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Why do the nodal, and principal points of a coaxial optical system coincide when the when the whole system is in the same medium?
But why does the refractive index of the medium outside the system become irrelevant?
The medium outside the system only affects the path of the rays that are incident upon the system's surfaces. However, when the entire system is immersed in a medium of uniform refractive index, the medium outside no longer exists, and the rays travel in a straight line through the system due to the coaxial symmetry.