SegmentDelaunayGraphTraits_2 Class Reference

#include <Concepts/SegmentDelaunayGraphTraits_2.h>

Definition

Refines

TriangulationTraits_2

The concept SegmentDelaunayGraphTraits_2 provides the traits requirements for the CGAL::Segment_Delaunay_graph_2<Gt,St,DS> and CGAL::Segment_Delaunay_graph_hierarchy_2<Gt,St,STag,DS> classes. In particular, it provides a type Site_2, which must be a model of the concept SegmentDelaunayGraphSite_2. It also provides constructions for sites and several function object types for the predicates.

Has models

CGAL::Segment_Delaunay_graph_traits_2<K,MTag>

CGAL::Segment_Delaunay_graph_traits_without_intersections_2<K,MTag>

CGAL::Segment_Delaunay_graph_filtered_traits_2<CK,CM,EK,EM,FK,FM>

CGAL::Segment_Delaunay_graph_filtered_traits_without_intersections_2<CK,CM,EK,EM,FK,FM>

See also

SegmentDelaunayGraphSite_2

CGAL::Segment_Delaunay_graph_2<Gt,St,DS>

CGAL::Segment_Delaunay_graph_hierarchy_2<Gt,St,STag,DS>

CGAL::Segment_Delaunay_graph_traits_2<K,MTag>

CGAL::Segment_Delaunay_graph_traits_without_intersections_2<K,MTag>

CGAL::Segment_Delaunay_graph_filtered_traits_2<CK,CM,EK,EM,FK,FM>

CGAL::Segment_Delaunay_graph_filtered_traits_without_intersections_2<CK,CM,EK,EM,FK,FM>

Types
typedef unspecified_type	Intersections_tag
	Indicates or not whether the intersecting segments are to be supported.
typedef unspecified_type	Site_2
	A type for a site of the segment Delaunay graph.
typedef unspecified_type	Point_2
	A type for a point.
typedef unspecified_type	Line_2
	A type for a line.
typedef unspecified_type	Ray_2
	A type for a ray.
typedef unspecified_type	Segment_2
	A type for a segment.
typedef unspecified_type	FT
	A type for the field number type of sites, points, etc...
typedef unspecified_type	RT
	A type for the ring number type of sites, points, etc.
typedef unspecified_type	Arrangement_type
	An enumeration type that indicates the type of the arrangement of two sites.
typedef unspecified_type	Object_2
	A type representing different types of objects in two dimensions, namely: Point_2, Site_2, Line_2, Ray_2 and Segment_2.
typedef unspecified_type	Assign_2
	Must provide template <class T> bool operator() ( T& t, Object_2 o) which assigns o to t if o was constructed from an object of type T.
typedef unspecified_type	Construct_object_2
	Must provide template <class T> Object_2 operator()( T t) that constructs an object of type Object_2 that contains t and returns it.
typedef unspecified_type	Construct_svd_vertex_2
	A constructor for a point of the segment Voronoi diagram equidistant from three sites.
typedef unspecified_type	Compare_x_2
	A predicate object type.
typedef unspecified_type	Compare_y_2
	A predicate object type.
typedef unspecified_type	Less_x_2
	A predicate object type.
typedef unspecified_type	Less_y_2
	A predicate object type.
typedef unspecified_type	Orientation_2
	A predicate object type.
typedef unspecified_type	Equal_2
	A predicate object type.
typedef unspecified_type	Are_parallel_2
	A predicate object type.
typedef unspecified_type	Oriented_side_of_bisector_2
	A predicate object type.
typedef unspecified_type	Vertex_conflict_2
	A predicate object type.
typedef unspecified_type	Finite_edge_interior_conflict_2
	A predicate object type.
typedef unspecified_type	Infinite_edge_interior_conflict_2
	A predicate object type.
typedef unspecified_type	Oriented_side_2
	A predicate object type.
typedef unspecified_type	Arrangement_type_2
	A predicate object type.

