Sampling grid — where it appears
Named by 17 essays across 6 fields — each of them below, with the objects they name alongside it.
A pixel is not a point
Where the sample sits inside a pixel is a convention, and getting it wrong shifts every mark by half a pixel in each axis. What that costs can be measured by recovering the camera from the picture — the answer is a principal point exactly 0.707 px from the truth with the focal length untouched, and the other half-pixel mistake does precisely the reverse.
Six flat pictures of everything
There is one way to photograph the whole sphere and keep every straight line straight, and it is to stop using one surface. Six flat pictures at ninety degrees cover everything, each of them a perfect pinhole, and the price is paid entirely at the seams — where a straight line does not bend but kinks, by an angle that reaches 45 degrees and is exactly zero for the lines lying in the seam's own plane.
A curved screen is eight flat ones
A projection matrix is a plane and nothing else, so a curved display cannot be rendered — it has to be driven as several planes and assembled. The gap between chord and arc is the whole error, it goes as the square of the angle each piece spans, and the piece count therefore goes as the inverse root of the tolerance — three for eight pixels, eight for one, fifteen for a quarter.
The evenness a curve buys
A curved screen is sold on evenness, and evenness turns out to be three quantities that disagree. On pixel pitch the curve wins from every seat; on the angle the glass is turned through it wins until three and a half metres along the sofa; on the plain distance from eye to glass — the reading the argument is usually made in — it gives up before half a metre.
A shadow map's texels land by two distances and two cosines
A renderer finds its shadows by taking a second picture from the lamp and storing a depth in every texel. Each texel reaches the screen through the surface it falls on, and how many pixels it covers there is a closed form — two focal lengths, two distances and two cosines. From a lamp beside the eye every texel lands at 0.79 px; from a lamp 40 m ahead facing back, the same map lands texels of 8.69 px on the floor 5 m out.
One warped shadow map, and what it cannot reach
The lamp facing the eye needed a shadow map 9,530 texels square — 90.8 million texels — for none of them to land on the near floor larger than a pixel. Fitted to the floor and warped by one projective parameter, the same map needs 805 thousand; nothing can do better than one texel per pixel of the eye's picture, 148 thousand. What stays out of reach is not the cosine of a surface, which a warp absorbs, but two surfaces that want different densities along one ray from the lamp.
A hole is not preserved
The shadow of a connected object is connected — always, at every lamp position, and for a reason with no geometry in it. A hole survives in neither direction: a ring's shadow closes up at a computable tilt, and an object with no hole in it casts a shadow that has one.
One homography makes a shadow map the eye's picture
A shadow map for a lamp facing the eye needed 706 thousand texels however its rows were re-spaced, nearly five times the one texel per pixel no map can beat. Warp the whole map by a projective transformation, not only its rows, and it needs 154 thousand — within five per cent of the bound — and every texel, carried into the eye's picture, lands at one pixel. The reason is exact: the eye's picture of a floor and the lamp's picture of the same floor are one homography apart.
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.
Ground and sky meet at the horizon without a crack
A ground of rate triangles and a sky of direction triangles share their vertices along the horizon, and rasterised as a graphics processor does it — positions snapped to a fraction of a pixel, every centre given to one triangle by the top-left rule — they lose no pixel and claim none twice, at any roll and any snapping. What the horizon does have is a sliver: a centre that falls within half a snapping step of it goes to whichever side the snapped edge puts it, and one that falls exactly on it, rolled one way, is given to the ground at a weight of zero. A ground stopped at a far plane leaves the crack the shared vertex never does: f·h/D rows.
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.
The eye that reaches the most
A higher eye buys the faces occlusion was hiding and loses design off the far end of the object, so "the best eye" is not a question with an answer until somebody says which of the two they are paying for. On three objects the answer is as high as possible; on a corridor with a doorway in it the two quantities cross and the best height is two and a bit metres.
A ground and a sky share an elevation, not a distance
Read from its own attribute — the ground's rate and weight, the sky's direction — each side of a rasterised horizon reports the elevation of its pixels' rays to 2.2 millionths of a radian, in 32-bit arithmetic as in 64, so a haze painted by elevation crosses the edge without a seam. A fog by distance cannot, and should not: the ground a row below a walker's horizon is 1.2 kilometres away, and at thirty kilometres of visibility the fog steps by 0.85 in one row. A camera's pixel, averaging its area, records a step of 0.69.
The even picture is fair to a centred room, and only to it
Judged by detail per degree of each seat's view, the corrected picture that is even on the wall is already the fairest possible for an audience centred on it: a layout drawn for that audience gains its worst seat nothing. For an audience on the far side it is also the best, because the far edge is where the panel is coarsest and the even picture already spends all of it there. Only an audience on the projector's side can be served better — thirty per cent better at a fifteen-degree turn — by putting the samples on the near edge, where the panel has them to give.
The stretch decides the band
A design band chosen by geometry — the rays that meet the object — includes rays that graze along it, and a grazing ray lands two design points twenty times further apart than the design says. Cap the stretch at four and a bare floor keeps 55 per cent of its design, a corner 80, and the top of a descending flight all of it.
Copying square by square
The taught grid workflow sets a pavement's cell corners out exactly and then fills each cell by eye, and the corners are right while the fill is not — 3.30 px on a picture 690 across at eight cells, falling as the square of the cell. On a wall square to the camera the same fill reads 3e-13 px, which is why the method feels reliable.
Where a shadow splits in two
A gantry's shadow is two pieces at a lamp height of 1.36 m, and the crossing to one piece happens at a tangency running the whole length of the beam rather than at a point — the same plane that meets a ball at an aspect of 1.00 to 1 meets the beam at 1736 to 1, and a grid finds the true crossing height to a fitted exponent of 1.00 as it is refined. A ring tipped 70° keeps its hole for a completely unrelated reason, closing only at 71.34°, which is the warning that a shadow's topology changes at a tangency names two different accidents rather than one.
Named alongside it
The objects these essays reach for when they reach for this one.
ForeshorteningProjective mapResolutionHomographyPicture surfacePoint lightShadow projectionCamera matrixView frustumJacobianKeystoneViewing position