## CGAL::Voronoi_intersection_2_traits_3<K>

### Definition

Voronoi_intersection_2_traits_3<K> is a model for the concept RegularTriangulationTraits_2 and InterpolationTraits. It can be used to instantiate the geometric traits class of a two-dimensional regular triangulation. A three-dimensional plane is defined by a point and a vector that are members of the traits class. The triangulation is defined on 3D points. It is the regular triangulation of the input points projected onto the plane and each weighted with the negative squared distance of the input point to the plane. It can be shown that it is dual to the power diagram obtained by intersecting the three-dimensional Voronoi diagram of the input points with the plane. All predicates and constructions used in the computation of the regular triangulation are formulated on the three dimensional points without explicitly constructing the projected points and the weights. This reduces the arithmetic demands. The traits class is templated by a kernel class K.

#include <CGAL/Voronoi_intersection_2_traits_3.h>

### Is Model for the Concept

RegularTriangulationTraits_2

### Types

typedef K::RT | Weight;
typedef K::FT | FT;
typedef K::Point_3 | Point_2;
typedef K::Segment_3
| Segment_2;
typedef K::Triangle_3
| Triangle_2;
typedef K::Line_3 | Line_2;
typedef K::Ray_3 | Ray_2;
typedef K::Vector_3
| Vector_2;
typedef K::Construct_triangle_3
| Construct_triangle_2;
typedef K::Construct_ray_3
| Construct_ray_2;
typedef K::Compare_distance_3
| Compare_distance_2;
Compute_area_3<Rep>
| Compute_area_2; | An instance of this function object class computes the square root of the result of K::Compute_squared_area_3. If the number type FT does not support the square root operation, the result is cast to double before computing the square root.
typedef Orientation_with_normal_plane_2_3<Rep>
| Orientation_2;
typedef Side_of_plane_centered_sphere_2_3<Point_2>
| Power_test_2;
typedef Construct_plane_centered_circumcenter_3<Point_2>
| Construct_weighted_circumcenter_2;
typedef Compare_first_projection_3<Point_2>
| Compare_x_2;
typedef Compare_second_projection_3<Point_2>
| Compare_y_2;

### Operations

### See Also

RegularTriangulationTraits_2
CGAL::Regular_triangulation_2<Gt, Tds>
CGAL::regular_neighbor_coordinates_2
CGAL::surface_neighbor_coordinates_3