On some special surfaces connected with convex surfaces of the Lobachevskii space

With a closed convex surface $\Phi$ in a Lobachevskii space we associate four special surfaces: the inscribed and circumscribed spheres, a sphere rolling freely over the inner side of $\Phi$ and an equidistant surface over whose inner side $\Phi$ rolls freely. We find an exact dependence between these four special surfaces.