Magnification changes with field angle is distortion. You can therefore derive exact magnification calculations via Seidel aberration coefficients and/or Zernike polynomials...if and only if you want to earn extra credit in a graduate level optical aberration course.

In the real world, lens designers write a macro in Zemax or Code V to generate plots and data tables of distortion vs field angle as the image plane moves through focus while they go on their lunch break, then copy the plots into power point after they get back from lunch. Doing it this way is more accurate than calculating, since the data is generated by the software’s physics-based ray trace algorithms.

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