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

Composition — where it appears

Applying one projection after another, which stays a projection when the surface in between is a plane and stops being one when it is not. That is what licenses a rectification of a photograph of a photograph, and what forbids the same argument once a curved surface is in the chain.

Named by 5 essays across 5 fields — each of them below, with the objects they name alongside it.

an anchorcorrect from 19 cm, at 160 mm wideflat 1e-13 px · rolled 30.5 px

A projection of a projection

Photograph a photograph and the composite map is a homography — four marks determine it and every other lands where they say, to 1e-13 px. Roll the print and the same four mispredict the rest by 30.5 px, which is why straightening a page of a thick book never quite works.

foundations · Composition 0.6× the page48 mm×8.3 at 4000.8× the page64 mm×6.3 at 4001.0× the page80 mm×5.0 at 4001.4× the page112 mm×3.6 at 4002.0× the page160 mm×2.5 at 4003.0× the page240 mm×1.7 at 4004.5× the page360 mm×1.1 at 4007.0× the page560 mm×0.7 at 400distance the picture is correct from, shown 160 mm widea rule about the paperwhich is a rule about the reader

Both vanishing points on the paper

Putting the two vanishing points on the sheet is presented as a composition rule. It is a statement about the reader: with the two points one page-width apart the picture is a 90° view, correct from 80 mm, and a reader holding it at arm's length is shown a room five times as deep as the one drawn. The layout that is honest at arm's length puts both points four and a half pages off the sheet.

wrong · Station 0102030400123how wide the audience is (m)the worst seat, in pixels of departure14.1 px per metrea flat panel of the same sizecurved television, the best single pre-warpthe flat panel's line is the axis

One picture and three people

A curved screen can be pre-warped for one seat, and the search over which seat to choose returns the middle one to a quarter of a per cent — there is nothing to be clever about. What the correction buys the sofa as a whole is six per cent, and the worst seat grows at fourteen pixels for every metre of audience, with no width at which it is zero except one person.

screen · Curvedscreen correct from 19 cm, at 160 mm wideratio 1.2509 · carries one onto the other to 1e-15 m

Two lamps and one map

A flat object lit by two lamps casts two shadows, and one is the other scaled about a point — ratio 1.2509 here, carrying every point of the first outline onto the second to 1e-15 m. No rotation and no shear is available to it, because a projection between two parallel planes has its axis at infinity. And the ratio is exactly 1 when the two lamps are at the same height, which makes a pair of shadows a measurement of the lamps.

light · Twolamps where each reader's picture landsgoing up: risers67%going up: treads33%coming down: risersnothingcoming down: treads100%a flight of 9, eye at 1.65 m at each endthe two sets of faces do not overlap

One flight, two pictures

From the top of a descending flight every riser faces away, so the design lands entirely on treads. From the foot of the same flight the risers take 68 per cent of the design at a median stretch of 1.4, and the treads take the rest at a median of 2.6. Give each reader the faces the other cannot use and one staircase carries two pictures, with no face asked to hold both.

viewing · Anamorph

Named alongside it

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

HomographyPicture planePicture surfacePlanar homologyProjective mapRectificationResidualStation pointVanishing pointViewing positionAnamorphosisAxis of a homology

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