Pascal's line, and the theorem underneath Pappus

There is a small family of constructions on this site that use no measurement whatever. They join pairs of points, they mark where lines cross, and they stop. That restriction is exactly the condition for a construction to survive a projection, which makes it the condition for the construction to be carried out on a photograph of a thing rather than on the thing — the same reason a straightedge can divide a receding row of posts and find a point’s polar with respect to a drawn conic.

Pascal’s theorem is the largest member of that family. Take six points of a conic in any order, treat them as a hexagon, and take the three pairs of opposite sides — the first against the fourth, the second against the fifth, the third against the sixth. Each pair meets somewhere. The three meeting points are collinear, and the line through them is the Pascal line.

Six joins, three meets, one line. Nothing is measured anywhere in it, so a reader with a photograph and a straightedge can run it on the print.

Six joins and three meets #

The construction deserves a sentence on its arithmetic, because the smallness of the residual is otherwise easy to over-read.

A join of two image points is a cross product of their homogeneous coordinates. A meet of two image lines is a cross product of theirs. So the three meeting points are each four cross products deep, computed in double precision, and the collinearity of three points is one determinant. A residual of 1.3e-12 pixels over a span of 109 is what a chain of that length costs in rounding; it is not evidence that the theorem holds to a trillionth of a pixel, but evidence that it holds exactly and the arithmetic is the only thing left to measure.

Two properties of the construction are worth being explicit about, because they are what make it more than a curiosity.

The first is that it is projective. The hexagon in these figures is drawn on a photographed conic, not on a world conic — the camera has already been applied, the six marks are wherever the projection put them, and the sides are drawn in the picture. That the residual is at the floor anyway is not a separate result: a projection carries lines to lines and intersections to intersections, so a construction made of joins and meets cannot be damaged by one. The same statement that makes the cross-ratio survive makes Pascal survive, and for a shorter reason.

The second is that the theorem does not care about the order of the six points. A hexagon is a choice of cyclic order, and six points admit sixty of them, so one set of six points on a conic has sixty Pascal lines. Every one of them is genuine. The figure here takes one order and holds it fixed while the conic moves; a reader working on a photograph could take any.

Those sixty lines are not sixty unrelated facts either. They fall into twenty triples that are themselves concurrent, and the twenty points of concurrency lie by fours on fifteen further lines, and so on for several more storeys — the whole arrangement being the classical hexagrammum mysticum, which is one of the largest pieces of pure incidence structure anybody has found in a plane. None of it is measured here and none of it needs to be for the argument below; it is worth naming only because it makes the point that a conic and six marks on it are a far richer object than the curve alone, and every storey of it is joins and meets, so every storey survives a photograph.

Flatten the conic and Pascal becomes Pappus #

Pappus’s theorem is older than Pascal’s by fourteen centuries and is usually stated separately. Three points on one line, three on another, the same six joins, the same three meets, and the three meets are collinear.

Set the two statements beside each other and the only difference is the curve the six points are on. Pascal needs them on a conic; Pappus needs them on a pair of lines, three on each. A pair of lines is a conic — a degenerate one, whose quadratic form factors — so the second statement is the first evaluated at a degenerate member of the family.

That is the standard remark, and it is worth doing something with rather than repeating. The family here rides the six points down a one-parameter set of conics ending in two lines. Six points sit on the hyperbola $uv = s$ in asymptotic coordinates on the ground, three at fixed $u$ and three at fixed $v$, interleaved round the hexagon. As $s$ falls the hyperbola opens out towards its own asymptotes; at $s = 0$ the curve is the two asymptotes, and the first three points are on one of them and the second three on the other. Pappus’s configuration has been reached without any of the six points moving off the curve at any moment.

The limit is reached continuously #

Two pictures at the ends of a family show that the theorem holds at both ends. They do not show that the passage between them is smooth, and that is the claim worth measuring, because a degenerate case can perfectly well be a place where a construction breaks down and is rescued by a separate argument.

The four lines drawn together are the answer. The Pascal line rotates and translates as the conic flattens, and it arrives at the Pappus line the way a continuous function arrives at its value rather than the way a special case is substituted in. There is no setting at which the construction produces something other than a line, and no setting at which the residual jumps.

