Characterize of the surface of revolution generated by the cycloid in terms of the mean curvature and the Gaussian curvature as a function of the distance $x> 0$ to the axis of revolution.
I did a deep research and only get that there is a line $CD$ and a point $H$ that satisfies the uniqueness of the cycloid and the only curve that satisfies the conditions of the vertical tangent angle proposition, and passes through the point $H$ is the cycloid, but I not get the theorem to prove uniqueness. I need help I've been researching for weeks.