Projective plane — where it appears
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
Where parallel lines meet
They meet at a point that is not in the plane, and the horizon is the line made of all such points. Treating that as bookkeeping misses what it buys — a vanishing point becomes an ordinary point of the picture, findable from the drawn lines, checkable, and enterable into any calculation the others are.
A point and a line are one object
Every projective statement has a partner got by swapping the words point and line, and the partner is true whenever the original is. Run on this collection's own Desargues configuration, all thirty of its incidences survive the exchange to 5 × 10⁻¹⁶ — and three of the dual's ten points land at infinity, which is a fact about where the drawing sits on the page rather than about the theorem.
Desargues read the other way
The theorem's converse is not a second theorem. Exchange points and lines in the ten-point configuration and every one of its thirty incidences holds to 5 × 10⁻¹⁶, the axis becomes a point, the three axis points become lines through it — and what has been written down is the converse, read off the drawing rather than proved.
Named alongside it
The objects these essays reach for when they reach for this one.
point at infinityDesarguesDualityHomogeneous coordinatesIncidenceline at infinityPencilProjective dualitydegrees of freedomHorizonleast squaresleast-squares intersection