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What's the surface joining two points in a family of concentric spheres? Shown below is the general idea; it's actually optical. Two rays meet at P from P1 and P2, respectively, where each point comes from a different sphere. How do I find surface S if I know the coordinates of P1 and P2?

My best bet is that one can describe S as

[tex] (x-h)^2+(y-k)^2=r^2 (\phi _i), \qquad R_1 \leq r (\phi _i ) \leq R_2 \mbox{ and } \phi_2 \leq \phi_i \leq \phi_1 [/tex]

but that seems too abstract and 2D. I'm looking for something like an even asphere description with radius of curvature and coefficients if I know P1 and P2. How can I do that?

Thanks