I'm looking for a kind of generalisation of an ellipse; a shape with a more complicated optical property. I'm not sure how to rigorously define this shape, or prove that it exists, or find an equation for it. So any comments or ideas in that direction would be appreciated.
The ellipse with foci A, B has the property that every billiard trajectory from A will reflect once and then reach B. Equivalently, for any point P in the ellipse, we have $$|P - A| + |P - B| = l$$ for some constant $l$.
What I'm looking for is a shape S (probably piecewise defined by two different curves), such that every trajectory from A will reflect twice in S and then go to B.
If the shape is defined by two curves $\gamma_1$ and $\gamma_2$ (as I suspect it will be), then I think my problem is equivalent to the following:
For any $P$ in $\gamma_1$, there exists $Q$ in $\gamma_2$ such that $$|P - A| + |Q - P| + |Q - B| = l$$ Furthermore, for any $Q$ in $\gamma_2$, there exists $P$ in $\gamma_1$ such that $$|Q - A| + |P - Q| + |B - Q| = l$$ Note that these conditions are not equivalent.
Can anyone see any way to turn this definition into a more traditional description of a shape? Maybe an implicit equation for the curves $\gamma_i$?
If $A=B$ just choose a circle around the point.
Otherwise, assume without loss of generality that the points are located at $A(-1,0)$, $B(1,0)$. Then, the billiard consists of three arcs, semicircles of radius $1$ centred at $A$ and $B$ and lying above the $x$-axis, and a semi-ellipse with foci at $A$ and $B$ below the $x$-axis.