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

design matrix — where it appears

The matrix of a linear system's coefficients, whose smallest singular values say whether the data determines an answer or merely permits one. Its smallest singular values are what separate a system that determines an answer from one that merely permits one, which no residual reveals.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

0.5120.1110100how finely each point is read off the picture (px, log scale)worst epipolar error against the exact geometry (px, log scale)raw pixelscentred and scaled1.1× apart at 0.25 px, 29.9× at 4 pxspread 102 against 2.5e+5

Eight points and the basis they are read in

The linear system that recovers a fundamental matrix is written in whatever coordinates the marks were read in, and pixel coordinates are a bad choice. Centring and scaling them first is worth nothing at a quarter-pixel reading and a factor of thirty at four.

twoviews · Eightpoint the lenstwo marks, a straightedge, no arithmetic2.8e-13 px

Two matches are enough

A general fundamental matrix has seven free numbers and needs eight correspondences. A mirror pair's has two, and two correspondences fix it — with a straightedge, on a print, by drawing the line from each mark to its reflection and marking where the two cross. Given the same sixteen marks read to four tenths of a pixel, the constrained fit lands 4.8 times closer to the truth than the eight-point algorithm.

mirrors · Mirrorpair horizon25withheldfive marks fitted, one withheldcorrect from 26 cm, at 160 mm widethe withheld mark is 2e-13 px off the fitted conic

Five marks and the sixth

Five points determine a conic exactly — five coefficients up to scale, five equations, nothing left over — so a fit through five marks on a photograph is not a fit at all. The sixth mark, withheld, lands on the curve to 1.9e-13 px. And the moment a sixth mark is used, the arithmetic changes character completely: it becomes a least-squares problem, and the residual starts telling you something the five could never say.

foundations · Fiveconic horizoncorrect from 21 cm, at 160 mm wide8/21 faces · 73% of the surface

What an eye can paint

A flight of steps has eighteen faces and no eye reaches more than fifteen. Pointed at a cluster of blocks, a seating rake and a corridor with a doorway in it, the same measurement finds 8 of 21, 7 of 13 and 6 of 7 — and the plane, which offers its whole self to every eye, is the control that makes the law a law rather than a fact about stairs.

viewing · Anamorph 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.

AnamorphosisConditioningCorrespondencedegrees of freedomeight-point algorithmForeshorteningFundamental matrixOcclusionPicture surfacePiecewise mapPlanar homologyProjective map

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