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CASSINIAN SURFACE
Multipolar equation: .
Algebraic surface of degree 2n. 
The Cassinian curves with n foci (or n poles), 3 dimensional analogues of the Cassinian curves, are the loci of the points in space for which the geometric mean of their distances to n points is a constant.
The case n = 2 gives the surface of revolution with meridian a Cassinian oval. 
If the poles are coplanar, then the section by the plane is a Cassinian curve.
Below are some examples of Cassinian surfaces the poles of which are located at the vertices of a regular polyhedron with radius a.
Poles at the vertices of a tetrahedron.
Various views when b/a increases from 0,9 to 2. 



Poles at the vertices of an octahedron.
Various views when b/a increases from 1 to 1.6. 



Poles at the vertices of a cube.
Various views when b/a increases from 1 to 2. 



Poles at the vertices of an icosahedron; for the dodecahedron, see the figure at the top. 

See also the Cassini surfaces.
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© Robert FERRÉOL
2017