This appendix describes a ray tracing scheme that can be used to trace a meridional ray by hand. It is a slight modification of one of T. Smith's ray tracing procedures [74]. In this calculation a ray approaching a surface is specified by the sine of the angle ψ with the axis, and the perpendicular distance h from the vertex of the surface to the ray. The calculation yields the sine of the angle ψ′ with the axis after refraction, and the perpendicular distance h′ from the vertex to the refracted ray. An additional calculation determines the change in h′ as the ray travels to the next surface.

Table A3.1 shows a ray traversing a cemented doublet which happens to have a unit focal length. The radii, refractive indices, and thicknesses are found on lines 1, 3, 4, and 5. The ray enters the lens parallel to the axis; so ψ = 0 for the first surface, as shown on line 11. The entrance height is 0.125, corresponding to an F/4 aperture; it is shown on line 6.

The change in direction is calculated in two steps. Lines 12 through 15 yield an approximate value sinθ for sin ψ′. Paraxially this value would be correct, but to obtain exact results a correction must be made. This correction is calculated in lines 18 through 26, followed by lines 16 and 17. The change from h to h′ is generated at the same time (lines 7 and 8).