Four points on a conic look the same from anywhere on it

The cross-ratio is the one number this collection trusts. Four points on a line keep it through any projection, and four concurrent lines have one of their own that any transversal reads off unchanged. Both are statements about four things, and both are the reason a photograph can be measured at all.

Chasles’ theorem is the moment that machinery attaches itself to a curve. Take four points of a conic. Join them to a fifth point of the same conic, giving a pencil of four rays, and read the pencil’s cross-ratio. That number does not depend on which fifth point of the conic was chosen — and it therefore belongs to the four marks and the curve together, with nothing left in it about where the reader stood.

That is a stronger claim than it sounds, and the strength is easiest to see by asking what would break it. A pencil at a fifth point is a very concrete thing: move the vertex and every one of the four rays swings. Nothing in the picture stays still. The four angles change, the four slopes change, and the four crossings on any drawn transversal change. One combination of them does not.

The theorem is about the conic, not about the four marks #

The derivation is short enough to give, and it explains where the invariance comes from rather than merely asserting it.

A conic has a rational parameterisation. Every point of it can be written as a quadratic in one parameter $t$, and for a circle the usual choice is the half-angle tangent. Fix four points of the conic at parameters $t_1$ to $t_4$, and fix a fifth at $t_0$. The line joining the point at $t_0$ to the point at $t_i$ has coefficients that are linear in $t_i$ once $t_0$ is fixed, because the quadratic terms cancel against the common factor the two points share. A pencil of lines whose coefficients are linear in a parameter is a projective copy of the parameter line — so the cross-ratio of the four rays equals the cross-ratio of the four parameters $t_1, t_2, t_3, t_4$.

And $t_0$ has gone. It cancelled in the step that made the coefficients linear, which is why the fifth point does not appear in the answer.

So the value is a function of the four marks alone, computed on the conic’s own parameter. The figure’s closed form says it had to be 1.627695 before any ray was drawn, and the drawn pencils agree with it at every one of the twenty-two positions. Two routes to one number, one of them geometric and one of them algebraic, is the shape this collection uses whenever a claim could otherwise be an artefact of how it was computed.

Notice what is absent from the derivation. No length, no angle, no midpoint, and no camera. The parameterisation is a property of the conic as a curve in a projective plane, and a projection carries a conic to a conic and a parameterisation to a parameterisation. That is the whole reason the theorem survives being photographed, and it is the same reason the polar of a point can be built with a straightedge and a receding row of posts can be divided without a ruler: a construction made only of joins and meets is a construction a projection cannot damage.

Walking the fifth point, and what stays still #

The two pictures above are the argument in its cheapest form. They are the same circle, the same camera and the same four marks; the only thing that changed is which point of the conic the pencil is drawn at, and the title above them is a single sentence that is true of both.

It is worth being precise about how much did change. The fifth point moved from position 5 of 22 to position 16 of 22, which is nearly half way round a closed curve. In the photograph that curve is an ellipse of appreciable eccentricity, so the two vertices are not related by any symmetry of the drawn figure — one sits on the near side of the imaged circle and the other on the far side, where the marks are crowded together by the foreshortening the ground plane imposes. The four rays at the second vertex are much closer in angle than at the first. Their cross-ratio is the same to every digit printed.

Twenty-two positions were used rather than four or five for a reason that is not thoroughness. The reading passes through infinity as the fifth point walks: whenever the vertex crosses one of the four marks’ antipodal positions, one of the four rays sweeps through the transversal’s direction and the cross-ratio runs off the end of the real line and comes back from the other end. A sparse sample would step over those events and report a small range by luck. A dense one walks through them, which is what forces the question of how the agreement should be measured at all.

The reading lives on a circle, so its spread is an angle #

This is the part that cost the instrument its first version, and it is worth a section because it is a general lesson about measuring agreement between quantities that can be infinite.

The cross-ratio of four rays is a point of a projective line, not a real number. The projective line is a circle: the values $+\infty$ and $-\infty$ are one point of it, and two readings of $10^{6}$ and $-10^{6}$ are neighbours rather than opposites. Taking the largest reading minus the smallest therefore measures the wrong thing. It reports an enormous number whenever a sample happened to land near a pole, and a modest one whenever the samples straddled the pole symmetrically — which means the arithmetic range is not monotone in how far the vertex is from the conic. A vertex further off the curve can read a smaller range than one nearer it, purely by where the samples fell.

