Imagine an elliptical billiard table, and a ball on its edge; we hit the ball so that it starts bouncing on the walls. Now, the study of the trajectories seems to be a fairly recent topic (for instance, they are tangent to certain caustic conics), but there’s one result I cannot seem to find anywhere on the web:

If the ball at some point returns to its original position, will it then repeat the same trajectory?

Because nothing guarantees that, a priori, the last segment is the reflection of the first one.

I have looked at a small introduction page at Wolfram, and they do say the following:

On an elliptical billiard table, the envelope of a trajectory is a smaller ellipse, a hyperbola, a line through the foci of the ellipse, or a closed polygon (Steinhaus 1999, pp. 239 and 241; Wagon 1991). The closed polygon case is related to Poncelet's porism.

Poncelet’s porism seems to imply the existence of infinite such polygons, for every $n$, but I’m not really sure how it answers the question as posed. And I couldn’t find the proofs by the authors referenced without buying their books.

The only related question I could find on this site was this one, but it was dealing more with the existence of closed orbits for a given number of sides and seemed to assume that the trajectory would repeat itself.

I’m fairly sure there’s a bruteforce way to do this, maybe with vectors or even complex numbers - and though I’m willing to accept those, I was hoping to find a purely geometric approach to this problem. To be honest, The only thing I could think of was to try and project the problem so the ellipse became a circle, but I couldn’t find something to characterize the projected path (since the angles of reflection aren’t invariant by projection, the new trajectory wouldn’t be a bouncing ball anymore).

So my question is: if the ball returns to its starting position after a finite number of bounces, how do we prove that it will repeat that same trajectory henceforth?

  • $\begingroup$ It will not repeat its trajectory unless it is reflected in its initial direction. So I suppose what you asking is proof that if it does return to its original position then it does so only from the direction which results in it being reflected along its initial direction. $\endgroup$ Apr 10, 2020 at 19:07

1 Answer 1


Let $F_1$ and $F_2$ be the foci of the ellipse. According to Theorem 4.4 in "Geometry and Billiards" by Serge Tabachnikov:

A billiard trajectory inside an ellipse forever remains tangent to a fixed confocal conic. More precisely, if a segment of a billiard trajectory does not intersect the segment $F_1F_2$, then all the segments of this trajectory do not intersect $F_1F_2$ and are all tangent to the same ellipse with foci $F_1$ and $F_2$; and if a segment of a trajectory intersects $F_1F_2$, then all the segments of this trajectory intersect $F_1F_2$ and are all tangent to the same hyperbola with foci $F_1$ and $F_2$.

But at most two tangent lines can be drawn to an ellipse or hyperbola from a given point. Hence, if the starting point is reached again, the trajectory can only repeat.

  • 1
    $\begingroup$ Nice! I thought the caustic was a later result, or that it required analytical arguments. I checked the source and it’s a synthetic proof, and the OP follows immediately. Very good! $\endgroup$
    – user35013
    Apr 10, 2020 at 21:52

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