The traits class Env_plane_traits_3<Kernel> models the EnvelopeTraits_3 concept, and is used for the construction of lower and upper envelopes of planes and half planes in the space. It is parameterized by a CGAL-kernel model, which is parameterized in turn by a number type. The number type should support exact rational arithmetic, to avoid numerical errors and robustness problems. In particular, the number type should support the arithmetic operations , , , and without loss of precision. For optimal performance, we recommend instantiating the traits class with the predefined Exact_predicates_exact_constructions_kernel provided by CGAL. Using this kernel guarantees exactness and robustness, while it incurs only a minor overhead (in comparison to working with a fast, inexact number type) for most inputs.
Note that an entire plane has no boundaries, and the projection of a half-plane is an (unbounded) line. Naturally, rays and segments may occur as a result of overlaying projections of several half planes. Indeed, Env_plane_traits_3<Kernel> inherits from the Arr_linear_traits_2<Kernel> traits class, and extends it by adding operations on planes and half planes. The nested Xy_monotone_surface_3 and Surface_3 types refer to the same type. They are constructible from a Kernel::Plane_3 in case of an entire plane, or from Kernel::Plane_3 and Kernel::Line_2 in case of a half-plane. The line orientation determines which half is considered.