Definition

The concept Periodic_2TriangulationTraits_2 is the first template parameter of the classes Periodic_2_triangulation_2<Traits, Tds>. This concept provides the types of the geometric primitives used in the triangulation and some function object types for the required predicates on those primitives.

It refines the concept TriangulationTraits_2 from the CGAL 2D Triangulation Reference package. It redefines the geometric objects, predicates and constructions to work with point-offset pairs. In most cases the offsets will be (0,0) and the predicates from TriangulationTraits_2 can be used directly. For efficiency reasons we maintain for each functor the version without offsets.

Refines:: TriangulationTraits_2 In addition to the requirements described for the traits class TriangulationTraits_2, the geometric traits class of a Periodic triangulation must fulfill the following requirements:

Has Models:: CGAL::Periodic_2_triangulation_traits_2

See Also: TriangulationTraits_2; CGAL::Periodic_2_triangulation_2<Traits,Tds>

Types
typedef Hidden_type	Point_2
	The point type. More...

typedef Hidden_type	Segment_2
	The segment type. More...

typedef Hidden_type	Vector_2
	The vector type. More...

typedef Hidden_type	Triangle_2
	The triangle type. More...

typedef Hidden_type	Iso_rectangle_2
	A type representing an axis-aligned rectangle. More...

typedef Hidden_type	Periodic_2_offset_2
	The offset type. More...

Predicate types
typedef Hidden_type	Compare_x_2
	A predicate object that must provide the function operators. More...

typedef Hidden_type	Compare_y_2
	A predicate object that must provide the function operators. More...

typedef Hidden_type	Less_x_2
	Predicate object. More...

typedef Hidden_type	Less_y_2
	Predicate object. More...

typedef Hidden_type	Orientation_2
	A predicate object that must provide the function operators. More...

Constructor types:
Note that the traits must provide exact constructions in order to guarantee exactness of the following construction functors.
typedef Hidden_type	Construct_point_2
	A constructor object for `Point_2`. More...

typedef Hidden_type	Construct_segment_2
	A constructor object for `Segment_2`. More...

typedef Hidden_type	Construct_triangle_2
	A constructor object for `Triangle_2`. More...

Creation
Only a default constructor, copy constructor and an assignment operator are required. Note that further constructors can be provided.
	Periodic_2TriangulationTraits_2 ()
	default constructor.

	Periodic_2TriangulationTraits_2 (Periodic_2TriangulationTraits_2 gtr)
	Copy constructor.

Periodic_2TriangulationTraits_2	operator= (Periodic_2TriangulationTraits_2 gtr)
	Assignment operator.

Predicate functions
The following functions give access to the predicate and constructor objects.
Compare_x_2	compare_x_2_object ()

Compare_y_2	compare_y_2_object ()

Less_x_2	less_x_2_object ()

Less_y_2	less_y_2_object ()

Orientation_2	orientation_2_object ()

Construct_point_2	construct_point_2_object ()

Construct_segment_2	construct_segment_2_object ()

Construct_triangle_2	construct_triangle_2_object ()

Access Functions
void	set_domain (Iso_rectangle_2 domain)
	Sets the fundamental domain. More...

Iso_rectangle_2	get_domain () const
	Returns the fundamental domain.

Member Typedef Documentation

typedef Hidden_type Periodic_2TriangulationTraits_2::Compare_x_2

A predicate object that must provide the function operators.

Comparison_result operator()(Point_2 p, Point_2 q),

which returns EQUAL if the \( x\)-coordinates of the two points are equal and

Comparison_result operator()(Point_2 p, Point_2 q, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q),

which returns EQUAL if the \( x\)-coordinates and \( x\)-offsets of the two point-offset pairs are equal. Otherwise it must return a consistent order for any two points.

Precondition: p, q lie inside the domain.

typedef Hidden_type Periodic_2TriangulationTraits_2::Compare_y_2

A predicate object that must provide the function operators.

Comparison_result operator()(Point_2 p, Point_2 q),

which returns EQUAL if the \( y\)-coordinates of the two points are equal and

Comparison_result operator()(Point_2 p, Point_2 q, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q),

which returns EQUAL if the \( y\)-coordinates and \( y\)-offsets of the two point-offset pairs are equal. Otherwise it must return a consistent order for any two points.

Precondition: p, q lie inside the domain.

typedef Hidden_type Periodic_2TriangulationTraits_2::Construct_point_2

A constructor object for Point_2.

Provides:

Point_2 operator()(Point_2 p,Periodic_2_offset_2 p_o),

which constructs a point from a point-offset pair.

Precondition: p lies inside the domain.

typedef Hidden_type Periodic_2TriangulationTraits_2::Construct_segment_2

A constructor object for Segment_2.

Provides:

Segment_2 operator()(Point_2 p,Point_2 q),

which constructs a segment from two points and

Segment_2 operator()(Point_2 p,Point_2 q, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q),

which constructs a segment from the points (p,o_p) and (q,o_q).

typedef Hidden_type Periodic_2TriangulationTraits_2::Construct_triangle_2

A constructor object for Triangle_2.

Provides:

Triangle_2 operator()(Point_2 p,Point_2 q,Point_2 r ),

which constructs a triangle from three points and

Triangle_2 operator()(Point_2 p,Point_2 q,Point_2 r, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q, Periodic_2_offset_2 o_r),

which constructs a triangle from the three points (p,o_p), (q,o_q) and (r,o_r).

typedef Hidden_type Periodic_2TriangulationTraits_2::Iso_rectangle_2

A type representing an axis-aligned rectangle.

It must be a model of Kernel::Iso_rectangle_2.

typedef Hidden_type Periodic_2TriangulationTraits_2::Less_x_2

Predicate object.

Provides the operators:

bool operator()(Point p, Point q) and

bool operator()(Point p, Point q, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q)

which returns true if p is before q according to the \( x\)-ordering of points.

This predicate is only necessary if the insert function with a range of points (using Hilbert sorting) is used.

typedef Hidden_type Periodic_2TriangulationTraits_2::Less_y_2

Predicate object.

Provides the operators:

bool operator()(Point p, Point q) and

bool operator()(Point p, Point q, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q)

which returns true if p is before q according to the \( y\)-ordering of points.

This predicate is only necessary if the insert function with a range of points (using Hilbert sorting) is used.

typedef Hidden_type Periodic_2TriangulationTraits_2::Orientation_2

A predicate object that must provide the function operators.

Orientation operator()(Point_2 p, Point_2 q, Point_2 r),

which returns LEFT_TURN, RIGHT_TURN or COLLINEAR depending on \( r\) being, with respect to the oriented line pq, on the left side, on the right side or on the line. and

Orientation operator()(Point_2 p, Point_2 q, Point_2 r, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q, Periodic_2_offset_2 o_r),

which returns LEFT_TURN, RIGHT_TURN or COLLINEAR depending on (r,o_r) being, with respect to the oriented line defined by (p,o_p)(q,o_q) on the left side, on the right side or on the line.

typedef Hidden_type Periodic_2TriangulationTraits_2::Periodic_2_offset_2

The offset type.

It must be a model of the concept Periodic_2Offset_2.

typedef Hidden_type Periodic_2TriangulationTraits_2::Point_2

The point type.

It must be a model of Kernel::Point_2.

typedef Hidden_type Periodic_2TriangulationTraits_2::Segment_2

The segment type.

It must be a model of Kernel::Segment_2.

typedef Hidden_type Periodic_2TriangulationTraits_2::Triangle_2

The triangle type.

It must be a model of Kernel::Triangle_2.