CGAL 6.0 - 2D Periodic Triangulations
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The concept Periodic_2DelaunayTriangulationTraits_2
is the first template parameter of the class CGAL::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. | |
typedef unspecified_type | Construct_circumcenter_2 |
Constructor object. | |
typedef unspecified_type | Compare_distance_2 |
Predicate type. | |
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.