CGAL 5.4.4  3D Envelopes

#include <CGAL/Env_sphere_traits_3.h>
ConicTraits.
The traits class Env_sphere_traits_3
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 \( xy\)plane, the resulting curve is an ellipse. The traits class is therefore parameterized by an arrangementtraits class that is capable of handling conic curves  namely an instantiation of the Arr_conic_traits_2
classtemplate  and inherits from it.
The conictraits class defines a nested type named Rat_kernel
, which is a geometric kernel parameterized by an exact rational type. Env_sphere_traits_3
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 traitsclass simply ignores the upper hemisphere when it computes lower envelopes, and ignores the lower hemisphere when it computes upper envelopes.