Concept

Projective stratification — where it appears

The three-stage account of what a picture of a plane determines: projectively from the picture, affinely with the vanishing line, metrically with one more fact.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

the diagonals against a ruler, at 3.2 mthe diagonals — exactthe ruler — 22.7 px outcorrect from 23 cm, at 160 mm wideharmonic set -1.000000000 · construction 8e-14 px

The diagonals find the middle

Three taught methods for spacing a receding row are all wrong, and the best of them misplaces a post by three and a half metres. There is a fourth, it needs no measurement and no vanishing point, and it is exact at every camera and every depth — because 'the diagonals of a rectangle cross at its centre' is a statement about which lines meet where, and that is the one kind of statement a projection cannot damage.

foundations · harmonic
projectiveaffinemetricmidpointtwo equal sidesa right anglecross-ratioprojective1.333333333affine0.5000001.333333333metric0.5000001.00000090.000°1.333333333— means the stage does not determine it at allcross-ratio 1.333333 throughout

What one picture of a plane determines

A photograph of a tiled floor fixes that floor's geometry up to a projectivity, and no further. Supplying the horizon buys back the midpoint — exactly, and nothing else. Supplying the image of one circle buys the right angle and the ratio of two lengths at right angles. Three stages, three prices, and a quantity a stage does not determine has no value rather than a wrong one.

foundations · stratification
centre of the ellipseimage of the centrepole of the horizon — 1e-13 px awaycorrect from 22 cm, at 160 mm widepole 1e-13 px from the truth

The centre, got back out of the picture

The image of a circle's centre is not the centre of the image ellipse — this site measured the gap four phases ago and left the obvious question unanswered. It has an exact answer, it needs a straightedge and the horizon, and it is one sentence: the image of the centre is the pole of the vanishing line with respect to the image conic.

foundations · conic

Named alongside it

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

Complete quadrangleCross ratioHarmonic conjugateline at infinityMidpointProjective invariantRectificationVanishing pointAffine mapConicDepth divisiongauge freedom

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