The traits class Env_sphere_traits_3<ConicTraits> models the EnvelopeTraits_3 concept, and is used for the construction of lower and upper envelopes of spheres. Note that when projecting the intersection curve of two spheres (a circle in 3D) onto the -plane, the resulting curve is an ellipse. The traits class is therefore parameterized by an arrangement-traits class that is capable of handling conic curves - namely an instantiation of the Arr_conic_traits_2 class-template - and inherits from it.
The conic-traits class defines a nested type named Rat_kernel, which is a geometric kernel parameterized by an exact rational type. Env_sphere_traits_3<ConicTraits> defines its Surface_3 type to be constructible from Rat_kernel::Sphere_3. Namely, it can handle spheres whose center points have rational coordinates (i.e., of the type Rat_kernel::FT), and whose squared radius is also rational. The Surface_3 type is also convertible to a Rat_kernel::Sphere_3 object.
The Xy_monotone_surface_3 type is the same as the nested Surface_3 type. The traits-class simply ignores the upper hemisphere when it computes lower envelopes, and ignores the lower hemisphere when it computes upper envelopes.