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I don't follow.

I assume that if numerically solving for a correcting shape was burdensome you would model an element using a first surface and a 'dummy' second surface that your software just treats as magically correcting. ... then when you are happy with your design, you go ahead and compute the actual shape for that second surface... or similar.

I would expect a closed form solution to more useful for meta analysis of the problem-- looking at its behaviors (and especially derivatives) in various external cases may suggest interesting and novel optical system designs. Like, "oh, foo changes cubically with with an infinite focal length that means if we could make lenses with cherry flavored unobtanium we could bounce a gravitation particle beam off the main deflector dish!".




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