Access to predicate 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 ()
Equal_2	equal_2_object ()
Are_parallel_2	are_parallel_2_object ()
Oriented_side_of_bisector_2	oriented_side_of_bisector_test_2_object ()
Vertex_conflict_2	vertex_conflict_2_object ()
Finite_edge_interior_conflict_2	finite_edge_interior_conflict_2_object ()
Infinite_edge_interior_conflict_2	infinite_edge_interior_conflict_2_object ()
Oriented_side_2	oriented_side_2_object ()
Arrangement_type_2	arrangement_type_2_object ()

Access to constructor objects
Construct_object_2	construct_object_2_object ()
Construct_svd_vertex_2	construct_svd_vertex_2_object ()

Access to other objects
Assign_2	assign_2_object ()

Member Typedef Documentation

Are_parallel_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Are_parallel_2

A predicate object type.

Must provide bool operator()(Site_2 s1, Site_2 s2), which determines is the segments represented by the sites s1 and s2 are parallel.

Precondition

s1 and s2 must be segments.

Arrangement_type

typedef unspecified_type SegmentDelaunayGraphTraits_2::Arrangement_type

An enumeration type that indicates the type of the arrangement of two sites.

The possible values are DISJOINT, IDENTICAL, CROSSING, TOUCHING_1, TOUCHING_2, TOUCHING_11, TOUCHING_12, TOUCHING_21, TOUCHING_22, OVERLAPPING_11, OVERLAPPING_12, OVERLAPPING_21, OVERLAPPING_22, INTERIOR, INTERIOR_1, INTERIOR_2, TOUCHING_11_INTERIOR_1, TOUCHING_11_INTERIOR_2, TOUCHING_12_INTERIOR_1, TOUCHING_12_INTERIOR_2, TOUCHING_21_INTERIOR_1, TOUCHING_21_INTERIOR_2, TOUCHING_22_INTERIOR_1, TOUCHING_22_INTERIOR_2. A detailed description of the meaning of these values is shown the end of the reference manual for this concept.

Arrangement_type_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Arrangement_type_2

A predicate object type.

Must provide Arrangement_type operator()(Site_2 s1, Site_2 s2) that returns the type of the arrangement of the two sites s1 and s2.

Assign_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Assign_2

Must provide template <class T> bool operator() ( T& t, Object_2 o) which assigns o to t if o was constructed from an object of type T.

Returns true, if the assignment was possible.

Compare_x_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Compare_x_2

A predicate object type.

Must provide Comparison_result operator()(Site_2 s1, Site_2 s2), which compares the \( x\)-coordinates of the points represented by the sites s1 and s2.

Precondition

s1 and s2 must be points.

Compare_y_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Compare_y_2

A predicate object type.

Must provide Comparison_result operator()(Site_2 s1, Site_2 s2), which compares the \( y\)-coordinates of the points represented by the sites s1 and s2.

Precondition

s1 and s2 must be points.

Construct_object_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Construct_object_2

Must provide template <class T> Object_2 operator()( T t) that constructs an object of type Object_2 that contains t and returns it.

Construct_svd_vertex_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Construct_svd_vertex_2

A constructor for a point of the segment Voronoi diagram equidistant from three sites.

Must provide Point_2 operator()(Site_2 s1, Site_2 s2, Site_2 s3), which constructs a point equidistant from the sites s1, s2 and s3.

Equal_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Equal_2

A predicate object type.

Must provide bool operator()(Site_2 s1, Site_2 s2), which determines is the points represented by the sites s1 and s2 are identical.

Precondition

s1 and s2 must be points.

Finite_edge_interior_conflict_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Finite_edge_interior_conflict_2

A predicate object type.

Must provide bool operator()(Site_2 s1, Site_2 s2, Site_2 s3, Site_2 s4, Site_2 q, Sign sgn). The sites s1, s2, s3 and s4 define a Voronoi edge that lies on the bisector of s1 and s2 and has as endpoints the Voronoi vertices defined by the triplets s1, s2, s3 and s1, s4 and s2. The sign sgn is the common sign of the distance of the site q from the Voronoi circle of the triplets s1, s2, s3 and s1, s4 and s2. In case that sgn is equal to NEGATIVE, the predicate returns true if and only if the entire Voronoi edge is in conflict with q. If sgn is equal to POSITIVE or ZERO, the predicate returns false if and only if q is not in conflict with the Voronoi edge.

Precondition

the Voronoi vertices of s1, s2, s3, and s1, s4, s2 must exist.

