Ray-tracing formulas are developed in this paper for refraction or reflection at a plane surface, a spherical surface, or an aspheric surface with rotational symmetry. The formulas have similar forms for all types of surface and for both meridional and skew rays, and they are especially adapted for rapid automatic computers. Certain quantities enter the formulas implicitly, and an approximation formula is developed for computing these quantities with adequate precision on the first repetition.

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