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

CGAL 4.8 - 2D Apollonius Graphs (Delaunay Graphs of Disks)

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ApolloniusGraphTraits_2 Concept Reference
Concepts

## Definition

The concept `ApolloniusGraphTraits_2` provides the traits requirements for the `Apollonius_graph_2` class. In particular, it provides a type `Site_2`, which must be a model of the concept `ApolloniusSite_2`. It also provides constructions for sites and several function object types for the predicates.

Has Models:

  `CGAL::Apollonius_graph_traits_2<K,Method_tag>`

  `CGAL::Apollonius_graph_filtered_traits_2<CK,CM,EK,EM,FK,FM>`

See Also
  `CGAL::Apollonius_graph_2<Gt,Agds>`
  `CGAL::Apollonius_graph_traits_2<K,Method_tag>`
  `CGAL::Apollonius_graph_filtered_traits_2<CK,CM,EK,EM,FK,FM>`

## Types
typedef unspecified_type | Point_2
| A type for a point.
typedef unspecified_type | Site_2
| A type for an Apollonius site. More...
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 | 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 | FT
| A type for the field number type of sites.
typedef unspecified_type | RT
| A type for the ring number type of sites.
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_Apollonius_vertex_2
| A constructor for a point of the Apollonius diagram equidistant from three sites. More...
typedef unspecified_type | Construct_Apollonius_site_2
| A constructor for a dual Apollonius site (a site whose center is a vertex of the Apollonius diagram and its weight is the common distance of its center from the three defining 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 | Compare_weight_2
| A predicate object type. More...
typedef unspecified_type | Orientation_2
| A predicate object type. More...
typedef unspecified_type | Is_hidden_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 | Is_degenerate_edge_2
| A predicate object type. More...

## Creation
| ApolloniusGraphTraits_2 ()
| Default constructor.
| ApolloniusGraphTraits_2 (ApolloniusGraphTraits_2 other)
| Copy constructor.
ApolloniusGraphTraits_2 | operator= (ApolloniusGraphTraits_2 other)
| Assignment operator.

## Access to predicate objects
Compare_x_2 | compare_x_2_object ()
Compare_y_2 | compare_y_2_object ()
Compare_weight_2 | compare_weight_2_object ()
Orientation_2 | orientation_2_object ()
Is_hidden_2 | is_hidden_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 ()
Is_degenerate_edge_2 | is_degenerate_edge_2_object ()

## Access to constructor objects
Construct_object_2 | construct_object_2_object ()
Construct_Apollonius_vertex_2 | construct_Apollonius_vertex_2_object ()
Construct_Apollonius_site_2 | construct_Apollonius_site_2_object ()

## Access to other objects
Assign_2 | assign_2_object ()

## Member Typedef Documentation

typedef unspecified_type ApolloniusGraphTraits_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 ApolloniusGraphTraits_2::Compare_weight_2

A predicate object type.

Must provide `Comparison_result operator()(Site_2 s1, Site_2 s2)`, which compares the weights of `s1` and `s2`.

typedef unspecified_type ApolloniusGraphTraits_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 centers of `s1` and `s2`.

typedef unspecified_type ApolloniusGraphTraits_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 centers of `s1` and `s2`.

typedef unspecified_type ApolloniusGraphTraits_2::Construct_Apollonius_site_2

A constructor for a dual Apollonius site (a site whose center is a vertex of the Apollonius diagram and its weight is the common distance of its center from the three defining sites).

Must provide `Site_2 operator()(Site_2 s1, Site_2 s2, Site_2 s3)`, which constructs a dual site whose center \( c\) is equidistant from `s1`, `s2` and `s3`, and its weight is equal to the (signed) distance of \( c\) from `s1` (or `s2` or `s3`).

Must also provide `Line_2 operator()(Site_2 s1, Site_2 s2)`, which constructs a line bitangent to `s1` and `s2`. This line is the dual site of `s1`, `s2` and the site at infinity; it can be viewed as a dual Apollonius site whose center is at infinity and its weight is infinite.

typedef unspecified_type ApolloniusGraphTraits_2::Construct_Apollonius_vertex_2

A constructor for a point of the Apollonius 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 ApolloniusGraphTraits_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, bool b)`. The sites `s1`, `s2`, `s3` and `s4` define an Apollonius edge that lies on the bisector of `s1` and `s2` and has as endpoints the Apollonius vertices defined by the triplets `s1`, `s2`, `s3` and `s1`, `s4` and `s2`. The Boolean `b` denotes if the two Apollonius vertices are in conflict with the site `q` (in which case `b` should be `true`, otherwise `false`). In case that `b` is `true`, the predicate returns `true` if and only if the entire Apollonius edge is in conflict with `q`. If `b` is `false`, the predicate returns `false` if and only if `q` is not in conflict with the Apollonius edge.

