
Standard surface
The standard surface is a conicoid - a surface of revolution whose cross section in the r-z plane is a curve given by the following parametric equation [Kidger2002]

In this equation c is the curvature at r=0 and the parameter ε determines the shape. The surface in 3-d space is obtained by revolving the curve about the z-axis (simply by using the parametric form above with r2=x2+y2). The standard surface is often expressed terms of the conic constant k defined by k=ε-1.
The sag of the surface is given in explicit form by the equation

The conic constant determins the shape according to the following table:
k>0 | ε>1 | oblate ellipsoid |
k=0 | ε=1 | sphere |
-1<k<0 | 0<ε<1 | prolate ellipsoid |
k =-1 | ε=0 | paraboloid |
k<-1 | ε<0 | hyperboloid |