next surface previous surface 2D curves 3D curves surfaces fractals polyhedra

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.


Creation of Alain Esculier
next surface previous surface 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL 2017