The honest measure is the one the projective line actually carries. Map each reading through $\theta = \arctan(x)$, which sends the whole line onto a half-turn with the point at infinity in its proper place, and take the diameter of the resulting set of directions. That number is zero exactly when every reading is the same projective point, it is bounded by ninety degrees, and it increases when the readings genuinely disperse. It is what every figure here calls the projective spread.

On the conic the spread is 1.1e-13 degrees. That is not “small”; it is the floor of double-precision arithmetic on this configuration, and the same figure at a different circle size reads the same order of magnitude. There is no residual left to attribute to the geometry.

The control is six pixels away #

An invariance with no control beside it is not a measurement. Anything computed the same way twice agrees with itself, and a great deal of the value of Chasles’ theorem is that it stops holding the moment its hypothesis fails.

So the same twenty-two-position walk is run with the fifth point on a curve just inside the imaged conic instead of on it. At 0.97 of the radius the two curves are 6.1 pixels apart in the picture — closer than a careful hand can mark a photograph, and far closer than the width of the line the conic is drawn with. The readings run from 1.00 to 2.52 and the projective spread is 23.5 degrees against the conic’s 1.1e-13.

Two things follow. The first is that the theorem’s hypothesis is doing work: it is not a statement about four points and an arithmetic that happens to be stable, because a configuration differing from it by six pixels is not stable at all. The second is more useful. Since the reading is flat on the conic and steep off it, the reading is an instrument, and the essay ends by calibrating it.

It is worth naming the failure this control is guarding against, because this collection has shipped it before. A necessary condition evaluated at the one input where it cannot fail is not a test. If the control point had been left on the conic — an easy slip, since a control is usually written by copying the measurement and changing one number — the second series would have been flat too, the figure would have looked like a strong result, and it would have said nothing whatever.

Being nearly on a conic is not a weaker version of being on it #

The picture above is the one to keep. Both pencils are four rays from a point near a curve to four marks on it, both are drawn to the same standard, and there is no visual difference between the case in which the theorem holds exactly and the case in which it does not hold at all.

That asymmetry is characteristic of projective statements, and it is worth stating as a general warning rather than as a fact about this figure. Metric statements degrade gracefully: a length measured with a slightly bent ruler is slightly wrong. Incidence statements do not degrade at all — a point is on a curve or it is not, and the theorems that depend on incidence stop applying the moment it fails, with no intermediate regime in which they nearly apply. What replaces the theorem is a rate, and the rate is the subject of the last figure here.

The same thing is true one dimension up in the ambiguity two circles in different planes can share: configurations that a photograph cannot tell apart, and configurations one degree away that it separates immediately, sit next to each other with nothing in the drawing to mark the boundary.

A photograph does nothing to any of this #

The circle in every figure here is on the ground and is being seen from a camera 23 centimetres from the page at the width these are printed. None of that enters the theorem, and it is worth showing why with the four-point case rather than merely asserting it.

The chain is three links long. Four collinear points have a cross-ratio and a projection preserves it. Four rays through a point have a cross-ratio, equal to the one their four intersections with any transversal carry, so the pencil is as good a carrier as the line. And a projection carries a conic to a conic, a point of the conic to a point of the conic, and the parameterisation to a projectively equivalent parameterisation — so the four parameters change but their cross-ratio does not.

The consequence is worth stating plainly, because it is what makes the theorem usable on a photograph of something rather than on the thing. The reading 1.627695 is not the value in the world corrected for the camera. It is the value in the world, obtained from the picture, with no knowledge of the camera, the circle’s size, the plane it lies in, or where the photographer stood. It is in the same class as the angle a Laguerre reading recovers and unlike everything in the collection that needs the camera recovered first, such as the focal length three vanishing points give.

Read backwards, the theorem is a definition of a conic #

Chasles’ theorem is usually met as a property of a curve that is already there. It is more interesting read in the other direction, and the reverse reading is what makes it foundational rather than merely true.

Fix two points $P$ and $Q$ and a projective correspondence between the pencil of lines at $P$ and the pencil at $Q$ — a rule that turns each ray through $P$ into a ray through $Q$, preserving cross-ratio. Every such correspondence is determined by three pairs. Now let a ray turn about $P$ and mark where it crosses its partner at $Q$. The locus of those crossings is a conic through $P$ and $Q$, and every conic through the two arises this way.

