Generatrix of catenoid of space 3-form

Constant mean curvature surfaces of revolution in euclidean 3-space are known as surfaces of Ch. Delaunay. They possess one remarkable property: their profile curves (generatrices) are the trajectories of focuses of conic sections by its rolling along the straight line. Analogous construction is realized in the space forms $H^3$ and $S^3$ in the case of minimal surfaces of revolution and the following theorem is proved.
Theorem. Generatrix of catenoid of revolution of space form $H^3(S^3)$ is the trajectory of focus of hyperbolic (spherical) parabola by its rolling along the geodesic ray.