In an elliptical table every path hugs a hidden curve forever. Poncelet’s porism, caustics, and the 2022 proof that ellipses are special
A ball on a frictionless table travels in straight lines and bounces with the angle in equal to the angle out. In a circle or an ellipse, that rule produces perfect order: every path hugs a hidden curve, for ever.
Press inside the table and drag to launch a ball. However long it runs, its path stays tangent to one curve that shares the table’s foci.
Press inside the table to place the ball and drag to aim it. λ is the confocal parameter of the caustic x²/(a²−λ) + y²/(b²−λ) = 1, computed from the current chord; it never changes.
In a circle the story is short. Every bounce leaves at the same angle to the wall, so every chord is the same length and sits the same distance from the centre. The chords all touch a smaller circle, the caustic (the word for a curve that rays of light pile up along). If the angle is a rational fraction of a full turn the path closes into a star polygon; otherwise it never closes, and it fills the ring between the caustic and the wall.
An ellipse hides the same structure less obviously. Its caustics are the conics with the same foci: smaller ellipses, and hyperbolas. A path that passes between the two foci always passes between them, and stays tangent to one hyperbola; a path that passes outside the foci always passes outside, tangent to one smaller ellipse. A path through one focus reflects through the other, which is why an elliptical room whispers. The quantity λ in the demo picks out which conic it is, and the bounce rule never changes it, just as energy never changes for a planet.
A billiard is called integrable when, like this, there is a conserved quantity for every orbit. The circle and the ellipse are the only known convex tables that are integrable all the way out to the wall.
Tune the caustic so that one path closes after a few bounces. Then slide its starting point round the table: every other path tangent to the same caustic closes too.
The caustic is found once, by bisection, so that the path from the end of the long axis closes after 5 bounces. Every other starting point is then just launched along a tangent to the same caustic and followed honestly; it closes on its own.
Jean-Victor Poncelet was a French army engineer captured during Napoleon’s retreat from Moscow in 1812. As a prisoner in Saratov he rebuilt projective geometry from memory, and published it in 1822. One result is his porism: take two conics, one inside the other, and draw a polygon whose corners lie on the outer one and whose sides touch the inner one. If the polygon closes up for one starting corner, it closes up for every starting corner, with the same number of sides.
For two confocal conics this is a statement about billiards, because a billiard path in an ellipse is exactly such a polygon, its sides tangent to the caustic. The demo finds the caustic for a chosen rotation number once, by bisection, and then launches every other path honestly. The gap between start and finish stays at rounding level all the way round.
The modern explanation goes through elliptic curves. The pairs (corner, tangent line) form an elliptic curve, a torus, and one bounce is a fixed rotation of that torus. A rotation that returns after q steps from one point returns after q steps from every point.
Record each bounce as a point: where it hit, and the angle it left at. Orderly tables draw curves. Click the portrait to launch your own orbit.
Horizontal: where the ball hits, as a fraction of the way round the wall. Vertical: the cosine of the angle it leaves at, from gliding one way (top) through straight across (middle) to gliding the other way (bottom). Each orbit is one colour.
George Birkhoff made billiards a model problem for dynamics in the 1920s. The trick in the demo is his: forget the flight between bounces and keep only the sequence of impacts, as points of a cylinder (position round the wall, angle). In the circle each orbit is a horizontal line. In the ellipse the lines bend, and an eye-shaped island appears around the orbit that bounces along the short axis; the boundary between circling and crossing orbits is the orbit through the foci.
For any smooth, strictly convex table, Lazutkin showed in 1973 that some caustics survive near the wall, so the top and bottom of the portrait always contain curves. Birkhoff (with Poritsky) asked whether any table other than an ellipse is integrable throughout. Misha Bialy proved in 1993 that if every orbit lies on a curve, the table is a circle. Avila, De Simoi and Kaloshin (2016) and Kaloshin and Sorrentino (2018) proved that a table close enough to an ellipse and integrable near its wall must be an ellipse. In 2022 Bialy and Mironov proved the conjecture for centrally symmetric tables, under a condition on how far the curves extend. The general case is still open.
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