\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 2D Segment Delaunay Graphs

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SegmentDelaunayGraphTraits_2 Concept Reference
Concepts

## Definition

The concept `SegmentDelaunayGraphTraits_2` provides the traits requirements for the `CGAL::Segment_Delaunay_graph_2<Gt,DS>` and `CGAL::Segment_Delaunay_graph_hierarchy_2<Gt,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.

Refines:

  `DefaultConstructible`

  `CopyConstructible`

  `Assignable`

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,DS>`
  `CGAL::Segment_Delaunay_graph_hierarchy_2<Gt,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. More...
typedef unspecified_type | Site_2
| A type for a site of the segment Delaunay graph. More...
typedef unspecified_type | Point_2
| A type for a point.
typedef unspecified_type | Line_2
| A type for a line. More...
typedef unspecified_type | Ray_2
| A type for a ray. More...
typedef unspecified_type | Segment_2
| A type for a segment. More...
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. More...
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`. More...
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. More...
typedef unspecified_type | Compare_x_2
| A predicate object type. More...
typedef unspecified_type | Compare_y_2
| A predicate object type. More...
typedef unspecified_type | Less_x_2
| A predicate object type. More...
typedef unspecified_type | Less_y_2
| A predicate object type. More...
typedef unspecified_type | Orientation_2
| A predicate object type. More...
typedef unspecified_type | Equal_2
| A predicate object type. More...
typedef unspecified_type | Are_parallel_2
| A predicate object type. More...
typedef unspecified_type | Oriented_side_of_bisector_2
| A predicate object type. More...
typedef unspecified_type | Vertex_conflict_2
| A predicate object type. More...
typedef unspecified_type | Finite_edge_interior_conflict_2
| A predicate object type. More...
typedef unspecified_type | Infinite_edge_interior_conflict_2
| A predicate object type. More...
typedef unspecified_type | Oriented_side_2
| A predicate object type. More...
typedef unspecified_type | Arrangement_type_2
| A predicate object type. More...

## 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 contructor 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

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.

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.

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`.

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.

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.

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.

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`.

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.

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.

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.

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`.

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()`.

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()`.

typedef unspecified_type SegmentDelaunayGraphTraits_2::Line_2

A type for a line.

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

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.

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.

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`.

typedef unspecified_type SegmentDelaunayGraphTraits_2::Ray_2

A type for a ray.

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

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.

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`.

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).