TriangulationTraits_2 Concept Reference

Definition

Refines

SpatialSortingTraits_2

The concept TriangulationTraits_2 describes the set of requirements to be fulfilled by any class used to instantiate the first template parameter of the class CGAL::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.

Has models

All models of the CGAL concept Kernel

CGAL::Projection_traits_3<K>

CGAL::Projection_traits_xy_3<K>

CGAL::Projection_traits_yz_3<K>

CGAL::Projection_traits_xz_3<K>

See also

CGAL::Triangulation_2

Types
typedef unspecified_type	Point_2
	The point type.
typedef unspecified_type	Segment_2
	The segment type.
typedef unspecified_type	Triangle_2
	The triangle type.
typedef unspecified_type	Construct_point_2
	A function object to construct a Point_2.
typedef unspecified_type	Construct_segment_2
	A function object to construct a Segment_2.
typedef unspecified_type	Construct_triangle_2
	A function object to construct a Triangle_2.
typedef unspecified_type	Less_x_2
	A function object to compare the x-coordinate of two points.
typedef unspecified_type	Less_y_2
	A function object to compare the y-coordinate of two points.
typedef unspecified_type	Compare_x_2
	A function object to compare the x-coordinate of two points.
typedef unspecified_type	Compare_y_2
	A function object to compare the y-coordinate of two points.
typedef unspecified_type	Orientation_2
	A function object to compute the orientation of three points.
typedef unspecified_type	Side_of_oriented_circle_2
	A function object to perform the incircle test for four points.
typedef unspecified_type	Construct_circumcenter_2
	A function object to compute the circumcentr of three points.

Creation
Only a default constructor, copy constructor and an assignment operator are required. Note that further constructors can be provided.
	TriangulationTraits_2 ()
	default constructor.
	TriangulationTraits_2 (TriangulationTraits_2 gtr)
	Copy constructor.
TriangulationTraits_2	operator= (TriangulationTraits_2 gtr)
	Assignment operator.

Predicate Functions
The following functions give access to the predicate and constructor objects.
Construct_point_2	construct_point_2_object ()
Construct_segment_2	construct_segment_2_object ()
Construct_triangle_2	construct_triangle_2_object ()
Compare_x_2	compare_x_2_object ()
Compare_y_2	compare_y_2_object ()
Orientation_2	orientation_2_object ()
Side_of_oriented_circle_2	side_of_oriented_circle_2_object ()
	Required only if side_of_oriented_circle is called called.
Construct_circumcenter_2	construct_circumcenter_2_object ()
	Required only if circumcenter is called.

Member Typedef Documentation

Compare_x_2

typedef unspecified_type TriangulationTraits_2::Compare_x_2

A function object to compare the x-coordinate of two points.

Provides the operator:

Comparison_result operator()(Point p, Point q)

which returns SMALLER, EQUAL or LARGER according to the \( x\)-ordering of points p and q.

Compare_y_2

typedef unspecified_type TriangulationTraits_2::Compare_y_2

A function object to compare the y-coordinate of two points.

Provides the operator:

Comparison_result operator()(Point p, Point q)

which returns (SMALLER, EQUAL or LARGER) according to the \( y\)-ordering of points p and q.

Construct_circumcenter_2

typedef unspecified_type TriangulationTraits_2::Construct_circumcenter_2

A function object to compute the circumcentr of three points.

Provides the operator:

Point operator()(Point p, Point q, Point r)

which returns the circumcenter of the three points p, q and r. This type is required only if the function Point circumcenter(Face_handle f)is called.

Construct_point_2

typedef unspecified_type TriangulationTraits_2::Construct_point_2

A function object to construct a Point_2.

Provides:

Point_2 operator()(Point_2 p),

which simply returns p.

Note

It is advised to return a const reference to p to avoid useless copies.

This peculiar requirement is necessary because CGAL::Triangulation_2 internally manipulates points with a Point type that is not always Point_2.

Construct_segment_2

typedef unspecified_type TriangulationTraits_2::Construct_segment_2

A function object to construct a Segment_2.

Provides:

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

which constructs a segment from two points.

Construct_triangle_2

typedef unspecified_type TriangulationTraits_2::Construct_triangle_2

A function object to construct a Triangle_2.

Provides:

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

which constructs a triangle from three points.

Less_x_2

typedef unspecified_type TriangulationTraits_2::Less_x_2

A function object to compare the x-coordinate of two points.

Provides the operator:

bool operator()(Point p, Point q)

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

Less_y_2

typedef unspecified_type TriangulationTraits_2::Less_y_2

A function object to compare the y-coordinate of two points.

Provides the operator:

bool operator()(Point p, Point q)

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

Orientation_2

typedef unspecified_type TriangulationTraits_2::Orientation_2

A function object to compute the orientation of three points.

Provides the operator:

Orientation operator()(Point p, Point q, Point 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.

Side_of_oriented_circle_2

typedef unspecified_type TriangulationTraits_2::Side_of_oriented_circle_2

A function object to perform the incircle test for four points.

Provides the operator:

Oriented_side operator()(Point p, Point q, Point r, Point s) which takes four points \( p, q, r, s\) as arguments and returns ON_POSITIVE_SIDE, ON_NEGATIVE_SIDE or, ON_ORIENTED_BOUNDARY according to the position of points s with respect to the oriented circle through through \( p,q\) and \( r\). This type is required only if the function side_of_oriented_circle(Face_handle f, Point p) is called.