[ATM] Cylindirical Extension to Conicoid Modelling

[Richard F.L.R. Snashall]

I have been trying to extend the user interface for the
conicoid model:

   z - 0.5 * ( 1 + k ) * C * z^2 - 0.5 * C * ( x^2 + y^2 ) = 0

to include cylindrical lenses, maybe something like:

   z - 0.5 * ( 1 + k ) * C * z^2 - 0.5 * C * ( mx * x^2 + my * y^2 ) = 0

This extension still has the property that the intersection
of a ray with the surface is still the root of a quadratic
in the physical pathlength.

This model allows for cylindrical lenses and some general
astigmatic qualities, i.e.: different curvatures in the
x and y directions, but not different conic constants.

I thought of just allowing for just the extra curvature (or
difference in curvature); to model a strict cylindrical lens,
though, it would require some form of pick-up.

I also thought of of models based on a single parameter, m:


     mx = 1 + m/2 - 7*m^2/12 + m^4/12
     my = 1 - m/2 - 7*m^2/12 + m^4/12


This gives (in the parentheses of the last term):

    -2        - x^2 + y^2
    -1                y^2
     0          x^2 + y^2
     1          x^2
     2          x^2 - y^2

For m >= 0 , only the y-direction curvature changes, while for
m <= 0, only the x-direction curvature changes.

A simpler alternative is:

     mx = 1 + m/2 - m^2/2
     my = 1 - m/2 - m^2/2

but only works works in this way for the three middle values.

I thought , though, that perhaps all of this may sound kind of
"too cutesy".  Any alternative ideas?