\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 3D Envelopes

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CGAL::Env_plane_traits_3< Kernel > Class Template Reference
3D Envelopes Reference

`#include <CGAL/Env_plane_traits_3.h>`

## Inherits from

CGAL::Arr_linear_traits_2< Kernel >.

## Definition

The traits class `Env_plane_traits_3` 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 \( +\), \( -\), \( \times\), and \( \div\) 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` 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.

Is Model Of:
  `EnvelopeTraits_3`

Examples:
  Envelope_3/envelope_planes.cpp.