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

View frustum — where it appears

The truncated pyramid a camera draws through, given by four sides and two planes. The four sides are ratios that carry the focal length and the principal point; the near and far planes move nothing in the picture and decide only which points are kept and how well they are ordered.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

the eyenear 0.90 mfar 4.20 mfocal 740 pxprincipal 345, 210

Four numbers and a window

A projection matrix is built from six numbers and one of them is not a number at all. Four sides carry the focal length and the principal point; the near and far planes move nothing a reader can see; and the bottom row, (0, 0, 1, 0), is the only place the depth divides — set it to (0, 0, 0, 1) and the same machine draws a parallel projection.

pipeline · Frustum vertical difference up to 30.8 px, drawn 8× overbaseline 63 mmconvergence 2.0 m

Turning the cameras inwards

A stereo pair made by rotating two cameras toward a common point puts the same world point at different heights in the two pictures — up to thirty pixels here, on a frame of four hundred. Two eyes level with each other see every point at the same height, so a pair with vertical difference is a pair of pictures of no scene at all.

screen · Stereodisplay worst 1e-13 px outcorrect from 17 cm, at 160 mm wide3 × 3 tiles

A tile is an off-centre frustum

Rendering a picture in tiles is exact, and the way to do it is one line of arithmetic: a tile's sides are the whole frustum's sides read at the tile's own pixel bounds. Aiming the camera at each tile instead is defensible at every step and is a different picture, out by about a tenth of a tile whatever the tile size.

pipeline · Tiling 24 kept · 26 clipped · every mark unmovedcorrect from 17 cm, at 160 mm wideclip plane tilted 18°

The near plane can be any plane

Rewrite one row of a projection matrix and the near plane stops being perpendicular to the axis and becomes whatever plane is asked for. Every x and every y is untouched — it is the same projection of the same scene from the same eye — and the depth order is wrecked, which is a clean separation of the two things a projection matrix does.

pipeline · Clipping 01230123the plane moved by a factor of ten to the …depth resolution, relative (powers of ten)near plane, brought infar plane, pushed out1/near − 1/farone term does all the work

One plane is nearly free

The near and far planes enter a depth buffer's precision through 1/near − 1/far, and one of those reciprocals is enormous. Pushing the far plane out by a factor of a thousand costs a tenth of a per cent; bringing the near plane in by the same factor costs a factor of a thousand — and an infinite far plane is the limit of the first rather than a separate case.

pipeline · Zbuffer the seat3 chords against the arccurved monitor, 3 flat pieces6.80 mm of sag · 5.50 px at the seat

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.

pipeline · Tiling 5710142028400.31310distance along the floor from below the eye (m, log scale)pixels one shadow-map texel spans on screen (log scale)1 pxlamp beside the eyelamp 30 m overheadlamp 40 m ahead, facing backmeasured against the closed form: 4e-91024-texel map · eye 1.7 m · 50° field

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.

pipeline · Shadowmap the whole picture, one density9530²the frustum fitted to the floor7322²the best projective warp of its rows897²the best warp of its rows, any shape840²one texel per pixel of the floor385²the same, with a wall at every point502²facing lamp, floor to 60 m, no texel over a pixel · log scale÷113 by one warp

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.

pipeline · Shadowmap the frustum fitted to the floor53.61 millionits rows re-spaced, one parameter805 thousandits rows re-spaced, any shape706 thousanda true perspective, one parameter752 thousanda full projective warp154 thousandone texel per pixel of the floor148 thousandlamp 40 m ahead, facing back, floor to 60 m, no texel over a pixel · log scale÷4.6 over any row warp

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.

pipeline · Shadowmap

Named alongside it

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

Camera matrixHomographyProjective mapSampling gridClip spaceHomogeneous coordinatesPoint lightPrincipal pointQuantisationShadow projectionCamera tiltConditioning

All concepts