The function all_furthest_neighbors_2 computes all furthest neighbors for the vertices of a convex polygon P, i.e. for each vertex v of P a vertex fv of P such that the distance between v and fv is maximized.

#include <CGAL/all_furthest_neighbors_2.h>

template < class RandomAccessIterator, class OutputIterator, class Traits >
all_furthest_neighbors_2 (
RandomAccessIterator points_begin,
RandomAccessIterator points_end,
OutputIterator o,
Traits t = Default_traits)

computes all furthest neighbors for the vertices of the convex polygon described by the range [points_begin, points_end), writes their indices (relative to points_begin) to o1 and returns the past-the-end iterator of this sequence.


The points denoted by the non-empty range [points_begin, points_end) form the boundary of a convex polygon P (oriented clock- or counterclockwise).

The geometric types and operations to be used for the computation are specified by the traits class parameter t. This parameter can be omitted if RandomAccessIterator refers to a point type from a Kernel. In this case, the kernel is used as default traits class.


  1. If t is specified explicitly, Traits is a model for AllFurthestNeighborsTraits_2.
  2. Value type of RandomAccessIterator is Traits::Point_2 or - if t is not specified explicitly - K::Point_2 where K is a model for Kernel.
  3. OutputIterator accepts int as value type.

See Also



The implementation uses monotone matrix search[AKM+87]. Its runtime complexity is linear in the number of vertices of P.


The following code generates a random convex polygon p with ten vertices, computes all furthest neighbors and writes the sequence of their indices (relative to points_begin) to cout (e.g. a sequence of 4788911224 means the furthest neighbor of points_begin[0] is points_begin[4], the furthest neighbor of points_begin[1] is points_begin[7] etc.).

File: examples/Matrix_search/all_furthest_neighbors_2.cpp
#include <CGAL/Cartesian.h>
#include <CGAL/Polygon_2.h>
#include <CGAL/point_generators_2.h>
#include <CGAL/random_convex_set_2.h>
#include <CGAL/all_furthest_neighbors_2.h>
#include <CGAL/IO/Ostream_iterator.h>
#include <iostream>
#include <vector>

typedef double                                    FT;

struct Kernel : public CGAL::Cartesian<FT> {};

typedef Kernel::Point_2                           Point;
typedef std::vector<int>                          Index_cont;
typedef CGAL::Polygon_2<Kernel>                   Polygon_2;
typedef CGAL::Random_points_in_square_2<Point>    Generator;
typedef CGAL::Ostream_iterator<int,std::ostream>  Oiterator;

int main()
  // generate random convex polygon:
  Polygon_2 p;
  CGAL::random_convex_set_2(10, std::back_inserter(p), Generator(1));

  // compute all furthest neighbors:
  CGAL::all_furthest_neighbors_2(p.vertices_begin(), p.vertices_end(),
  std::cout << std::endl;

  return 0;
// ----------------------------------------------------------------------------
// ** EOF
// ----------------------------------------------------------------------------


 1  i.e. the furthest neighbor of points_begin[i] is points_begin[i-th number written to o]