## CGAL::random_polygon_2

### Definition

The function random_polygon_2constructs a random simple polygon from points that are drawn from a specific domain. Though each simple polygon defined on this set of points has a non-zero probability of being constructed, some polygons may have higher probabilities than others. The overall distribution of the generated polygons is not known since it depends on the generated points.

#include <CGAL/random_polygon_2.h>

template < class OutputIterator, class PointGenerator, class Traits >

OutputIterator

random_polygon_2 ( | int n, OutputIterator result, PointGenerator pg, Traits t = Default_traits)

|  | computes a random simple polygon by writing its vertices (oriented counterclockwise) to result. The polygon generated will have a number of vertices equal to the number of unique points in the first n points generated by pg.

### Requirements

1. Traits is a model of the concept RandomPolygonTraits_2
2. PointGenerator::value_type is equivalent to Traits::Point_2 and OutputIterator::value_type.

The default traits class Default_traits is the kernel in which Traits::Point_2 is defined.

### See Also

CGAL::Random_points_in_disc_2<Point_2, Creator>
CGAL::Random_points_in_square_2<Point_2, Creator>

### Implementation

The implementation is based on the method of eliminating self-intersections in a polygon by using so-called ``2-opt'' moves. Such a move eliminates an intersection between two edges by reversing the order of the vertices between the edges. No more than O(n3) such moves are required to simplify a polygon defined on n points [vLS82]. Intersecting edges are detected using a simple sweep through the vertices and then one intersection is chosen at random to eliminate after each sweep. The worse-case running time is therefore O(n4 logn).

### Example

The following program displays a random simple polygon with up to 50 vertices, where the vertex coordinates are drawn uniformly from the unit square centered at the origin.

    // file: examples/Generator/random_poly_example.C

    #include <CGAL/Cartesian.h>
    #include <CGAL/point_generators_2.h>
    #include <CGAL/random_polygon_2.h>
    #include <CGAL/Polygon_2.h>

    typedef CGAL::Cartesian< double >                  K;
    typedef K::Point_2                                 Point_2;
    typedef std::list<Point_2>                         Container;
    typedef CGAL::Polygon_2<K, Container>              Polygon_2;
    typedef CGAL::Random_points_in_square_2< Point_2 > Point_generator;

    int main() {
      Polygon_2 polygon;
      // create 50-gon and write it into a window:
      CGAL::random_polygon_2(50, std::back_inserter(polygon),
                             Point_generator(0.5));
      std::cout << polygon;
      return 0;
    }