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

Separation — where it appears

The relation holding when two pairs of points on a line interleave, each pair lying in a different arc cut by the other. It survives a projection where betweenness does not, and it is the sign of the cross-ratio rather than any relation among three points.

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

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-ratioFixed pointHomographyInvariantpoint at infinityProjective lineProjectivityVanishing pointInvolutionline at infinity

All concepts