If I have a lens with coordinates UV on the lens surface where U, V are [-1, 1] and I want to find the real-world (x,y,z) coordinates of the UV point, how would I do that if I have the following information:
-The lens is centered on the z-axis.
-You know the lens' z-intercept value z.
-The diameter of the lens is d.
-It has a spherical radius of curvature value 'c' where c>0, c<0 and c=0 are convex, concave and flat respectively.
The equation for the sag $z$ of a spherical lens of curvature $c$ is
$$z = \frac{1}{c} \left [ 1-\sqrt{1-c^2 \rho^2}\right ] = \frac{c \rho^2}{1+\sqrt{1-c^2 \rho^2}}$$
$\rho^2=x^2+y^2$ may be written in terms of your $(u,v)$ coordinates as
$$\rho^2=\frac14 d^2 (u^2+v^2)$$