Four lines have a cross-ratio

The cross-ratio is always introduced the same way. Four points on a line, a formula of four differences, a demonstration that the number survives projection while lengths and ratios do not. This site’s own first rung does it that way too.

That introduction leaves the fact looking like a small miracle: of all the quantities four points could have, one happens to be preserved. It is not a miracle, and the reason it is not is a change of subject.

The dual statement #

In the projective plane, points and lines are interchangeable. Every statement has a partner obtained by swapping the two words and turning lies on into passes through: three points on a line becomes three lines through a point; the line joining two points becomes the point joining two lines.

Apply that to the cross-ratio. Four points on a line becomes four lines through a point — a pencil — and the dual quantity is the cross-ratio of that pencil, computable from the four directions with no points on it at all.

Cut the pencil with any transversal and it makes four collinear points. Those four points have a cross-ratio. It is the pencil’s, whatever transversal was chosen: three quite different transversals across the same four rays give the same number to nine decimal places.

Which is why projection preserves it #

Now put an eye at the pencil’s vertex.

Four points in the world, seen from an eye, are four rays. Those rays are a pencil. The picture is a plane cutting the pencil, so the four image points are a transversal’s section of it — and so are the four world points, cut by a different plane.

Both sets of four points are sections of one pencil. The cross-ratio belongs to the pencil, so the two sections agree, and they agree not because of a coincidence in a formula but because they are two readings of the same object.

That reframing does something to the original statement. The question “why does the cross-ratio survive projection?” becomes “why does a section of a pencil not depend on where the cut is taken?” — and the second question answers itself once the pencil is the thing under discussion. The projection was never doing anything to the four points; it was choosing a second plane to read one pencil in.

The use, which is not decorative #

Duality reads like a formal elegance until it makes something measurable that otherwise is not, and here it does.

A cross-ratio measured on points needs the four points to be in the picture. The most useful fourth point in perspective work is a vanishing point, and vanishing points are routinely off the canvas — a mildly rotated building puts one of them several picture-widths away, and the rectify rung works with one that has run off the paper.

The pencil form needs no such thing. Take any vertex, take the four rays to whatever is being measured, and compute from the four directions. Directions do not run off the edge of anything.

So a fence whose far end is out of frame, a row of columns disappearing behind a wall, a set of divisions whose convergence point is off the negative: all of them still carry a measurable cross-ratio, because the quantity lives in the pencil and the pencil is complete as soon as the four rays exist.

Why there is exactly one invariant and not two #

The pencil reading answers a question the point reading leaves hanging: why is there one surviving quantity rather than three or none?

Count the freedom. Four collinear points have four coordinates along their line. The projective transformations of a line — the maps a projection can perform on it — form a three-parameter family. Four minus three is one, so there is exactly one function of the four points that no projective transformation of the line can change, and the cross-ratio is it.

Read through the pencil, the same count says something more concrete. Four rays from a point have four angles; the projective transformations available are the ones that change where the section is taken, which is again three parameters; and one shape parameter of the pencil survives. That parameter is what every section reports.

So the answer to “why one” is a subtraction, and the answer to “why that one” is that the cross-ratio is the standard coordinate on the space that is left. Any other invariant is a function of it — the harmonic condition is the cross-ratio equal to $-1$, the equal-spacing condition is $4/3$, and the six values a cross-ratio takes under permutations of the four points are six functions of one number.

That is worth having because it forecloses a natural hope. There is no second invariant waiting to be found, no additional quantity that a cleverer construction could rescue from a projection. One number, and the count says why.

What is being claimed, precisely #

It is worth stating the theorem in its exact form, because the loose version — “four lines have a cross-ratio” — invites a question the exact version answers.

The loose version suggests the number is a property of four lines considered as a set of four lines in the plane. It is not: four arbitrary lines have six intersections and no distinguished quantity. The number belongs to four lines through a common point, and the requirement of concurrency is what makes the quantity exist.

Stated fully: given four concurrent lines, every transversal not through their common point cuts them in four collinear points, and the cross-ratio of those four points is the same for every such transversal. The excluded case — a transversal through the vertex — is excluded because it cuts all four lines at the same place, and four coincident points have no cross-ratio.

That exclusion is worth noticing because it is the only case, and its rarity is a fact about the projective plane. In the drawn plane there are two apparent exceptions: a transversal parallel to one of the four lines never meets it, and a transversal parallel to two of them never meets either. Both are handled by the same promotion that makes the theory work at all — the missing intersection is a point at infinity, and a point at infinity enters the cross-ratio formula like any other. The number comes out finite and correct.

So the theorem has one hypothesis and one exclusion, and both are about degeneracy rather than about geometry. A statement with that shape is usually a statement about a structure rather than about a configuration, which is what duality is.

The harmonic case #

The cross-ratio of $-1$ is the harmonic case, and it duals too. Four collinear points in harmonic relation become four concurrent lines in harmonic relation — a harmonic pencil — and the constructions transfer.

