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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