A line is a space of its own

The first essay on this site says that a projection destroys length, angle, area and the ratio in which a point divides a segment, and that one quantity survives. Everything since has been built on that one quantity.

What has not been said is where it lives.

The cross-ratio is not a property of a picture, or of a camera, or of a scene. It is a property of four points of a line, and a line — a projective line — is a space in its own right, with its own points, its own maps and its own complete theory. That theory is small enough to state in a page and it turns out to be underneath a good deal of what the construction field does by hand.

What a point of a line is #

A point of the projective line is a pair of numbers up to scale: $[a : b]$, where $[3 : 1]$ and $[6 : 2]$ are the same point. The ordinary coordinate is $t = a/b$, and $[1 : 0]$ — the pair with a zero underneath — is the point at infinity.

That is not a trick for handling a special case. It is the same device the site uses for the plane, one size down: a point of the picture is a triple up to scale, a point of space is a quadruple, and a point of a line is a pair.

Writing it that way makes the cross-ratio an ordinary determinant expression, with the point at infinity as one of the four costing nothing. That matters more than it sounds: the fourth point of almost every construction in the construction field is the vanishing point, and a formula that has to special-case it is a formula that special-cases the most useful configuration there is.

What a map of a line is #

A projectivity of a line is a two-by-two matrix, up to scale, acting on the pairs:

$$t \mapsto \frac{a t + b}{c t + d}$$

Four entries, minus one for the scale, leaves three degrees of freedom. That number is the whole essay.

Three degrees of freedom means three conditions determine the map. Three conditions means three point-pairs. And once three pairs are given, every other point’s image is not a choice — it is a consequence.

The construction that produces the map is the standard one and it is worth naming because it is the reason three is the right number. There is exactly one projectivity carrying any three distinct points to $0$, $1$ and $\infty$. Compose one of those with the inverse of another and any three points go to any other three, once, and no further freedom is left over.

Three points are a coordinate system #

The count has a constructive reading that is the whole point of treating a line as a space.

Since exactly one projectivity carries any three distinct points to $0$, $1$ and $\infty$, any three points of a line define a coordinate on it — and the coordinate of a fourth point is precisely its cross-ratio with the three. So the invariant is not merely something that survives a projection; it is the line’s own coordinate, once three of its points have been nominated as the frame.

That is why $n$ points on a line carry $n-3$ independent numbers: three are spent on the frame and the rest are readings in it. And it is why a photograph of a receding row supplies a coordinate system on that row with nothing brought from outside — every construction this collection performs along a receding line, from a measuring point to a row of posts read as a map, is arithmetic in that coordinate.

The choice of frame is free and not costless. The map’s three parameters are determined by the three pairs through expressions carrying the reciprocals of the gaps between the source points, so three points close together determine the map badly — and the conventional choice of $0$, $1$ and $\infty$ is the maximally spread one, which is why it is conventional. On a drawing, that translates to using the near end, the far end and the vanishing point as the frame, rather than three consecutive divisions.

The three kinds of map, one dimension down #

The classification the plane has in three cases the line has in three too, and the two are related.

A projectivity of a line has fixed points where $ct^{2} + (d-a)t - b = 0$, a quadratic, so there are two distinct real ones, one double one, or a conjugate pair — hyperbolic, parabolic, elliptic. That is the same trichotomy a map of the plane has, arrived at from a quadratic instead of a cubic, and the plane’s cases restrict to the line’s: a homology restricted to a line through its centre is hyperbolic, an elation’s restriction is parabolic, and the perpendicularity involution is the elliptic case with the extra condition $a + d = 0$.

The hyperbolic case carries a number worth naming. Put the two fixed points at $0$ and $\infty$ and the map becomes $t \mapsto \lambda t$ — plain multiplication — and $\lambda$ is a projective invariant of the map. That multiplier is the characteristic ratio: the same number a homology of the plane reports as its ratio, seen as what the map does to any one line through its centre.

So the plane’s census and the line’s are one census. A shadow’s homology restricted to a line through the lamp’s image is multiplication by 0.6719; an anamorph’s is multiplication by $-1.4815$; a mirror’s is multiplication by $-1$, which is the involution. Reading them on a line rather than in the plane loses nothing and makes the ratio’s meaning immediate — it is a scaling, in the one coordinate where the map is a scaling.

That also settles what the three cases look like on a drawing. A hyperbolic map has two fixed points and everything slides between them; a parabolic map has one and everything slides toward it from both sides; an elliptic map has none real, so every point moves and nothing stays. A receding row of posts is the first, with the near end and the vanishing point as its two fixed points — which is why four concurrent lines and four collinear points keep coming out as the same measurement on the same object. The classification also says which drawings cannot happen: a row whose spacing tightens toward two different places at once would need a map with two attracting fixed points, and a projectivity of a line has at most two fixed points in total, both of which such a row is already using.

The claim, and how it fails #

“The cross-ratio is an invariant of projection” is a true sentence that says almost nothing. Every projection preserves plenty of things — incidence, straightness, the order in which points lie for some configurations — and preserving a number is common.

The claim worth making is the strong one:

The cross-ratio is the only invariant of four points of a line. Two quadruples with the same cross-ratio are carried to each other by some projectivity, and two with different cross-ratios are carried by none.

That is a claim which can be made to fail, and making it fail is what makes it a measurement.

Two refusals are not a pattern, and a routine that says no to everything says nothing. The third figure is the half that makes the other two mean something: it asks the same machinery for exactly the pair the map already delivers, and the answer comes back yes. A test with no accepting case is not a test.

Run the other way, that is a test on a drawing. Four marks claimed to be the images of four known points either have the right cross-ratio or they were not made by any projection of that line, and the deciding number is one subtraction.

