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CASSINI SURFACE

There are many generalizations to the space of the Cassinian ovals, sometimes named Cassini surfaces.

First generalization
 
 

Cartesian equation:.
Quartic surface.

 
This surface is a surface the level curves of which are Cassinian ovals: the level curve z is the Cassinian oval with parameter b = z with foci centered on the lines .

The section by the plane y = 0 is the reunion of the circle  and the conjugate hyperbola .

Second Generalization
 
Another surface the level curves of which are the Cassinian ovals is the surface .

Third generalization : 3D cassinian surface with two poles
The surfaces of revolution obtained when a Cassinian oval turns around its axis (here, the axis Ox):

Bipolar equation: 
the pole O being the middle of the bipole (F', F), with a = OF = OF'
Cartesian equation: .
Quartic surface, non rational for ab.

When , the section of the surface by the plane  gives the lemniscate of Booth.

Fourth généralization
 
This time, we rotate the oval around its small axis (here, the axis Oz).
Cartesian equation:.
                                           .
 For , the surface is used to model a red blood cell (see this paper).

 


 
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© Robert FERRÉOL 2020