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In addition to the requirements of the concept TriangulationTraits_2 described , the concept DelaunayTriangulationTraits_2 provide a predicate to check the empty circle property. The correponding predicate type is called type Side_of_oriented_circle_2.
The additional types Line_2, Ray_2 and the constructor objects Construct_ray_2 Construct_circumcenter_2, Construct_bisector_2, Construct_midpoint are used to build the dual Voronoi diagram and are required only if the dual functions are called. The additional predicate type Compare_distance_2 is required if calls to nearest_vertex(..) are issued.
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The line type. This type is required only if
some dual
functions are called.
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The type for ray. This type is required only if
some dual
functions are called.
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Predicate type.
Provides the operator : Oriented_side operator()(Point p, Point q, Point r, Point s) which takes four points 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 and .
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Predicate type. Provides
the operator : Comparison_result operator()(Point_2 p, Point_2 q, Point_2 r) which returns SMALLER, EQUAL or LARGER according to the distance between p and q being smaller, equal or larger than the distance between p and r. This type is only require if nearest_vertex queries are issued.
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Constructor object. Provides
the operator : Point_2 operator()(Point_2 p, Point_2 q, Point_2 r) which returns the circumcenter of the three points p, q anf r. This type is required only if functions relative to the dual Voronoi diagram are called.
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Constructor object. Provides
the operator : Line_2 operator()(Point_2 p, Point_2 q) which constructs the bisector line of points p and q. This type is required only if functions relative to the dual Voronoi diagram are called.
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A constructor object to build
a ray from a point and a line. Provides : Ray_2 operator() ( Point_2 p, Line_2 l);
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default constructor.
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copy constructor
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The following functions are required only if member functions of the Delaunay triangulation relative to the dual Voronoi diagram are called.
The following traits class provide everything except types and member
functions required for the dual Voronoi diagram ;
CGAL::Triangulation_euclidean_traits_xy_3<Rep>,
CGAL::Triangulation_euclidean_traits_yz_3<Rep>,
CGAL::Triangulation_euclidean_traits_zx_3<Rep>.