A line is a closed curve

A line is a space of its own established that the marks a row of posts leaves on a photograph are a one-dimensional geometry in their own right, with its own points and its own maps, and that the vanishing point belongs to it as an ordinary member rather than as a limit nobody reaches. The map a row of posts is then read the maps: advancing along the row by a fixed distance is one kind of transformation of that line and scaling the world coordinate is another.

This essay takes the shape of the line itself, and the shape has a consequence that is easy to state and easy to get wrong.

A projective line is topologically a circle, and the cost of that is betweenness. Adjoin the point at infinity to an ordinary line and the two ends join up; there is no longer an outside to fall off. So the sentence “this post is between those two” — which is the most elementary thing anybody says about a row of marks — is not a projective statement, and a projection is free to falsify it.

What survives is not betweenness but separation #

Something does survive, and it is worth naming precisely because it is so nearly the same thing.

Take two pairs of points rather than three points. Two pairs on a circle either interleave — going round, the order is A, B, C, D — or they do not, in which case one pair sits entirely inside one of the arcs the other cuts. That relation is separation, and it is a statement about a cyclic order rather than a linear one. It needs no end of the line, because it never asks which side of anything a point is on.

Drawn on the circle, separation is visible without any arithmetic: join A to C and B to D, and the two chords cross exactly when the pairs separate. The figure above draws both chords and they do cross. The cross-ratio says the same thing with a sign — at this setting it is −0.4211, and a negative cross-ratio is precisely the condition that the two pairs separate.

That gives two independent readings of one relation, which is the arrangement this collection prefers wherever it can get one: a chord crossing that a reader can see, and a number that can be computed from four coordinates with no picture at all.

Three points have no order to preserve, and four is where one appears #

The census below reports that betweenness fails 21.1 per cent of the time. That number is worth having, and it is worth knowing before reading it that the failure was not merely likely — it was unavoidable, and the reason says exactly why separation needs pairs.

A projectivity of the line is determined by what it does to three points, and any three distinct points can be carried to any other three. That is the whole content of the fundamental theorem in one dimension, and it has an immediate consequence that is rarely stated as bluntly as it deserves: no relation among three points can survive a projectivity, because there is a projectivity carrying any arrangement of three to any other. Betweenness is a relation among three points. It never had a chance.

So the 21.1 per cent is not a frequency that might have come out differently with better luck. It is the fraction of randomly drawn maps whose pole happens to land inside the interval, and the only reason it is not far higher is that the interval is small compared with the range the poles were drawn from. A population of maps concentrated near the points would break the order almost always; the sweep further down shows 77.2 per cent at the worst placement.

Four points are where this stops. Three can be normalised away — sent to zero, one and infinity, and there is exactly one projectivity doing it — so a fourth point has nowhere left to hide, and what is left of it is a number. That number is the cross-ratio, and it is the first projective invariant of a line because four is the first count at which an invariant can exist.

Separation appears at the same count, and that is not a coincidence. It is one bit of the cross-ratio — its sign — so the first relation of order that survives a projection arrives with the first invariant that survives one, for the same reason and at the same moment. A reader who wants an order fact about a drawn line has to phrase it about two pairs, because two pairs is four points and four points is where the geometry begins to hold anything at all.

That also explains the shape of what a projection destroys. It does not degrade order gradually, the way a measurement degrades with noise; it either has the information or it does not, and for three points it does not.

The map that throws a point out of the middle #

The way betweenness fails is not gradual and it is not a distortion. Nothing gets stretched. What happens is that the end of the line moves.

A projectivity of the line has a pole — the point it sends to infinity. While that pole sits outside the interval the three points occupy, the map is monotone across that interval, and a monotone map cannot change which of three points is in the middle. The order survives, and it survives for a reason that has nothing to do with projective geometry.

Slide the pole in between two of the points and the situation changes completely, and it changes at an instant rather than over a range. The point the map sends to infinity is now inside the interval, so one of the three is carried out past the end of the line and comes back on the other side.

The invariant is untouched in both cases, and that is the point of quoting it twice. A reader who watched only the cross-ratio would see nothing happen. A reader who watched only the drawn order would think the map had done violence to the middle point. Both are looking at the same map, and only one of them is looking at something projective.

Twenty-one per cent, and a zero that had to be earned #

The two figures above are two settings of one family, chosen to show the transition. What a reader should want next is the rate: over maps drawn at random rather than arranged, how often does each reading survive?

The affine column is the first control and it is a genuine one. An affinity is a projectivity that holds the point at infinity still — it has no pole in the finite plane at all — so it cannot move an end of the line, and betweenness is safe under it. That is why the ordinary intuition works: everybody’s first geometry is affine, and in an affine world “between” is a fact.

The second control is the one that matters more, and it is there because a zero is the easiest number in this collection to report by accident. A counter that never fires because it is broken and a counter that never fires because nothing broke print the same digit. So the same instrument is handed a map that is not a projectivity — a fold of the circle, which sends two points to one and is therefore not invertible — and it finds 3,522 breaks in ten thousand. The zero above is a measurement made by an instrument that has been shown to be capable of a non-zero reading.

That distinction is worth dwelling on because it is not a formality. Separation is preserved by every one-to-one map of the circle, projective or not: it is a fact about cyclic order and nothing else. So a projectivity preserving it is not evidence that the map is projective, and an essay that offered the zero as though it were would be offering an identity. What separates a projectivity from the merely continuous maps is that it holds the cross-ratio’s value, not merely its sign — and that is a different measurement, made elsewhere in this collection when four points are carried through a projection and compared.

Where the failures are, and why the answer is a threshold rather than a slope #

The census gives a rate over a population. A more useful question is where in the family the failures live, because that says what a reader has to check.