That is Steiner’s construction, and it is Chasles read from right to left. The forward statement says that the pencils at two points of a conic are projectively related, with the correspondence being “same point of the curve”. The backward statement says that being projectively related is all a pair of pencils needs, and the conic appears as a consequence.

Two things follow that are worth having early. The first is that a conic is not primarily a quadratic equation; it is what two projectively related pencils draw, and the equation is a fact about coordinates rather than about the object. The second is procedural: since the whole construction is joins, meets and a projectivity — and every projectivity is composed of perspectivities — a conic through five given points can be drawn on a photograph, point by point, with nothing but a straightedge. The five points fix the two pencils and the three pairs that determine the correspondence between them, and the sixth point of the curve is then a construction rather than a computation.

Four marks and a fifth point are a test for conicity #

Since the reading is flat on a conic and steep off it, a reader with a photograph and a pencil has a test: mark four points on a suspected conic, pick a fifth on it, read the cross-ratio, pick another fifth, read it again. If the two disagree, the curve is not a conic.

The useful form of that is a threshold, and the last figure is the calibration.

The interesting end of that scale is the bottom. Two pixels of error already produces eight degrees of spread against a floor of a ten-thousand-billionth of one, which means the test has an enormous dynamic range near the hypothesis and is not limited by arithmetic anywhere a reader could work. What limits it is the marking: a spread of eight degrees is only evidence if four marks and a fifth can be placed to better than two pixels, and on a real photograph of a real curve they usually cannot.

The top end is worth reading too, because it says when the instrument stops being one. By twenty-four pixels the spread is within a degree and a half of the ninety that means “as far apart as readings can be”. Past that point the test still says not a conic and can no longer say how far from one, which is a saturation rather than a failure and is the ordinary fate of a bounded statistic.

What the reading does not settle #

Three limits, and the third is the one that matters.

The spread is a statement about the marks, not about the curve between them. Four points and a fifth are five points, and five points determine a conic exactly — so a walk that returns a constant reading is evidence that the marked points lie on some conic, and says nothing about whether the drawn curve between them does. A curve made of conic arcs glued together would pass a test whose five marks all fell in one arc.

The reading is a projective invariant and therefore carries no metric information at all. It does not say which conic, it does not say where the plane is, and it does not distinguish a photographed circle from a photographed ellipse — the two circles that draw one picture read identically, as they must. Everything metric costs a further fact from outside the picture.

And the reading is a statistic computed from marked points, which means it inherits their errors in a way this collection has already measured and found asymmetric. A cross-ratio estimated from noisy marks is biased, not merely noisy, because the estimator is a ratio and the expectation of a ratio is not the ratio of expectations. The bars above are computed from exact arithmetic on exactly placed marks, so they are the instrument’s ceiling rather than its performance. A reader working on a photograph should expect the floor to sit well above 1.1e-13, and the honest way to find out where is to run the same walk on a curve known to be a conic.

What this is the bottom rung of #

Chasles’ theorem is the entry point to everything this collection does with conics, and it is worth naming the flight of stairs it starts.

Five points determine a conic because five is the number of degrees of freedom, and the sixth mark that was withheld from the fit lands on the curve as a prediction. The dual statement is that five tangents determine the same conic by the same algebra with the roles exchanged. Chasles sits underneath both, because the projective parameterisation it exposes is what makes a conic a one-dimensional projective object at all — and a hexagon inscribed in a conic has a Pascal line for exactly this reason, the three meets of its opposite sides being collinear as a consequence of the same cancellation that removed the fifth point above.

The Pascal line is the first construction that uses the theorem rather than restating it, and it is a construction of the restricted kind: six joins and three meets, no measurement anywhere, so it can be carried out on the photograph. Its degenerate case, where the conic has flattened into a pair of lines, is Pappus’s theorem — which is the sense in which Chasles is underneath both.

Above that sit the things a conic in a picture buys: the centre that a projection destroys and a polar construction recovers, the two hidden points that upgrade a photographed plane to a metric one, and the fact that a photographed circle can be a hyperbola and calibrate exactly as well.

None of that would be worth much if the object underneath it were fragile, and the whole point of the control here is that it is not fragile in the way that matters. The invariance is exact where it holds and it fails immediately where it does not, which is precisely the behaviour a foundation should have.