Precondition
  the Apollonius 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, bool b)`. The sites `s1`, `s2`, `s3` and the site at infinity \( s_\infty\) define an Apollonius edge that lies on the bisector of `s1` and `s2` and has as endpoints the Apollonius vertices defined by the triplets `s1`, `s2`, `s3` and `s1`, \( s_\infty\) and `s2` (the second Apollonius vertex is actually at infinity). The Boolean `b` denotes if the two Apollonius vertices are in conflict with the site `q` (in which case `b` should be `true`, otherwise `false`). In case that `b` is `true`, the predicate returns `true` if and only if the entire Apollonius edge is in conflict with `q`. If `b` is `false`, the predicate returns `false` if and only if `q` is not in conflict with the Apollonius edge.

Precondition
  the Apollonius vertex of `s1`, `s2`, `s3` must exist.

Must finally provide `bool operator()(Site_2 s1, Site_2 s2, Site_2 q, bool b)`. The sites `s1`, `s2` and the site at infinity \( s_\infty\) define an Apollonius edge that lies on the bisector of `s1` and `s2` and has as endpoints the Apollonius vertices defined by the triplets `s1`, `s2`, \( s_\infty\) and `s1`, \( s_\infty\) and `s2` (both Apollonius vertices are actually at infinity). The Boolean `b` denotes if the two Apollonius vertices are in conflict with the site `q` (in which case `b` should be `true`, otherwise `false`). In case that `b` is `true`, the predicate returns `true` if and only if the entire Apollonius edge is in conflict with `q`. If `b` is `false`, the predicate returns `false` if and only if `q` is not in conflict with the Apollonius edge.

typedef unspecified_type ApolloniusGraphTraits_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, bool b)`. The sites \( s_\infty\), `s1`, `s2` and `s3` define an Apollonius edge that lies on the bisector of \( s_\infty\) and `s1` and has as endpoints the Apollonius vertices defined by the triplets \( s_\infty\), `s1`, `s2` and \( s_\infty\), `s3` and `s1`. The Boolean `b` denotes if the two Apollonius vertices are in conflict with the site `q` (in which case `b` should be `true`, otherwise `false`. In case that `b` is `true`, the predicate returns `true` if and only if the entire Apollonius edge is in conflict with `q`. If `b` is `false`, the predicate returns `false` if and only if `q` is not in conflict with the Apollonius edge.

typedef unspecified_type ApolloniusGraphTraits_2::Is_degenerate_edge_2

A predicate object type.

Must provide `bool operator()(Site_2 s1, Site_2 s2, Site_2 s3, Site_2 s4)`. It returns `true` if the Apollonius edge defined by `s1`, `s2`, `s3` and `s4` is degenerate, `false` otherwise. An Apollonius edge is called degenerate if its two endpoints coincide.

Precondition
  the Apollonius vertices of `s1`, `s2`, `s3`, and `s1`, `s4`, `s2` must exist.

typedef unspecified_type ApolloniusGraphTraits_2::Is_hidden_2

A predicate object type.

Must provide `bool operator()(Site_2 s1, Site_2 s2)`, which returns `true` if the circle corresponding to `s2` is contained in the closure of the disk corresponding to `s1`, `false` otherwise.

typedef unspecified_type ApolloniusGraphTraits_2::Line_2

A type for a line.

Only required if access to the dual of the Apollonius graph is required or if the primal or dual diagram are inserted in a stream.

typedef unspecified_type ApolloniusGraphTraits_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 the centers of the three sites `s1`, `s2` and `s3`.

Must also provide `Orientation operator()(Site_2 s1, Site_2 s2, Site_2 s3, Site_2 p1, Site_2 p2)`, which performs the usual orientation test for the Apollonius vertex of `s1`, `s2`, `s3` and the centers of `p1` and `p2`.

Precondition
  the Apollonius vertex of `s1`, `s2` and `s3` must exist.

typedef unspecified_type ApolloniusGraphTraits_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 ApolloniusGraphTraits_2::Ray_2

A type for a ray.

Only required if access to the dual of the Apollonius graph is required or if the primal or dual diagram are inserted in a stream.

typedef unspecified_type ApolloniusGraphTraits_2::Segment_2

A type for a segment.

Only required if access to the dual of the Apollonius graph is required or if the primal or dual diagram are inserted in a stream.

typedef unspecified_type ApolloniusGraphTraits_2::Site_2

A type for an Apollonius site.

Must be a model of the concept `ApolloniusSite_2`.

typedef unspecified_type ApolloniusGraphTraits_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 dual Apollonius site of `s1`, `s2`, `s3`.

Precondition
  the dual Apollonius site 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 dual Apollonius site, with its center at infinity).