[math-fun] Nomenclature of spherical surface objects ?

Is there any standard terminology for the projections of the faces onto the circumsphere of the regular polyhedra? I.e., is there a standard name for the projection of one of the faces of a regular tetrahedron? One of the faces of a cube? Etc. What about the locations (lat/long) of the vertices of these regular polyhedra? Since there are only a few of these, I thought that someone might have come up with specific names for them.