TITLE: On reflection properties of convex regions
QUESTION [6 upvotes]: It is well known that any ray of light passing thru a focus of an ellipse will pass thru the other focus after a single reflection from the ellipse boundary. If $A$ and $B$ are the foci of an ellipse, this property of rays holds both ways (those passing thru $A$ meet at $B$ and vice versa).

Is there a closed convex region $C$ with the property: there exists a pair of points $A$ and $B$ within $C$ such that all rays thru $A$ will reflect once on $C$ and pass thru $B$ but not all rays thru $B$ will pass thru $A$ after one reflection from $C$?

Is there a closed convex region $C$ such that: there is a pair of points $A$ and $B$ in the interior such that all rays thru $A$ pass thru $B$ after exactly 2 reflections from $C$? It is sufficient for the ray going directly from $A$ to $B$ to get reflected exactly twice from somewhere on $C$ and then pass thru $B$.


Note 1: Question 2 can have 'one-way' (convergence only of rays thru $A$ at $B$) and 'both-ways' variants.
Note 2: If not explicit constructions, even existence/non-existence arguments could be sought as answers to these questions.
One can also ask if relaxing convexity has any implications.

REPLY [2 votes]: The answer to question 1) is no:
Take any ray R passing through B. Since C is convex and B lies in the interior of C, R will intersect C at a point O. Now the ray $\overrightarrow {AO}$ lies in the interior of C and its reflected ray passes through B. But this means we have a path of light $\overrightarrow {AO},\overrightarrow {OB}$. Since C is smooth the tangent at O is unique. Hence we can reverse this to give a light path $\overrightarrow {BO},\overrightarrow {OA}$ and hence the ray R from B will reflect off C at O and pass through A.
Since R was arbitrary, any ray passing through B will reflect off C and pass through A.
For question 2) a good example is to take two parabolas facing each other with a common axis, where the parabolas intersect at points A and A':

Rays through the focus of either parabola, not passing through A or A' or hitting the other parabola first will be reflected twice and then pass through the other focus. If we move the foci apart we can get an arbitrarily large percentage of directions to work.
Thus the answer to 2) is yes if we allow an arbitrarily small percentage of rays from each focus to be excluded. This is the 2-way version.
Note that we can prove that such any example has to be 2-way in the same way as we did for question 1).