Must also provide bool operator()(Site_2 s1, Site_2 s2, Site_2 s3, Site_2 q, Sign sgn). The sites s1, s2, s3 and the site at infinity \( s_\infty\) define a Voronoi edge that lies on the bisector of s1 and s2 and has as endpoints the Voronoi vertices \( v_{123}\) and \( v_{1\infty{2}}\) defined by the triplets s1, s2, s3 and s1, \( s_\infty\) and s2 (the second vertex is actually at infinity). The sign sgn is the common sign of the distance of the site q from the two Voronoi circles centered at the Voronoi vertices \( v_{123}\) and \( v_{1\infty{2}}\). In case that sgn is NEGATIVE, the predicate returns true if and only if the entire Voronoi edge is in conflict with q. If sgn is POSITIVE or ZERO, the predicate returns false if and only if q is not in conflict with the Voronoi edge.

Precondition

the Voronoi vertex \( v_{123}\) of s1, s2, s3 must exist.

Must finally provide bool operator()(Site_2 s1, Site_2 s2, Site_2 q, Sign sgn). The sites s1, s2 and the site at infinity \( s_\infty\) define a Voronoi edge that lies on the bisector of \( v_{12\infty}\) and \( v_{1\infty{}2}\) s1 and s2 and has as endpoints the Voronoi vertices defined by the triplets s1, s2, \( s_\infty\) and s1, \( s_\infty\) and s2 (both vertices are actually at infinity). The sign sgn denotes the common sign of the distance of the site q from the Voronoi circles centered at \( v_{12\infty}\) and \( v_{1\infty{}2}\). If sgn is NEGATIVE, the predicate returns true if and only if the entire Voronoi edge is in conflict with q. If POSITIVE or ZERO is false, the predicate returns false if and only if q is not in conflict with the Voronoi edge.

FT

typedef unspecified_type SegmentDelaunayGraphTraits_2::FT

A type for the field number type of sites, points, etc...

Infinite_edge_interior_conflict_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Infinite_edge_interior_conflict_2

A predicate object type.

Must provide bool operator()(Site_2 s1, Site_2 s2, Site_2 s3, Site_2 q, Sign sgn). The sites \( s_\infty\), s1, s2 and s3 define a Voronoi edge that lies on the bisector of \( s_\infty\) and s1 and has as endpoints the Voronoi vertices \( v_{\infty{}12}\) and \( v_{\infty{}31}\) defined by the triplets \( s_\infty\), s1, s2 and \( s_\infty\), s3 and s1. The sign sgn is the common sign of the distances of q from the Voronoi circles centered at the vertices \( v_{\infty{}12}\) and \( v_{\infty{}31}\). If sgn is NEGATIVE, the predicate returns true if and only if the entire Voronoi edge is in conflict with q. If sgn is POSITIVE or ZERO, the predicate returns false if and only if q is not in conflict with the Voronoi edge.

Intersections_tag

typedef unspecified_type SegmentDelaunayGraphTraits_2::Intersections_tag

Indicates or not whether the intersecting segments are to be supported.

The tag must either be CGAL::Tag_true or CGAL::Tag_false.

Less_x_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Less_x_2

A predicate object type.

Must provide bool operator()(Point_2 p1, Point_2 p2), which returns true if p1.x() < p2.x().

Less_y_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Less_y_2

A predicate object type.

Must provide bool operator()(Point_2 p1, Point_2 p2), which returns true if p1.y() < p2.y().

Line_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Line_2

A type for a line.

Only required if the segment Delaunay graph is inserted in a stream.

Object_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Object_2

A type representing different types of objects in two dimensions, namely: Point_2, Site_2, Line_2, Ray_2 and Segment_2.

Orientation_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Orientation_2

A predicate object type.

Must provide Orientation operator()(Site_2 s1, Site_2 s2, Site_2 s3), which performs the usual orientation test for three points. s1, s2 and s3.

Precondition

the sites s1, s2 and s3 must be points.

Oriented_side_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Oriented_side_2

A predicate object type.