That is what makes the diagonals of a rectangle work in both directions at once. The near end, the far end, the midpoint and the vanishing point are a harmonic set of points along a receding side. The two diagonals, the side and the line to the vanishing point are a harmonic set of lines through a corner. Same fact, two readings, and a draughtsman can use whichever one has its ingredients in the picture.

Desargues, which is its own dual #

The most quoted example of duality is Desargues’ theorem, and it earns the status: dualising the theorem produces its own converse.

Two triangles in perspective from a point are in perspective from a line dualises to two triangles in perspective from a line are in perspective from a point, which is the converse. So the theorem and its converse are one statement seen twice, and proving either proves both.

Where duality stops being available #

Two limits, and stating them is what keeps this from being mysticism.

Duality is a fact about the projective plane, not about the drawn plane. Its statements hold with points and lines at infinity included, and a construction that dualises perfectly on paper may dualise to something requiring a line at infinity that cannot be drawn. Nothing is wrong when that happens; the dual statement is still true and its ingredients are directions rather than marks.

And it does not dualise measurements. Length has no dual, angle has no dual, and the reason is that neither is projective. Only the incidence structure dualises, which is precisely the structure a projection preserves — so the class of statements that dualise is the same class that survives photography, and the two facts are the same fact.

That coincidence is worth carrying. This site keeps finding that its exact constructions are the incidence-only ones: the diagonals, Desargues, the pole of the horizon. Those are exactly the constructions with duals, and it is not three properties coinciding — it is one property, seen three times, that a projective statement is one made of nothing but which lines meet where.

How the number is actually computed #

The formula for four collinear points is a ratio of four differences of coordinates along the line. For four concurrent lines there is no coordinate to take differences of, so the dual form has to be computed from angles — and the naive translation does not work, because the cross-ratio of four angles is not the cross-ratio of the four points they cut.

What does work is the tangent. Measure each ray’s angle from the first, take the tangent of each, and feed the four tangents into the same four-difference formula. That is not an approximation and not a convention: a transversal at unit distance from the vertex cuts the four rays at positions equal to those tangents, so the tangents are a set of positions the four points take on one particular transversal, and the cross-ratio of the pencil is the cross-ratio of that section.

Which is a small piece of machinery with a large moral attached. The dual quantity is not a new definition; it is the old definition applied to a section that always exists. Every dual statement on this site has that shape — it is the original statement about an object that was always there and had not been named.

The check that the implementation is right is the one the theorem suggests: cut the pencil with three transversals of quite different slopes and positions and require the four points each makes to carry the same number. They agree to fourteen digits. A formula that got the tangent step wrong would produce a perfectly plausible number that varied with the transversal, and nothing about a single reading would say so.

What a vanishing point is, dually #

The vanishing ladder promotes a direction to a point: parallel lines meet at a point of the projective plane, that point is findable in a drawing from the drawn lines, and it enters calculations like any other point. The dual promotion is worth stating because it is used constantly on this site without being named.

The dual of a point at infinity is a line through the origin of directions — which is to say, the dual of a direction is a pencil. So the statement “these lines are parallel” and the statement “these lines are concurrent” are one statement in the projective plane, differing only in where the common point is. A bundle of drawn edges converging at 40 px from the frame’s corner and a bundle running exactly parallel on the page are the same configuration described twice.

That is why vanishingPointOf can be written once and used for both. It fits a common intersection in the least-squares sense and reports the residual; a genuinely parallel bundle produces a nearly-singular system, which the function refuses rather than answering. The refusal is the honest half — a direction is not a place, and a routine that returns a very distant place for a bundle that has none has substituted one for the other.

Where the site already uses the dual form #

Three places, and none of them says so.

The measuring point. The construction lays out correct depths by drawing rays from a point on the horizon through the divisions of a measuring line. Those rays are a pencil and the receding line is a transversal; the construction is a pencil section, and the reason it produces correct depths is that the pencil’s cross-ratio is what the eye’s own rays carry.

The three-vanishing-point recovery. Each bundle of a box’s parallel edges is a pencil in the drawing whose vertex is the direction’s vanishing point. The camera falls out of the three vertices by an incidence computation, and nothing about the individual edges matters — only which pencil each belongs to.

And the anamorphic construction. A figure drawn to be correct from one point is a pencil from that point sectioned by a surface that is not the picture plane. Changing the surface changes the drawing and not the pencil, which is why the same anamorph is correct on a flat sheet, on a cylinder and on a floor, and why the point it is correct from is the one thing all three have in common.

The sentence to keep #

The cross-ratio does not survive projection. It was never in the four points to begin with: it is a property of the pencil of rays from the eye, and the four points — in the world, in the picture, in any other picture — are places where somebody chose to cut.

That is a better sentence than the usual one because it says why there is exactly one invariant rather than several. A pencil of four rays has one shape parameter, up to the projective transformations of the line. Four points cut from it can carry that parameter and cannot carry more, no matter how many quantities are written down about them.