Definition

template<typename Kernel, typename ArrSegmentTraits = Arr_segment_traits_2<Kernel>>
class CGAL::Env_triangle_traits_3< Kernel, ArrSegmentTraits >

The traits class template Env_triangle_traits_3 models the EnvelopeTraits_3 concept, and is used for the construction of lower and upper envelopes of triangles in the space.

When the template is instantiated, the template parameter Kernel must be substituted by a CGAL-kernel model, the number type of which 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.

The second template parameter, namely ArrSegmentTraits, should be substituted by a 2D arrangement geometry traits class that supports segments. By default it is substituted by Arr_segment_traits_2<Kernel>.

Note that when we project the boundary of a triangle, or the intersection of two triangles, onto the \( xy\)-plane, we obtain line segments. Indeed, Env_triangle_traits_3 inherits from the traits class that substitutes ArrSegmentTraits, and extends it by adding operations on 3D objects, namely spatial triangles. Note that the traits class does not define Kernel::Triangle_3 as its surface (and \( xy\)-monotone surface) type, as one may expect. This is because the traits class needs to store extra data with the triangles in order to efficiently operate on them. Nevertheless, the nested Xy_monotone_surface_3 and Surface_3 types are however constructible from a Kernel::Triangle_3 instance and are also convertible to a Kernel::Triangle_3 object. Both types, Xy_monotone_surface_3 and Surface_3, refer to the same class, as every triangle is (weakly) \( xy\)-monotone).

Is model of: EnvelopeTraits_3

Examples: Envelope_3/envelope_triangles.cpp.