Periodic light rays in ellipse

Question

I was just fiddling with GeoGebra when I found this: imagine a light ray in an elliptical mirror; whenever I choose a point $A$ on the ellipse and a positive integer $n$, I can adjust $B$ so that the ray starting from $A$ and going towards $B$ returns to $A$ and touches the ellipse in exactly $n$ points. Here are some examples. The last picture shows 28 reflections.

First of all, is this true? If so, is it possible to prove it with not too advanced tools? Is this a peculiarity of the ellipse only, or there are other curves with the same property?

Answer 1

I'm going out on a limb and suggesting that this is true and easy (which means it doesn't start addressing the really hard questions about billiards on elliptical tables).

I argue from continuity.

If you start at $A$ and go only a short distance to $B$ then you will bounce around a convex polygon and return to a point past $A$ in a finite number of steps. That point depends continuously on the position of $B$. The nearer $B$ to $A$ the more steps that will take. Now given $n$, imagine moving $B$ away until you reach the transition that just hits $A$ in $n$ steps.

I suspect that's the strategy the OP used to construct his examples in GeoGebra.

In fact, the ball will continue to follow this trajectory:

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). (http://mathworld.wolfram.com/Billiards.html)

(I found that in responding to the OP's comment below and edited question demonstrating the fact.)

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Leaving this for historical reasons:

If this argument is wrong I'm sure a commenter will point that out. Then I can decide whether to delete it, or leave it as an instructive failure. If it's right it probably works for many smooth convex sets .