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CASSINI SURFACE
 

Cartesian equation:.
Quartic surface.

 
The Cassini 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 .


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

To generalize to space the Cassinian ovals, we can also consider 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.

See also the Cassinian surfaces.
 
 
 
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© Robert FERRÉOL 2017