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