A spacer defines the air-gap distance between a lens and a flat surface. The corner of the spacer can be modeled as a circle with radius, r1. First a circular lens is used, and the "sag," or amount that the lens bulges out on the axis, defines the air gap distance by: sag = R - (R2 - (chord)2) ½
In this case the contact point Δy above the axis is pretty straight forward to find. If now the lens is replaced with an asphere, sag defined as "z" below
Z =sag height
r =radial distance from vertex (Chord, or "half-chord")
c= (1/R) = curvature at the vertex
Find the correction to the 1st order in r in the relation between Δy and H, where H is the difference between the full width of the spacer and the air gap
Contact of a sphere and a sphere surface is investigated using derivatives. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.