal pointk₁ = -0.320, k₂ = 0.110 — barrel distortioncentre line 0e+0 px of sag, outermost 17.8 px

    Straight lines that are not

    Everybody says the edges of a wide-angle frame bow. Nothing is special about the edge. A radial map moves every point along its own radius, so the only line it leaves straight is one through the principal point, and the bend of every other is decided by how far it passes from that one place.

    part 1 · lens
  2. 00.50011.502-0.400-0.20000.200k₁departure of the cross-ratio from the world's value (%)pinhole1.29%the pinhole's own error, on the same four points2e-16 — the control

    A lens destroys the invariant

    The cross-ratio is the one thing a projection preserves, and nearly everything checkable about a photograph is checked with it. A lens returns it one and a third per cent out where the pinhole is exact to fifteen digits — and the height error that follows tracks a quantity nobody would guess.

    part 2 · lens
  3. k₁ = -0.28 · the round trip closes to 8.0e-13 pxpale: what the eye receives · dark: what the renderer drawsthe inner 86% of a 72° frame, where the inverse is exact

    The render is distorted on purpose

    A headset renders a bent picture so its lens can straighten it, which is a lens's distortion polynomial run backwards, and the one case in which distortion is introduced deliberately. The round trip closes to a thousandth of a millionth of a pixel, and the price is that one rendered pixel becomes 0.493 delivered pixels at the edge of the field and one at the centre.

    part 3 · screen
  4. 00.50011.50020406080field angle off the axis (degrees)picture radius, in focal lengthsfolds at 47.49°43.91°50.54°the radial factor reaches zeropinholefolds at 47.49°43.91° and 50.54° share one radius

    A barrel model folds at a radius it sets itself

    The polynomial every calibration fits to a wide lens stops increasing at a radius fixed by its own first coefficient — 47.49° of field at k₁ = −0.28 — and past it two directions land on one picture radius. The routine that undistorts pictures with it does not refuse there. It hands back wrong directions from 46.75°, by as much as 106.5°, and refuses only at 65.5°: a fifth of the field returned silently wrong.

    part 4 · lens
  5. principal point moved 22.30 px · focal length 0.584 px shorterlines straight to 1e-13 px

    A tilted sensor is not a distortion

    Tilt a sensor 3° out of square with its lens and every point of the picture moves — up to 7.5 px on the frame drawn here — yet every straight line stays straight to 10⁻¹³ px and the cross-ratio survives to 10⁻¹⁶. The picture is an ordinary pinhole picture whose principal point has moved 22.30 px. A calibration that frees its principal point absorbs it exactly; one that holds the principal point and reaches for tangential distortion terms leaves 1.87 px, and used as a correction it bends straight rows by 4 px.

    part 5 · lens
  6. 00.50011.5020123where a pinhole would put the point, in focal lengths from the centrewhere the model puts itwhere the polynomial foldsthe division model's horizonboth at k = -0.42fold 42° · horizon 1.54

    A model that inverts has a horizon instead of a fold

    The polynomial every calibration fits turns around at a finite radius and stops being a map from direction to picture. The division model, chosen because it inverts in closed form, never turns around — it rises for ever toward a horizon at one over the root of its own coefficient, so the whole hemisphere of directions lands inside a finite disc. Fitted to the four fisheye laws over seventy-five degrees it follows every one of them three to five times more closely, and below sixty the polynomial is still the better model.

    part 6 · lens

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