如何打印 Voronoi 图的面？

下面的代码假设输入是点，而不是线段（这是错误的）。

按照此2D Voronoi 图适配器 http://doc.cgal.org/latest/Voronoi_diagram_2/index.html#secvda2examples例如，我正在尝试编写一个程序，该程序将线段作为输入并打印 Voronoi 图的面的顶点。

这是我的尝试（保留示例的包含/typedef）：

// standard includes
#include <iostream>
#include <fstream>
#include <cassert>
// includes for defining the Voronoi diagram adaptor
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_2.h>
#include <CGAL/Voronoi_diagram_2.h>
#include <CGAL/Delaunay_triangulation_adaptation_traits_2.h>
#include <CGAL/Delaunay_triangulation_adaptation_policies_2.h>
// typedefs for defining the adaptor
typedef CGAL::Exact_predicates_inexact_constructions_kernel                  K;
typedef CGAL::Delaunay_triangulation_2<K>                                    DT;
typedef CGAL::Delaunay_triangulation_adaptation_traits_2<DT>                 AT;
typedef CGAL::Delaunay_triangulation_caching_degeneracy_removal_policy_2<DT> AP;
typedef CGAL::Voronoi_diagram_2<DT,AT,AP>                                    VD;
// typedef for the result type of the point location
typedef AT::Site_2                    Site_2;
typedef AT::Point_2                   Point_2;
typedef VD::Locate_result             Locate_result;
typedef VD::Vertex_handle             Vertex_handle;
typedef VD::Face_handle               Face_handle;
typedef VD::Halfedge_handle           Halfedge_handle;
typedef VD::Ccb_halfedge_circulator   Ccb_halfedge_circulator;

int main()
{
    std::ifstream ifs("data.cin");
    assert( ifs );
    VD vd;
    Site_2 t;
    while ( ifs >> t ) { vd.insert(t); }
    ifs.close();
    assert( vd.is_valid() );
    Face_handle* f = boost::get<Face_handle>(vd);
    std::cout << "Exiting...\n";
    return 0;
}

这会收到编译错误：

/home/gsamaras/CGAL-4.7/code/voronoi_adaptor/voronoi_adaptor.cpp:46:48: error: no matching function for call to ‘get(VD&)’
     Face_handle* f = boost::get<Face_handle>(vd);
                                                ^
/home/gsamaras/CGAL-4.7/code/voronoi_adaptor/voronoi_adaptor.cpp:46:48: note: candidates are:
In file included from /usr/include/boost/variant.hpp:22:0,
                 from /home/gsamaras/CGAL-4.7/code/voronoi_adaptor/../../include/CGAL/Object.h:36,
                 from /home/gsamaras/CGAL-4.7/code/voronoi_adaptor/../../include/CGAL/kernel_basic.h:33,
                 from /home/gsamaras/CGAL-4.7/code/voronoi_adaptor/../../include/CGAL/basic.h:46,
                 from /home/gsamaras/CGAL-4.7/code/voronoi_adaptor/../../include/CGAL/Cartesian/Cartesian_base.h:28,
                 from /home/gsamaras/CGAL-4.7/code/voronoi_adaptor/../../include/CGAL/Simple_cartesian.h:28,
                 from /home/gsamaras/CGAL-4.7/code/voronoi_adaptor/../../include/CGAL/Exact_predicates_inexact_constructions_kernel.h:28,
                 from /home/gsamaras/CGAL-4.7/code/voronoi_adaptor/voronoi_adaptor.cpp:6:
/usr/include/boost/variant/get.hpp:141:1: note: template<class U, class T0, class T1, class T2, class T3, class T4, class T5, class T6, class T7, class T8, class T9, class T10, class T11, class T12, class T13, class T14, class T15, class T16, class T17, class T18, class T19> typename boost::add_pointer<T>::type boost::get(boost::variant<T0, T1, T2, T3, T4, T5, T6, T7, T8, T9, T10, T11, T12, T13, T14, T15, T16, T17, T18, T19>*)
 get(
 ^
/usr/include/boost/variant/get.hpp:141:1: note:   template argument deduction/substitution failed:
/home/gsamaras/CGAL-4.7/code/voronoi_adaptor/voronoi_adaptor.cpp:46:48: note:   mismatched types ‘boost::variant<T0, T1, T2, T3, T4, T5, T6, T7, T8, T9, T10, T11, T12, T13, T14, T15, T16, T17, T18, T19>*’ and ‘CGAL::Voronoi_diagram_2<CGAL::Delaunay_triangulation_2<CGAL::Epick>, CGAL::Delaunay_triangulation_adaptation_traits_2<CGAL::Delaunay_triangulation_2<CGAL::Epick> >, CGAL::Delaunay_triangulation_caching_degeneracy_removal_policy_2<CGAL::Delaunay_triangulation_2<CGAL::Epick> > >’
     Face_handle* f = boost::get<Face_handle>(vd);
                                                ^
...

