Pline — the series
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A line is a space of its own
Most of what is said about projective geometry in pictures is 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.
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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.
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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.