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. See more details on this page.

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.

Creation of Alain Esculier

© Robert FERRÉOL 2017