The shape is a threshold and not a slope, and that is the practical content of the whole essay. There is no such thing as a projectivity that mildly disturbs the order of three points. Either its pole is outside the interval a reader is looking at, in which case the order is exactly safe, or it is inside, in which case one point has been thrown to the far end and the order is exactly wrong.

For a photograph the pole is a specific and familiar thing. A camera maps a world line to a drawn line, and the point it sends to infinity on the paper is the point of the world line that lies in the plane through the eye parallel to the picture — the point whose ray runs parallel to the image plane and therefore has no image. Everything on the far side of that point is behind the picture plane. So the rule for a photograph is short: the order of marks along a drawn line is trustworthy exactly when the whole stretch of the world line being read is on one side of the eye.

That is nearly always true, which is why nobody notices, and it is the reason the assumption is invisible rather than absent.

It is worth being exact about what a reader has to check, because “the pole is outside the interval” is a condition about a point that is not drawn anywhere.

The pole of a camera’s map from a world line to its drawn line is the point of that world line lying in the plane through the eye parallel to the picture. It is not the vanishing point, and confusing the two is the natural mistake: the vanishing point is the image of the world line’s point at infinity, while the pole is the pre-image of the drawn line’s. The vanishing point is on the paper and can be pointed at; the pole is in the room, beside the camera, and has no image at all. A reader with only the photograph cannot see it, and can only reason about where it must be.

Which is why the practical rule is stated as a fact about the scene rather than about the picture: everything being read has to be on one side of the eye. That is a condition anybody can check from where they were standing, and it cannot be checked from the print.

The vanishing point is where the arithmetic already puts it #

The clearest evidence that the point at infinity is ordinary is that a construction can use it without any special case, and the site has already leaned on this without saying it in these terms. Where parallel lines meet puts the vanishing point on the paper as the image of a direction. What the closed line adds is that the image is not a limit that the posts approach — it is a point of the same drawn line as the posts, available to be joined and measured with.

The cross-ratio makes that concrete. Take posts at one, two and four metres along a row together with the point infinitely far along it: in the room that cross-ratio is 1.500000. Take the three marks those posts leave together with the vanishing point: on the paper it is 1.500000, agreeing to fifteen places. Nothing was done to the vanishing point to make it behave. It is where the arithmetic puts the image of a post infinitely far away, and it then behaves like every other point because it is one.

This is the fact that makes the harmonic construction work on a receding row and the fact that lets a straightedge reach exactly the rational points of a drawn line. Both constructions use the vanishing point as one of their inputs, and neither would be available if it were a limit rather than a point.

What a closed line does to the maps of it #

Once the line is a circle, the transformations of it are transformations of a circle, and that constrains what they can do in a way an open line does not.

A projectivity of the line has two fixed points, or one, or none — and the third case is only possible because the line closes. On an open interval a continuous, increasing, invertible map that moves everything must push things toward one end, and the ends are fixed. Close the line and there is no end to be pushed toward, so a map can rotate the whole thing and fix nothing at all.

That case is not a curiosity. It is drawn by a turning stair: the tread edges of a spiral stair have vanishing points that walk along the horizon, and the map carrying one tread’s vanishing point to the next has no fixed point anywhere on that line. A reader who thinks of the horizon as an ordinary line with two ends has nowhere to put that map. On the circle it is an ordinary rotation, and it returns to its starting point after a definite number of steps. Which map a picture produces, and what each does to a row of posts, is the classification that follows directly from this shape.

Two ends that are one point, from the camera’s side #

There is a second route to the same fact and it comes from the camera rather than from the algebra, which is the arrangement this collection likes best.

Follow a world line away from the camera in one direction and its marks crowd toward the vanishing point. Follow the same line the other way, behind the eye, and the marks crowd toward the vanishing point again — the same mark, from the other side. A direction and its reverse have one image, because a pinhole maps a direction and the two opposite rays through the eye are the same ray drawn twice. The picture contains what is behind the camera measures that directly.

So the two ends of a photographed line are one point of the drawn line, and they are one point for a physical reason rather than a definitional one. The closure is not an adjoined convenience. It is what a lens does.

What this does not settle #

Two limits are worth stating plainly.

The first is that none of this says the marks on a real photograph are hard to order. They are not. A reader looking at a row of fence posts has no difficulty saying which is which, and the threshold above explains why: the pole is behind them, out of the picture entirely, so the order is exactly safe and the affine intuition is exactly right. The claim is about what is guaranteed by the projection, not about what is usually true of photographs.

The second is that separation, having survived, is not worth very much on its own. It is one bit about four points where the cross-ratio is a real number about the same four, and the census shows why: separation is kept by maps that are not projective at all, so knowing it survived tells a reader almost nothing about what map was applied. It is the cross-ratio that identifies the map, and the amount of it that a real measurement can actually reach is a separate and harder question.

What the closed line does buy is a vocabulary in which nothing is a special case: one kind of point, one kind of map, no ends, no limits, and a vanishing point that can be joined to things. Every construction in this collection that uses a vanishing point as an ordinary mark — and there are a great many — is spending that.

One object, and the two facts it forces #

The circle at the top of this essay is not an illustration of the projective line. It is the projective line, drawn at its own topology instead of at the one an ordinary rule imposes, and everything above follows from taking that seriously rather than treating the point at infinity as an accounting entry.

Two things follow immediately, and both are measurements rather than definitions. Betweenness is not projective and fails at a threshold, 21.1 per cent of the time over random maps and 77.2 per cent at the worst pole placement. Separation is projective, fails never, and is preserved by so much else that its zero had to be earned against a fold before it meant anything.

The second of those is the one worth carrying, because it is a habit rather than a fact. A reading that never changes is not evidence until something has been found that changes it.