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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]

c  ( r^2 + \varepsilon z^2  \right ) - 2  z =0

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

z={ cr^2 \over 1 + \sqrt{ 1 - (1+k)c^2r^2}}

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

See also