Must provide Oriented_side operator()(Site_1 s1, Site_2 s2, Site_2 s3, Site_2 s, Site_2 p). Determines the oriented side of the line \( \ell\) that contains the point site p, where \( \ell\) is the line that passes through the Voronoi vertex of the sites s1, s2, s3 and is perpendicular to the segment site s.

Precondition

s must be a segment and p must be a point.

Oriented_side_of_bisector_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Oriented_side_of_bisector_2

A predicate object type.

Must provide Oriented_side operator()(Site_2 s1, Site_2 s2, Point_2 p), which returns the oriented side of the bisector of s1 and s2 that contains p. Returns ON_POSITIVE_SIDE if p lies in the half-space of s1 (i.e., p is closer to s1 than s2); returns ON_NEGATIVE_SIDE if p lies in the half-space of s2; returns ON_ORIENTED_BOUNDARY if p lies on the bisector of s1 and s2.

Point_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Point_2

A type for a point.

Ray_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Ray_2

A type for a ray.

Only required if the segment Delaunay graph is inserted in a stream.

RT

typedef unspecified_type SegmentDelaunayGraphTraits_2::RT

A type for the ring number type of sites, points, etc.

Segment_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Segment_2

A type for a segment.

Only required if if the segment Delaunay graph is inserted in a stream.

Site_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Site_2

A type for a site of the segment Delaunay graph.

Must be a model of the concept SegmentDelaunayGraphSite_2.

Vertex_conflict_2

typedef unspecified_type SegmentDelaunayGraphTraits_2::Vertex_conflict_2

A predicate object type.

Must provide Sign operator()(Site_2 s1, Site_2 s2, Site_2 s3, Site_2 q), which returns the sign of the distance of q from the Voronoi circle of s1, s2, s3 (the Voronoi circle of three sites s1, s2, s3 is a circle co-tangent to all three sites, that touches them in that order as we walk on its circumference in the counter-clockwise sense).

Precondition

the Voronoi circle of s1, s2, s3 must exist.

Must also provide Sign operator()(Site_2 s1, Site_2 s2, Site_2 q), which returns the sign of the distance of q from the bitangent line of s1, s2 (a degenerate Voronoi circle, with its center at infinity).

Member Function Documentation

are_parallel_2_object()

Are_parallel_2 SegmentDelaunayGraphTraits_2::are_parallel_2_object

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arrangement_type_2_object()

Arrangement_type_2 SegmentDelaunayGraphTraits_2::arrangement_type_2_object

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assign_2_object()

Assign_2 SegmentDelaunayGraphTraits_2::assign_2_object

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compare_x_2_object()

Compare_x_2 SegmentDelaunayGraphTraits_2::compare_x_2_object

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compare_y_2_object()

Compare_y_2 SegmentDelaunayGraphTraits_2::compare_y_2_object

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construct_object_2_object()

Construct_object_2 SegmentDelaunayGraphTraits_2::construct_object_2_object

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construct_svd_vertex_2_object()

Construct_svd_vertex_2 SegmentDelaunayGraphTraits_2::construct_svd_vertex_2_object

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equal_2_object()

Equal_2 SegmentDelaunayGraphTraits_2::equal_2_object

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finite_edge_interior_conflict_2_object()

Finite_edge_interior_conflict_2 SegmentDelaunayGraphTraits_2::finite_edge_interior_conflict_2_object

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infinite_edge_interior_conflict_2_object()

Infinite_edge_interior_conflict_2 SegmentDelaunayGraphTraits_2::infinite_edge_interior_conflict_2_object

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less_x_2_object()

Less_x_2 SegmentDelaunayGraphTraits_2::less_x_2_object

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less_y_2_object()

Less_y_2 SegmentDelaunayGraphTraits_2::less_y_2_object

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orientation_2_object()

Orientation_2 SegmentDelaunayGraphTraits_2::orientation_2_object

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oriented_side_2_object()

Oriented_side_2 SegmentDelaunayGraphTraits_2::oriented_side_2_object

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oriented_side_of_bisector_test_2_object()

Oriented_side_of_bisector_2 SegmentDelaunayGraphTraits_2::oriented_side_of_bisector_test_2_object

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vertex_conflict_2_object()

Vertex_conflict_2 SegmentDelaunayGraphTraits_2::vertex_conflict_2_object

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