Why the test is weaker than it looks #

The site has already been caught by this and the finding is worth repeating here, where the reason for it is visible.

Dividing depth by eye measures three taught methods for spacing a receding row and finds all three wrong in metres. The obvious check — take four consecutive divisions and compute their cross-ratio — gives one of the three a perfect score.

The reason is exactly the three degrees of freedom. Four points chosen as “the first four of a sequence” are not four independent points: any four equally spaced points can be carried to any other four equally spaced points by a projectivity, because three pairs were enough and the fourth came free. A test run on four points that a map could carry anywhere is a test on nothing.

That is the practical rule this section exists for. Four points of which three were free tell nobody anything; the fourth has to come from somewhere the first three did not.

A perspectivity is a projectivity, and not conversely #

Two lines in a plane and a point not on either: joining each point of the first line to the centre and reading where the join meets the second is a perspectivity. It is the simplest map of a line to a line and it is what every ray in every figure on this site performs.

Every perspectivity is a projectivity. The converse is false, and the gap is instructive: a perspectivity between two distinct lines fixes their point of intersection, so it has only two free parameters rather than three. A general projectivity is a composition of perspectivities — two of them suffice — and the extra parameter is the intermediate line’s freedom.

The map, written down #

It is worth seeing the answer once as numbers, because the shape of it explains what a projectivity of a line can and cannot do.

Three pairs — nought to 3.1, one to minus two, four to a half — give a map with four entries, and applied to 2.70 it returns 0.2780. That number was not fitted to anything. It is the value the three arrows above force, and the cross-ratio of the four points on the left agrees with the cross-ratio of their four images to sixteen decimal places.

A map with three parameters is very free. It can reverse the order of points, it can send a finite point to infinity and bring infinity back to a finite place, and it can compress an unbounded range into a short interval. What it cannot do is move four points independently, and every constraint on a drawing that this site has ever checked is a case of that one limitation.

The one place a projectivity of a line does misbehave is at its pole — the point $t = -d/c$, which it sends to infinity. Anything near there is thrown a long way, and a drawing whose useful range straddles that point is a drawing whose measurements are conditioned by how close it came.

What lives on this line, on this site #

Almost everything, once it is looked at.

The horizon is a projective line, and the vanishing points of horizontal directions are its points. The vanishing point of a direction and the vanishing point of the direction at right angles to it are paired by a map of that line, which is the subject of a rung further along.

A receding row of posts sets up a map of the drawn line to itself, and which map it is decides how the spacings behave — the subject of the rung immediately after this one.

The pencil of rays through the eye is a projective line’s worth of directions, and the cross-ratio of four of them is the number that survives every section of the pencil.

What a measuring point is, said this way #

The drawing office has a construction for laying out true distances along a receding line: mark the distances at true scale along a line that runs parallel to the picture plane, join each mark to a fixed point on the horizon, and read where the join crosses the receding line.

Said in this essay’s vocabulary it is one sentence. The measuring line and the receding line are two lines of one plane; the joins are a pencil through a point; so the correspondence between them is a perspectivity, and a perspectivity carries the cross-ratio. The measuring line carries true distances because it is parallel to the picture plane and therefore drawn at one uniform scale; the perspectivity carries them across; and the receding line ends up with the projected positions because a projection and a perspectivity are the same operation seen from two sides.

The construction’s exactness is then not a property of the draughtsman’s care. It is a property of the object: two lines and a centre, and there is nothing in that arrangement for care to affect. That is the same reading the bay repeated by a straightedge gives its own construction, and it is the reason a straightedge construction can be exact where a rule about a paper angle is not.

Why one dimension is worth the trouble #

There is a fair objection to all of this: the plane’s theory contains the line’s, so why write the line’s down separately.

Three answers, and each of them has produced something on this site.

The counting is different, and the counting is what gets used. Three pairs against four correspondences is the difference between a construction a draughtsman can perform on a horizon and one that needs a whole plane’s worth of marks. Where a question can be phrased on a line, it is cheaper — and several of the site’s constructions turn out to be line questions dressed as plane ones.

The classification is different. A map of a plane is sorted by its fixed structure: a line of fixed points and a fixed centre, with the census this collection runs on such maps. A map of a line is sorted by its fixed points, of which there are two, or one, or none — and none is a real case with real consequences, which has no plane analogue that reads the same way.

And the degenerate case is different. A projectivity of a line with no real fixed point is perfectly ordinary; the site’s own metric machinery rests on exactly such a map. Nothing in the plane’s census prepares a reader for a map whose fixed set is a pair of complex conjugates and whose two numbers are the focal length and the centre of the picture.

What this does not say #

It says nothing about which four points to use. That is the whole difficulty in practice, and the section above says only what makes a choice worthless rather than what makes one good.

It says nothing about noise. Every number here is exact; a cross-ratio read off a real drawing has a reading error in each of four positions, and its sensitivity to those errors depends on the configuration in a way that is worth an essay of its own. What can be said now is that the sensitivity is a fact about the arrangement rather than about the arithmetic, which is a distinction this site has had to make three times in three different fields.

And it says nothing about the plane. A projectivity of the plane is eight degrees of freedom and needs four correspondences; a projectivity of space is fifteen and needs five. The counting is the same counting and the answers are different numbers.

The transferable form #

The useful half of this rung is a habit rather than a result.

Count the degrees of freedom before running the test. A quantity computed from as many free points as the map has parameters is preserved by every map of that kind, and preserving it proves nothing at all.

That is what the four-consecutive-divisions check got wrong, and it is what makes the vanishing-point version right. It generalises: a homography of the plane has eight parameters, so any check computed from four points of a plane is passed by every plane map there is, and a check on a plane needs a fifth point — which is exactly why five marks fix a conic and why the conic is the object a plane’s checks are made of.