CGAL::natural_neighbor_coordinates_2()

The functions natural_neighbor_coordinates_2() compute natural neighbor coordinates, also called Sibson's coordinates, for 2D points provided a two-dimensional triangulation and a query point in the convex hull of the vertices of the triangulation.

Requirements

1. Dt are equivalent to the class Delaunay_triangulation_2<Traits, Tds>.
2. The traits class Traits of Dt is a model of the concept DelaunayTriangulationTraits_2. Only the following members of this traits class are used:
   • Construct_circumcenter_2
   • FT
   • Point_2
   • construct_circumcenter_2_object
   • Additionally, Traits must meet the requirements for the traits class of the polygon_area_2() function.
3. The value type of OutputIterator is equivalent to std::pair<Dt::Point_2, Dt::Geom_traits::FT>, i.e., a pair associating a point and its natural neighbor coordinate.

Implementation

This function computes the area of the sub-cells stolen from the Voronoi cells of the points in dt when inserting p. The total area of the Voronoi cell of p is also computed and returned by the function. If p lies outside the convex hull, the coordinate values cannot be computed and the third value of the result triple is set to false.

See Also

CGAL::linear_interpolation()

CGAL::sibson_c1_interpolation()

3D Surface Neighbor Coordinates Functions

CGAL::regular_neighbor_coordinates_2()

Functions
template<class Dt , class OutputIterator >
CGAL::Triple< OutputIterator, typename Dt::Geom_traits::FT, bool >	CGAL::natural_neighbor_coordinates_2 (const Dt &dt, const typename Dt::Geom_traits::Point_2 &p, OutputIterator out, typename Dt::Face_handle start=typename Dt::Face_handle())
	computes the natural neighbor coordinates for p with respect to the points in the two-dimensional Delaunay triangulation dt. More...
template<class Dt , class OutputIterator , class EdgeIterator >
CGAL::Triple< OutputIterator, typename Dt::Geom_traits::FT, bool >	CGAL::natural_neighbor_coordinates_2 (const Dt &dt, const typename Dt::Geom_traits::Point_2 &p, OutputIterator out, EdgeIterator hole_begin, EdgeIterator hole_end)
	The same as above. More...
template<class Dt , class OutputIterator >
CGAL::Triple< OutputIterator, typename Dt::Geom_traits::FT, bool >	CGAL::natural_neighbor_coordinates_2 (const Dt &dt, typename Dt::Vertex_handle vh, OutputIterator out)
	computes the natural neighbor coordinates of the point vh->point() with respect to the vertices of dt excluding vh->point(). More...

Function Documentation

template<class Dt , class OutputIterator >

CGAL::Triple<OutputIterator, typename Dt::Geom_traits::FT, bool > CGAL::natural_neighbor_coordinates_2	(	const Dt &	dt,
		const typename Dt::Geom_traits::Point_2 &	p,
		OutputIterator	out,
		typename Dt::Face_handle	start = typename Dt::Face_handle()
	)

computes the natural neighbor coordinates for p with respect to the points in the two-dimensional Delaunay triangulation dt.

Template Parameters

Dt	must be of type Delaunay_triangulation_2<Traits, Tds>.
OutputIterator	must have the value type std::pair<Dt::Point_2, Dt::Geom_traits::FT>.

The sequence of point/coordinate pairs that is computed by the function is placed starting at out. The function returns a triple with an iterator that is placed past-the-end of the resulting sequence of point/coordinate pairs, the normalization factor of the coordinates and a Boolean value which is set to true iff the coordinate computation was successful.

#include <CGAL/natural_neighbor_coordinates_2.h>

Examples:

Interpolation/linear_interpolation_2.cpp, Interpolation/nn_coordinates_2.cpp, and Interpolation/sibson_interpolation_2.cpp.

template<class Dt , class OutputIterator , class EdgeIterator >

CGAL::Triple< OutputIterator, typename Dt::Geom_traits::FT, bool > CGAL::natural_neighbor_coordinates_2	(	const Dt &	dt,
		const typename Dt::Geom_traits::Point_2 &	p,
		OutputIterator	out,
		EdgeIterator	hole_begin,
		EdgeIterator	hole_end
	)

The same as above.

The iterator range [hole_begin, hole_end) determines the boundary edges of the conflict zone of p in the triangulation. It is the result of the function dt.get_boundary_of_conflicts(p,std::back_inserter(hole), start).

#include <CGAL/natural_neighbor_coordinates_2.h>

template<class Dt , class OutputIterator >

CGAL::Triple< OutputIterator, typename Dt::Geom_traits::FT, bool > CGAL::natural_neighbor_coordinates_2	(	const Dt &	dt,
		typename Dt::Vertex_handle	vh,
		OutputIterator	out
	)

computes the natural neighbor coordinates of the point vh->point() with respect to the vertices of dt excluding vh->point().

The same as above for the remaining parameters.

#include <CGAL/natural_neighbor_coordinates_2.h>