A "'paraxial ray "'is a ray which makes a small angle ( " ? " ) to the optical axis of the system, and lies close to the axis throughout the system.
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For paraxial rays, the sphere gives the same results as the paraboloid or the exact fourth-order surface, as can be seen by Taylor-expanding the respective equations in terms of distance " y " from the axis ( the series for the paraboloid and the fourth-order surface are of course trivial ).
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