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.
 
A type for a point.
 
 
A type for an Apollonius site. Must be a model
of the concept ApolloniusSite_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.
 
 
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.
 
 
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.
 
 
A type representing different types of objects
in two dimensions, namely: Point_2, Site_2,
Line_2, Ray_2 and Segment_2.
 
 
A type for the field number type of sites.
 
 
A type for the ring number type of sites.
 
 
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.
 
 
Must provide template <class T> Object_2 operator()( T t) that constructs an object of type
Object_2 that contains t and returns it.
 
 
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.
 
 
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.
 
 
A predicate object type. Must
provide Comparison_result operator()(Site_2 s1, Site_2 s2), which compares the $$xcoordinates of the centers of
s1 and s2.
 
 
A predicate object type. Must
provide Comparison_result operator()(Site_2 s1, Site_2 s2), which compares the $$ycoordinates of the centers of
s1 and s2.
 
 
A predicate object type. Must
provide Comparison_result operator()(Site_2 s1, Site_2 s2), which compares the weights of s1
and s2.
 
 
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.
 
 
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.
 
 
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 halfspace of s1 (i.e., p is closer to s1 than
s2); returns ON_NEGATIVE_SIDE if p lies in the
halfspace of s2; returns ON_ORIENTED_BOUNDARY if p
lies on the bisector of s1 and s2.
 
 
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).
 
 
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 $$s_{ } define an Apollonius edge that lies on the bisector of s1 and s2 and has as endpoints the Apollonius vertices defined by the triplets s1, s2, s3 and s1, $$s_{ } 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 $$s_{ } define an Apollonius edge that lies on the bisector of s1 and s2 and has as endpoints the Apollonius vertices defined by the triplets s1, s2, $$s_{ } and s1, $$s_{ } 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.
 
 
A predicate
object type. Must provide bool operator()(Site_2 s1, Site_2 s2, Site_2 s3, Site_2 q, bool b). The
sites $$s_{ }, s1, s2 and s3 define an
Apollonius edge that lies on the bisector of $$s_{ } and s1
and has as endpoints the Apollonius vertices defined by the triplets
$$s_{ }, s1, s2 and $$s_{ }, 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.
 
 
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.

 
Default constructor.
 
 
Copy constructor.

 
 Assignment operator. 

 
 
 
 



CGAL::Apollonius_graph_traits_2<K,Method_tag>
CGAL::Apollonius_graph_filtered_traits_2<CK,CM,EK,EM,FK,FM>