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

Keystone — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 15° of yaw, 6° of pitch, 1.50 throw ratio81.3% of the panel reaches the corrected rectangleouter: the thrown quadrilateral · inner: what correction can keep18.7% of the panel discarded

    A projector is a camera run backwards

    Turn a projector fifteen degrees from square and it throws a trapezium; keystone correction cannot add light outside it, so it shrinks the picture until it fits and discards a sixth of the panel. And the instrument itself comes back out of the picture it threw — 2880 panel pixels recovered against 2880, by the function the wrong field wrote for hand-drawn cubes.

    part 1 · screen
  2. -7.50-5-2.5000123how far the projector stands from the eye, log₁₀ mmwhat the seat is left with, log₁₀ pixels1 pxcurved television as a wall, the seat at 3 ma pixel by 50 mm · exact at zero

    A projector in the viewer's eye

    A projector paints a wall along its own rays, so from the projector's own position the wall's shape is invisible — exactly, on a cylinder, on a dome and on a plane alike. Move a hand's breadth away and the residual is pixels. The one place a projector can stand and ignore the shape of what it is throwing onto is the eye of the person watching.

    part 2 · screen
  3. corrected from the picture — 0e+0° between the verticalscorrect from 20 cm, at 160 mm wideagrees with a level camera at the same eye to 3e-13 px

    Straightening does not move the eye

    Correct a photograph's converging verticals and what comes out agrees with a level camera at the same point — one the correction was never shown — to 3e-13 px, with the verticals parallel to 0e+0°. The cross-ratio of four points along a ground line reads 1.3333 before and after, so the corrected picture measures exactly what the original measured, from exactly where the original was taken and nowhere else.

    part 4 · construction
  4. near edge: 0.98far edge: 0.76turned 15°, corrected to 16:9, source at panel resolutiondark: more panel per source pixel

    Keystone correction spends the panel unevenly

    Turn a projector fifteen degrees and keystone correction throws away 18 per cent of its panel — the number a specification quotes. It is the smaller half of the price. What remains is spread unevenly: the near edge of the corrected picture gets a panel pixel for every source pixel, the far edge three-quarters of one, so the far edge carries 58 per cent of the source's detail. A tipped projector keeps more than a turned one, and a lens shift that places the same picture loses nothing at all.

    part 5 · screen
  5. near edgefar edgeturned 15°, every 96th panel column and row82.2% of a frame

    An even spend of the panel is an uneven picture

    A keystone correction that spends the panel evenly exists: render the picture straight into the panel pixels that reach the corrected rectangle, one sample each. At fifteen degrees that is 82.2 per cent of a frame's samples, and on the wall it delivers exactly what a full frame does — 0.99 of the source's detail at the near edge, 0.76 at the far — because the panel, not the source, was setting the detail everywhere. It removes no unevenness. On fewer samples it makes the picture worse where it was worst; the even picture is a uniform source of 57.6 per cent of a frame.

    part 6 · screen

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