Definition

The concept Periodic_3TriangulationTraits_3 is the first template parameter of the class Periodic_3_triangulation_3. It refines the concept TriangulationTraits_3 from the CGAL 3D Triangulations. It redefines the geometric objects, predicates and constructions to work with point-offset pairs. In most cases the offsets will be (0,0,0) and the predicates from TriangulationTraits_3 can be used directly. For efficiency reasons we maintain for each functor the version without offsets.

Refines:: TriangulationTraits_3

Has Models:: CGAL::Periodic_3_triangulation_traits_3

See Also: Periodic_3DelaunayTriangulationTraits_3; Periodic_3RegularTriangulationTraits_3

In addition to the requirements described for the traits class TriangulationTraits_3, the geometric traits class of a Periodic triangulation must fulfill the following requirements.

Types
typedef unspecified_type	Point_3
	The point type. More...

typedef unspecified_type	Vector_3
	The vector type. More...

typedef unspecified_type	Periodic_3_offset_3
	The offset type. More...

typedef unspecified_type	Iso_cuboid_3
	A type representing an axis-aligned cuboid. More...

The following three types represent geometric primitives in \( \mathbb R^3\). They are required to provide functions converting primitives from \( \mathbb T_c^3\) to \( \mathbb R^3\), i.e. constructing representatives in \( \mathbb R^3\).
typedef unspecified_type	Segment_3
	A segment type. More...

typedef unspecified_type	Triangle_3
	A triangle type. More...

typedef unspecified_type	Tetrahedron_3
	A tetrahedron type. More...

typedef unspecified_type	Compare_xyz_3
	A predicate object that must provide the function operators. More...

typedef unspecified_type	Orientation_3
	A predicate object that must provide the function operators. More...

Note that the traits must provide exact constructions in order to guarantee exactness of the following construction functors.
typedef unspecified_type	Construct_point_3
	A constructor object that must provide the function operator. More...

typedef unspecified_type	Construct_segment_3
	A constructor object that must provide the function operators. More...

typedef unspecified_type	Construct_triangle_3
	A constructor object that must provide the function operators. More...

typedef unspecified_type	Construct_tetrahedron_3
	A constructor object that must provide the function operators. More...

Creation
	Periodic_3_triangulation_traits_3 ()
	Default constructor.

	Periodic_3_triangulation_traits_3 (const Periodic_triangulation_traits_3 &tr)
	Copy constructor.

Access Functions
void	set_domain (const Iso_cuboid_3 &domain)
	Set the fundamental domain. More...

const Iso_cuboid_3 &	get_domain () const
	Returns the fundamental domain.

Operations
The following functions give access to the predicate and construction objects:
Compare_xyz_3	compare_xyz_3_object ()

Orientation_3	orientation_3_object ()

Construct_segment_3	construct_segment_3_object ()

Construct_triangle_3	construct_triangle_3_object ()

Construct_tetrahedron_3	construct_tetrahedron_3_object ()

Member Typedef Documentation

typedef unspecified_type Periodic_3TriangulationTraits_3::Compare_xyz_3

A predicate object that must provide the function operators.

Comparison_result operator()(Point_3 p, Point_3 q),

which returns EQUAL if the two points are equal and

Comparison_result operator()(Point_3 p, Point_3 q, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q),

which returns EQUAL if the two point-offset pairs are equal. Otherwise it must return a consistent order for any two points chosen in a same line.

Precondition: p, q lie inside the domain.

typedef unspecified_type Periodic_3TriangulationTraits_3::Construct_point_3

A constructor object that must provide the function operator.

Point_3 operator()(Point_3 p, Periodic_3_offset_3 o_p),

which constructs a point from a point-offset pair.

Precondition: p lies inside the domain.

typedef unspecified_type Periodic_3TriangulationTraits_3::Construct_segment_3

A constructor object that must provide the function operators.

Segment_3 operator()(Point_3 p, Point_3 q),

which constructs a segment from two points and

Segment_3 operator()(Point_3 p, Point_3 q, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q),

which constructs a segment from two point-offset pairs.

Precondition: p, q lie inside the domain.

typedef unspecified_type Periodic_3TriangulationTraits_3::Construct_tetrahedron_3

A constructor object that must provide the function operators.

Tetrahedron_3 operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s),

which constructs a tetrahedron from four points and

Tetrahedron_3 operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s),

which constructs a tetrahedron from four point-offset pairs.

Precondition: p, q, r, s lie inside the domain.

typedef unspecified_type Periodic_3TriangulationTraits_3::Construct_triangle_3

A constructor object that must provide the function operators.

Triangle_3 operator()(Point_3 p, Point_3 q, Point_3 r ),

which constructs a triangle from three points and

Triangle_3 operator()(Point_3 p, Point_3 q, Point_3 r, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r),

which constructs a triangle from three point-offset pairs.

Precondition: p, q, r lie inside the domain.

typedef unspecified_type Periodic_3TriangulationTraits_3::Iso_cuboid_3

A type representing an axis-aligned cuboid.

It must be a model of Kernel::Iso_cuboid_3.

typedef unspecified_type Periodic_3TriangulationTraits_3::Orientation_3

A predicate object that must provide the function operators.

Orientation operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s),

which returns POSITIVE, if s lies on the positive side of the oriented plane h defined by p, q, and r, returns NEGATIVE if s lies on the negative side of h, and returns COPLANAR if s lies on h and

Orientation operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s),

which returns POSITIVE, if the point-offset pair (s,o_s) lies on the positive side of the oriented plane h defined by (p,o_p), (q,o_q), and (r,o_r), returns NEGATIVE if (s,o_s) lies on the negative side of h, and returns COPLANAR if (s,o_s) lies on h.