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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):
Bifocal equation:
,
the pole O being the middle of the foci (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