The cyclides are images of
under inversion in a sphere. The animation moves the center of the sphere. (Inversion in the unit sphere maps vectors x in R^3 to x/|x|^2.) Just like tori, cyclides can be touched, from inside and from outside, along parameter lines, by 1-parameter families of spheres.
Cyclides, artistic rendering, by Luc Benard.
3DXM Surface Gallery