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CGAL 5.0.1 - 2D Periodic Triangulations
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CGAL 5.0.1 - 2D Periodic Triangulations
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The concept Periodic_2DelaunayTriangulationTraits_2 is the first template parameter of the class Periodic_2_Delaunay_triangulation_2. It refines the concepts Periodic_2TriangulationTraits_2 and DelaunayTriangulationTraits_2. It redefines the geometric objects, predicates and constructions to work with point-offset pairs. In most cases the offsets will be (0,0) and the predicates from DelaunayTriangulationTraits_2 can be used directly. For efficiency reasons we maintain for each functor the version without offsets.
In addition to the requirements of the concepts Periodic_2TriangulationTraits_2 and DelaunayTriangulationTraits_2, the concept Periodic_2DelaunayTriangulationTraits_2 provides a predicate to check the empty circle property. The corresponding predicate type is called type Periodic_2DelaunayTriangulationTraits_2::Side_of_oriented_circle_2.
The additional constructor object Periodic_2DelaunayTriangulationTraits_2::Construct_circumcenter_2 is used to build the dual Voronoi diagram and are required only if the dual functions are called. The additional predicate type Periodic_2DelaunayTriangulationTraits_2::Compare_distance_2 is required if calls to nearest_vertex(..) are issued.
CGAL::Periodic_2_Delaunay_triangulation_traits_2<Traits, Offset>DelaunayTriangulationTraits_2 Types | |
| typedef unspecified_type | Side_of_oriented_circle_2 |
| Predicate object. More... | |
| typedef unspecified_type | Construct_circumcenter_2 |
| Constructor object. More... | |
| typedef unspecified_type | Compare_distance_2 |
| Predicate type. More... | |
Predicate Functions | |
| Side_of_oriented_circle_2 | side_of_oriented_circle_2_object () |
Required only if side_of_oriented_circle is called called. | |
| Construct_circumcenter_2 | construct_circumcenter_2_object () |
Required only if circumcenter is called. | |
| Compare_distance_2 | compare_distance_2_object () |
Required only if compare_distance is called. | |
Predicate type.
Provides the operators:
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.
Comparison_result operator()(Point_2 p, Point_2 q, Point_2 r, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q Periodic_2_offset_2 o_r) which returns SMALLER, EQUAL or LARGER according to the distance between (p, o_p), and (q, o_q) being smaller, equal or larger than the distance between (p, o_p) and (r, o_r).
This type is only require if nearest_vertex queries are issued.
Constructor object.
Provides the operators:
Point operator()(Point p, Point q, Point r)
which returns the circumcenter of the three points p, q and r.
Point operator()(Point p, Point q, Point r, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q Periodic_2_offset_2 o_r)
which returns the circumcenter of the three points (p, o_p), (q, o_q) and (r, o_r).
This type is required only if the function Point circumcenter(Face_handle f)is called.
Predicate object.
Must provide the operators
Oriented_side operator()(Point p, Point q, Point r, Point s) which takes four points \( p, q, r, s\) 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 through \( p,q\) and \( r\) and
Oriented_side operator()( Point p, Point q, Point r, Point s, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q Periodic_2_offset_2 o_r, Periodic_2_offset_2 o_s) which takes four points \( (p, o_p), (q, o_q), (r, o_r), (s, o_s)\) as arguments and returns ON_POSITIVE_SIDE, ON_NEGATIVE_SIDE or, ON_ORIENTED_BOUNDARY according to the position of points (s, o_s) with respect to the oriented circle through through (p, o_p), (q, o_q) and (r, o_r).
This type is required only if the function side_of_oriented_circle(Face_handle f, Point p) is called.
1.8.13