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CGAL 4.7 - 2D Periodic Triangulations
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Periodic_2TriangulationTraits_2 Concept Reference
2D Periodic Triangulations Reference » Concepts

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 unspecified_type Point_2
 The point type. More...
 
typedef unspecified_type Segment_2
 The segment type. More...
 
typedef unspecified_type Vector_2
 The vector type. More...
 
typedef unspecified_type Triangle_2
 The triangle type. More...
 
typedef unspecified_type Iso_rectangle_2
 A type representing an axis-aligned rectangle. More...
 
typedef unspecified_type Periodic_2_offset_2
 The offset type. More...
 

Predicate types

typedef unspecified_type Compare_x_2
 A predicate object that must provide the function operators. More...
 
typedef unspecified_type Compare_y_2
 A predicate object that must provide the function operators. More...
 
typedef unspecified_type Less_x_2
 Predicate object. More...
 
typedef unspecified_type Less_y_2
 Predicate object. More...
 
typedef unspecified_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 unspecified_type Construct_point_2
 A constructor object for Point_2. More...
 
typedef unspecified_type Construct_segment_2
 A constructor object for Segment_2. More...
 
typedef unspecified_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. More...
 
 Periodic_2TriangulationTraits_2 (Periodic_2TriangulationTraits_2 gtr)
 Copy constructor. More...
 
Periodic_2TriangulationTraits_2 operator= (Periodic_2TriangulationTraits_2 gtr)
 Assignment operator. More...
 

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. More...
 

Member Typedef Documentation

typedef unspecified_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 unspecified_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 unspecified_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 unspecified_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 unspecified_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 unspecified_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 unspecified_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 unspecified_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 unspecified_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 unspecified_type Periodic_2TriangulationTraits_2::Periodic_2_offset_2

The offset type.

It must be a model of the concept Periodic_2Offset_2.

typedef unspecified_type Periodic_2TriangulationTraits_2::Point_2

The point type.

It must be a model of Kernel::Point_2.

typedef unspecified_type Periodic_2TriangulationTraits_2::Segment_2

The segment type.

It must be a model of Kernel::Segment_2.

typedef unspecified_type Periodic_2TriangulationTraits_2::Triangle_2

The triangle type.

It must be a model of Kernel::Triangle_2.

typedef unspecified_type Periodic_2TriangulationTraits_2::Vector_2

The vector type.

It must be a model of Kernel::Vector_2.

Constructor & Destructor Documentation

Periodic_2TriangulationTraits_2::Periodic_2TriangulationTraits_2 ( )

default constructor.

Periodic_2TriangulationTraits_2::Periodic_2TriangulationTraits_2 ( Periodic_2TriangulationTraits_2  gtr)

Copy constructor.

Member Function Documentation

Compare_x_2 Periodic_2TriangulationTraits_2::compare_x_2_object ( )
Compare_y_2 Periodic_2TriangulationTraits_2::compare_y_2_object ( )
Construct_point_2 Periodic_2TriangulationTraits_2::construct_point_2_object ( )
Construct_segment_2 Periodic_2TriangulationTraits_2::construct_segment_2_object ( )
Construct_triangle_2 Periodic_2TriangulationTraits_2::construct_triangle_2_object ( )
Iso_rectangle_2 Periodic_2TriangulationTraits_2::get_domain ( ) const

Returns the fundamental domain.

Less_x_2 Periodic_2TriangulationTraits_2::less_x_2_object ( )
Less_y_2 Periodic_2TriangulationTraits_2::less_y_2_object ( )
Periodic_2TriangulationTraits_2 Periodic_2TriangulationTraits_2::operator= ( Periodic_2TriangulationTraits_2  gtr)

Assignment operator.

Orientation_2 Periodic_2TriangulationTraits_2::orientation_2_object ( )
void Periodic_2TriangulationTraits_2::set_domain ( Iso_rectangle_2  domain)

Sets the fundamental domain.

This is necessary to evaluate predicates correctly.

Precondition
domain represents a square.