A parabolic cyclide is a cubic surface obtained by inversion of a standard torus with positive major and minor radii and the inversion center at a point in the smooth locus of the torus. In the nondegenerate case, the two sphere-center loci of this cyclide are parabolas (Alcázar et al. 2018).
The figure above shows the ring torus 
, with major radius
2 and minor radius 1, together with a bounded view of
its inversion. The orange marker is the inversion
center. Invert in the unit sphere centered at
, a point on the torus.
Writing 
for coordinates relative to this inversion center
and 
,
substitute 
into the torus equation. Clearing denominators
gives the parabolic ring cyclide
 |
The smooth locus hypothesis excludes inversions that reduce the polynomial degree further. For example,
the horn torus 
has a point of self-contact at the
origin. Inversion in the
unit sphere centered there gives the circular cylinder
 |
