An optical lens for focusing two pairs of points

We construct an optical lens in the (x, y)-plane which focuses two pairs of points, i.e., all the rays from a given point X _i are focused by the lens at a given point y _i , for i = 1, 2. The points X ₁, X ₂, Y ₁, Y ₂ lie on the x-axis and the lens has the form $$\left\{ {\gamma _{\text{1}} {\text{ }} + {\text{ }}f_{\text{1}} {\text{(}}y{\text{) }}\underline \leqslant {\text{ }}x{\text{ }}\underline \leqslant {\text{ }}\gamma _{\text{2}} {\text{ }} + {\text{ }}f_{\text{2}} {\text{(}}y{\text{)}},{\text{ }}\left| y \right|{\text{ }}\underline \leqslant {\text{ }}y_{\text{0}} } \right\}$$ where γ ₁, γ ₂ are given, and f _i (0) = 0, f _i (?y) = f _i (y). We then let X ₂ → X ₁, Y ₂ → Y ₁ and investigate the limiting lens. We show that this limit is generally not a symmetric lens, i.e., f ₁ + f ₂ ? 0.