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

Invariant — where it appears

A quantity a class of transformations cannot change, which is what a measurement made through such a transformation can be built on. The cross-ratio is the only invariant of four points of a line under projection, and stating it as the *only* one is what makes it a claim that can fail.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

horizonABCDon the groundin the picturelength AB1.00011.3930ratio AB:CD0.56670.6837cross-ratio1.31681.3168correct from 26 cm, at 160 mm wide34° across

What a projection destroys

A projection loses length, angle, area and the ratio in which a point divides a segment. Exactly one quantity comes through untouched, and almost everything that can be checked about a picture is checked with it — including, as it turns out, some things it cannot check at all.

foundations · Crossratio view 1, heldview 6one standard deviation, drawn 5× actual size — held: the first camerapoints 52.1–158.0 mm · cameras 0.0–104.8 mmmarks read to 1 px

An uncertainty is quoted from something

One adjustment, one set of marks read to a pixel, and its uncertainty written four ways. Held at the first camera, the last camera is 105 mm from certain; held at nothing, every camera is within 4 to 6 mm. A ratio of two distances carries 0.1465 per cent in all four, to two parts in a billion.

manyviews · Gauge tt′0.003.101.00-2.004.000.502.700.28predicted, not fittedthree pairs givencross-ratio 0.839506

A line is a space of its own

Most of what is said about projective geometry in pictures is said about the plane. One dimension down there is a smaller object with a complete theory: a point of a line is one ratio, a map of a line is three numbers, three pairs fix it, and the cross-ratio is not merely an invariant but the only one — which is a claim that can be made to fail.

foundations · Pline the point at infinity0ABCDA-1.40B-0.30C0.80the same four, on a straight line1 of the four is off the endthe chords crosscross-ratio -0.9469

A line is a closed curve

The point at infinity is an ordinary point, so a projective line is a circle — and the consequence is about order. Betweenness broke in 21.1 per cent of ten thousand random projectivities and separation in none of them, and the zero is a reading rather than a blind instrument because a fold of the same circle breaks it 3,522 times.

foundations · Pline horizontread 1correct from 14 cm, at 160 mm wideelliptic · trace² 2.0000 · back after 4

Three kinds of map on a row of posts

A projectivity of a line has two fixed points, one, or none, and every one of the three is a picture this collection already draws. The three orbits are told apart by where they go — one piles onto a fixed point, one crawls, and the third returns after six steps and is 1.9e-11 pixels from where it started.

foundations · Classify

Named alongside it

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

Cross-ratiopoint at infinityProjective lineProjectivityVanishing pointFixed pointHomographySeparationbundle adjustmentCollineationConicControl point

All concepts