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

Pencil — where it appears

The family of all lines through one point, which is a projective line's worth of them. Four lines of a pencil carry a cross-ratio of their own and any transversal reads it back, which is what lets a cross-ratio be measured when the points themselves have left the picture.

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

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

Four lines have a cross-ratio

The cross-ratio is introduced as a property of four points on a line. Its dual — four lines through a point carry the same number, and any transversal reads it back — is not a second theorem. It is why the first one is true: four rays from an eye are a pencil, every picture of them is a section of that pencil, and a quantity belonging to the pencil cannot depend on which section was taken.

foundations · duality 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 ratioDemonstrationdegrees of freedomDesarguesFree parameterHarmonic conjugateHomogeneous coordinatesInvariantline at infinityPerspectivityPinholepoint at infinity

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