 |  |
The quartic surface obtained by replacing the constant 
in the equation of the Cassini ovals with 
, obtaining
![[(x-a)^2+y^2][(x+a)^2+y^2]=z^4.](/images/equations/CassiniSurface/NumberedEquation1.svg) |
(1)
|
As can be seen by letting 
to obtain
 |
(2)
|
 |
(3)
|
the intersection of the surface with the 
plane is a circle
of radius 
.
The Gaussian curvature of the surface is given implicitly by
![K(x,y,z)=(a^2(a^2+x^2-y^2))/(z^2[a^4+2a^2(-x^2+y^2)-2z^2(x^2+y^2+z^2)]).](/images/equations/CassiniSurface/NumberedEquation4.svg) |
(4)
|
Let a torus of tube radius 
be cut by a plane perpendicular to the plane of the torus's
centroid. Call the distance of this plane from the center of the torus hole 
, let 
, and consider the intersection of this plane with the torus
as 
is varied. The resulting curves are Cassini ovals,
and the surface having these curves as cross sections
is the Cassini surface
 |
(5)
|
which has a scaled 
on the right side instead of 
.