我需要将 Voronoi 图的面打印为众所周知的文字 https://en.wikipedia.org/wiki/Well-known_text多边形。这需要迭代面的边缘并为无限点提供有限的表示。我是这样做的，如下所示。

//Generate WKT polygons from Voronoi cells using CGAL
//Compile with: g++ main.cpp -Wall -lCGAL -lgmp
//Author: Richard Barnes (rbarnes.org)
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Regular_triangulation_filtered_traits_2.h>
#include <CGAL/Regular_triangulation_adaptation_traits_2.h>
#include <CGAL/Regular_triangulation_adaptation_policies_2.h>
#include <CGAL/Regular_triangulation_2.h>
#include <CGAL/Voronoi_diagram_2.h>
#include <CGAL/intersections.h>
#include <CGAL/bounding_box.h>
#include <CGAL/Polygon_2.h>
#include <iostream>
#include <cstdint>

typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Regular_triangulation_filtered_traits_2<K>  Traits;

typedef CGAL::Regular_triangulation_2<Traits> RT2;
typedef CGAL::Regular_triangulation_adaptation_traits_2<RT2>         AT;
typedef CGAL::Regular_triangulation_degeneracy_removal_policy_2<RT2> DRP;
typedef CGAL::Voronoi_diagram_2<RT2, AT, DRP> VD;

int main(int argc, char **argv){
  std::vector<RT2::Weighted_point> wpoints;

  std::cout.precision(4);
  std::cout.setf(std::ios::fixed);

  //Generated random points
  for(int i=0;i<100;i++)
    //Weight of 0 gives a Voronoi diagram. Non-zero weight gives a power diagram
    wpoints.push_back(RT2::Weighted_point(K::Point_2(rand()%100,rand()%100), 0)); 

  //Create a Regular Triangulation from the points
  RT2 rt(wpoints.begin(), wpoints.end());
  rt.is_valid();

  //Wrap the triangulation with a Voronoi diagram adaptor. This is necessary to
  //get the Voronoi faces.
  VD vd(rt);

  //CGAL often returns objects that are either segments or rays. This converts
  //these objects into segments. If the object would have resolved into a ray,
  //that ray is intersected with the bounding box defined above and returned as
  //a segment.
  const auto ConvertToSeg = [&](const CGAL::Object seg_obj, bool outgoing) -> K::Segment_2 {
    //One of these will succeed and one will have a NULL pointer
    const K::Segment_2 *dseg = CGAL::object_cast<K::Segment_2>(&seg_obj);
    const K::Ray_2     *dray = CGAL::object_cast<K::Ray_2>(&seg_obj);
    if (dseg) { //Okay, we have a segment
      return *dseg;
    } else {    //Must be a ray
      const auto &source = dray->source();
      const auto dsx     = source.x();
      const auto dsy     = source.y();
      const auto &dir    = dray->direction();
      const auto tpoint  = K::Point_2(dsx+1000*dir.dx(),dsy+1000*dir.dy());
      if(outgoing)
        return K::Segment_2(
          dray->source(),
          tpoint
        );
      else
        return K::Segment_2(
          tpoint,
          dray->source()
        );
    }
  };

  std::cout<<"\"id\",\"geom\"\n";

  int fnum = 0;
  //Loop over the faces of the Voronoi diagram in some arbitrary order
  for(VD::Face_iterator fit = vd.faces_begin(); fit!=vd.faces_end();++fit,fnum++){
    CGAL::Polygon_2<K> pgon;

    //Edge circulators traverse endlessly around a face. Make a note of the
    //starting point so we know when to quit.
    VD::Face::Ccb_halfedge_circulator ec_start = fit->ccb();

    //Find a bounded edge to start on
    for(;ec_start->is_unbounded();ec_start++){}

    //Current location of the edge circulator
    VD::Face::Ccb_halfedge_circulator ec = ec_start;

    do {
      //A half edge circulator representing a ray doesn't carry direction
      //information. To get it, we take the dual of the dual of the half-edge.
      //The dual of a half-edge circulator is the edge of a Delaunay triangle.
      //The dual of the edge of Delaunay triangle is either a segment or a ray.
      // const CGAL::Object seg_dual = rt.dual(ec->dual());
      const CGAL::Object seg_dual = vd.dual().dual(ec->dual());

      //Convert the segment/ray into a segment
      const auto this_seg = ConvertToSeg(seg_dual, ec->has_target());

      pgon.push_back(this_seg.source());

      //If the segment has no target, it's a ray. This means that the next
      //segment will also be a ray. We need to connect those two rays with a
      //segment. The following accomplishes this.
      if(!ec->has_target()){
        const CGAL::Object nseg_dual = vd.dual().dual(ec->next()->dual());
        const auto next_seg = ConvertToSeg(nseg_dual, ec->next()->has_target());
        pgon.push_back(next_seg.target());
      }
    } while ( ++ec != ec_start ); //Loop until we get back to the beginning

    std::cout<<fnum<<", "
    "\"POLYGON ((";
    for(auto v=pgon.vertices_begin();v!=pgon.vertices_end();v++)
      std::cout<<v->x()<<" "<<v->y()<<", ";
    std::cout<<pgon.vertices_begin()->x()<<" "<<pgon.vertices_begin()->y()<<"))\"\n";
  }

  return 0;
}