I'm looking for a general equation/function z = f(x, y, radius1, radius2, p1, p2) for an atoric surface. p1 and p2 could be either eccentricity or conic constant values. Can anyone help me with that?


In case it is not clear what kind of surface it is, it's similar to a toric surface (so with spherical cross sections in two perpendicular axes where each has its own radius), but now the cross section won't be spherical anymore. Instead, each principal axis has each its own eccentricity or conic constant, causing the cross section to be a parabola, hyperbola, ellipse or a circle (when conic constant equals 0).


I'm not necessarily looking for the complete 3D surface of a solid. A surface with the above mentioned properties is good enough for me.


If I picked the wrong tags, please tell me.



Atoric equation
For reference, here is the description of the atoric surface described in [Spectacle lenses incorporating atoric surfaces](https://patents.google.com/patent/US7111937).


The profile lies in the $x$–$z$ plane and is parameterised by two variables $u,v$:
$$x(u,v)=u+a(u)q(v),\;\;y(u,v)=v,\;\;z(u,v)=p(u)+\gamma(u)q(v)$$
$$p(u)=\frac{c\_p u^2}{1+\sqrt{1-\epsilon c\_p^2 u^2}}+{\cal O}(u^4),\;\;q(v)=\frac{c\_q v^2}{1+\sqrt{1-\epsilon c\_q^2 v^2}}+{\cal O}(v^4),$$
and $n=(a(u),0,\gamma(u))$ is an inward unit normal vector of the $p(u)$ profile, as indicated in the figure:


![](https://ilorentz.org/beenakker/MO/atoric.png)


