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 2 essays across one field — 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 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

Everything this site has said about projective geometry has been 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

Named alongside it

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

Cross ratiopoint at infinityCollineationConicdegrees of freedomDemonstrationFree parameterHarmonic rangeHomogeneous coordinatesHorizonPencilPerspectivity

All concepts