Projective line — where it appears
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
Also named here as projectivity — the same set of essays touches all of them, so they are one junction rather than several.
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.
The map a row of posts is
Walking one bay further down a row is a map of the drawn line to itself, and which map it is settles everything about how the spacings behave. It is parabolic — one fixed point, counted twice, and that point is the vanishing point — which is why the drawn posts crowd toward it and never arrive. Doubling a distance instead gives a hyperbolic map whose multiplier a straightedge can read.
Perpendicular is a pairing
On a horizon, the vanishing point of a direction and the vanishing point of the direction at right angles to it are joined by a map that is its own inverse. Such a map has two degrees of freedom rather than three, so two pairs determine it — and its two imaginary fixed points are the focal length and the centre of the picture, handed back from two rectangles on one floor with nothing assumed.
Named alongside it
The objects these essays reach for when they reach for this one.
degrees of freedomDemonstrationProjectivityCross ratioFixed pointInvolutionpoint at infinityVanishing pointAbsolute conicCamera calibrationCircular pointsDepth division