It is worth being exact about what “the same six points” means across the family, because the phrase could describe two quite different experiments. The six are not held fixed in the picture while the curve is swapped underneath them — that would put them off the conic at every setting but one, and the residual would be the control rather than the measurement. They are held fixed in the family’s own coordinates, three at one value of $u$ and three at one value of $v$, so each of the six slides along its own ruling as $s$ falls and every one of them is on the current conic exactly. What is constant is the combinatorics: which point is opposite which, and therefore which three meets are being tested.

This matters more than the tidiness of it. A theorem stated for conics and a theorem stated for line pairs are two theorems if the second has to be proved separately, and one theorem if the first specialises. The measurement says which, and it says one — with the qualification that a family and a limit are being used to say it, so what has been shown is that this passage to the degenerate case is smooth, not that every passage is.

The residual along the whole family, and its control #

Four settings are a sample. Twenty-six are a curve, and a curve is what shows whether anything happens in between.

The upper line is the whole reason the lower one is a measurement. Move one of the six marks off the conic — here by eleven centimetres on the ground plane, which is a few pixels in the picture — and the three meets stop being collinear, everywhere along the family, by an amount three orders of magnitude above the level the theorem holds at. The construction is not returning a small number because small numbers are what it returns.

That makes Pascal’s theorem a test for conicity, and a test of an unusually clean kind: it takes six marks on a photograph, uses no measurement, uses nothing about the camera, and answers whether the marks lie on a conic. It is the same class of instrument as the cross-ratio four marks subtend at a fifth point of the curve, and the two have complementary shapes: that one needs five marks and a walk, this one needs six marks and a single construction.

Why the family flattens to two lines and not to one #

An obvious alternative family gives the wrong answer, and the reason it does is worth a paragraph because it explains what Pappus’s configuration actually requires.

The natural way to degenerate a conic is to flatten an ellipse onto its major axis. Do that and the six points do land on a line — but on one line, all six of them. Pappus’s theorem has nothing to say about six collinear points: the three pairs of opposite sides are pairs of lines through pairs of points of one line, so every side is that same line, and the three meets are undefined. The construction has not degenerated, it has collapsed.

What Pappus needs is three points on each of two lines, which means the degenerate conic must be a pair of distinct lines rather than a doubled one. A hyperbola opening onto its asymptotes does exactly that, which is why the family here is $uv = s$ rather than an ellipse losing a semi-axis.

The distinction is the same one the outline of a degenerate quadric runs into, where a cone’s dual collapses to a single point and forgets which two lines pass through it. A degenerate object is not simply a smaller object; it is an object whose rank has dropped, and what it forgets depends on how it dropped.

What the reading is worth per pixel #

A test that answers yes or no is less useful than one that answers by how much, and this one does the second if its scale is known.

A slope of about a fifth is the number to carry away, and it is a slope rather than a threshold because that is what the geometry supplies. Displacing a mark by one pixel moves two of the six sides, moves two of the three meets, and tilts the line through them by an amount proportional to the displacement for small displacements. Nothing in that chain is discontinuous, so the residual is linear near zero with a coefficient set by the particular hexagon.

Which mark is displaced changes the coefficient. The six vertices do not enter the construction symmetrically — each is on two sides, and those two sides are in different opposite-pairs — so a hexagon has six slopes rather than one, and a reader calibrating on a real drawing would want the worst of them. The figure displaces the fourth.

The practical reading is that a fifth of a pixel of residual is a pixel of marking error, and marking error on a photograph is rarely under a pixel. So the test discriminates a curve that is not a conic at all from one that is, and it does not discriminate a conic from a carefully drawn near-conic — which is exactly the regime the four-arc ellipse the drawing office uses turns out to fail in anyway, at 1.7 per cent of a figure’s width.

The dual is the version a draughtsman can run #

Pascal’s theorem has a dual, obtained by exchanging every point for a line and every join for a meet, and the dual is a theorem in its own right with a separate name.

