Dupin cyclide as the stereographic projection of a Hopf torus

Question

Let $H \colon S^3 \to S^2$ be the Hopf map and let $\gamma$ be a curve on $S^2$. Then $H^{-1}(\gamma)$ is called the Hopf cylinder or the Hopf torus when $\gamma$ is closed, with profile curve $\gamma$.

When $\gamma$ is a circle on $S^2$, the stereographic projection of the corresponding Hopf torus highly looks like a Dupin ring cyclide. Is it really a Dupin cyclide and how to prove it? I know three definitions of a Dupin ring cyclide: 1) definition by means of parametric equations, 2) definition as an isosurface, 3) definition as an inverted torus. I don't see how to check with any of these definitions.

Answer 1

Yes, Hopf tori (coming from round circles) are Dupin cyclides. We see this using your third definition.

Stereographic projection sends round spheres to round spheres (and planes). Thus the projection conjugates inversion in spheres to inversion/reflection in spheres/planes. Thus rotations in the three-sphere are conjugated to Möbius transformations. In particular this holds for elements of $\mathrm{SU}(2)$. Since you started with a round circle in the two-sphere, the rotation making it standard (centre at the north pole) is conjugated to a Möbius transformation making its Hopf torus standard (a surface of revolution).