The exchange is exact and free — a point and a line are one object in this plane, and a conic’s tangent lines are a conic in the dual plane, which is what makes five tangents determine the same curve five points do. What is not free is which version is usable on a given drawing, and the two differ sharply there. A point marked on a drawn curve is a guess about where the curve is; a tangent laid against a drawn curve with a straightedge is a physical operation that a hand can do well. So the dual is the better instrument on a real drawing even though the two are the same theorem.

This collection has both, and the division of labour between them is the point. Pascal is measured here down a family, because the family is where the relationship to Pappus lives. Brianchon is measured against a curve that is not a conic, because rejecting a real drawing is where a straightedge test earns its keep.

The same six points carry a self-polar triangle #

One more structure lives in this configuration, and naming it connects the whole of it back to the constructions the collection already has.

Take any four of the six points. They form a complete quadrangle: four points, six joins, and three further crossings where opposite joins meet. Those three crossings are the quadrangle’s diagonal points, and the triangle they form is self-polar with respect to the conic — the polar of each vertex is the opposite side, exactly.

That is not a decorative fact. It is the reason the polar of a point can be built with two secants and a straightedge, where the construction is measured against the matrix product and agrees to 4.3e-13; it is the reason the diagonals of a photographed rectangle find its midpoint and a quadrilateral finds the middle of a receding side; and a self-polar triangle is what an ordinary camera’s three perpendicular vanishing points form with respect to the conic that calibrates it. One configuration, one relation, and three constructions that look unrelated until the relation is named.

So a Pascal hexagon is not a special arrangement. It is six points of a conic, and six points of a conic carry fifteen quadrangles, sixty Pascal lines and a great deal of incidence structure that a straightedge can reach and a ruler cannot.

What the test does not settle #

Three limits, stated in the order a reader would run into them.

The construction says the six marks lie on a conic. It says nothing about which conic, and nothing metric at all. Five marks determine the curve exactly and a sixth is a prediction; Pascal is the straightedge form of that prediction, and like it, it fixes shape in the picture and no size anywhere.

The residual is a picture-plane quantity and so depends on the drawing’s scale. Doubling the size of the print doubles the span, doubles the displacement a given marking error produces and doubles the residual, so a bare residual is not comparable between two drawings. What is comparable is the residual against the span — 1.3e-12 against 109, and 5.7e-14 against 140 — which is the form the figures print it in for that reason.

The construction also has a configuration it must refuse rather than answer badly, and the machinery here refuses it. At one setting of the family two of the six vertices coincide, which leaves a “side” joining a point to itself — an undefined line, not a nearly-defined one. Returning a Pascal line there would mean returning the cross product of a point with itself, which is the zero vector, and a determinant taken against it is zero for any three points whatever. That is the shape of false pass this collection watches for, so the family parameter that collapses the two is rejected outright while the settings either side of it are accepted.

And the theorem is a statement about six marks, not about the curve drawn between them. Six points chosen from two different conic arcs glued together will pass, provided the six happen to satisfy one conic; the test cannot see the curve it was not given. That is the same limitation the cross-ratio test has and it comes from the same place: a projection preserves incidence and nothing else, so a construction built only of incidences can only report incidences.

What Pappus being underneath Pascal is worth #

The last thing to say is why the continuity measured here matters beyond neatness, and it is about where these theorems sit rather than what they state.

Pappus’s theorem is the axiom that makes a projective plane coordinatisable by a commutative field. It is not a consequence of the incidence axioms; there are projective planes in which it fails, and in those planes the coordinates multiply in the wrong order. So Pappus is a structural statement about the plane, and Pascal is a statement about conics in it — and the family here shows that the second contains the first as the value it takes at a degenerate conic.

That is a satisfying direction of containment: the general theorem about curves reduces, continuously and at the arithmetic floor, to the axiom about the plane. It also puts Desargues’s theorem in the right relation, since Desargues is the weaker axiom that holds in more planes and follows from Pappus, and reads the same both ways for reasons of duality rather than of curvature.

Underneath all three is the same restriction this essay opened with. Joins, meets, and no measurement — which is the only kind of statement a photograph can be trusted to carry, and which is why the collection keeps arriving at constructions of exactly this shape.