This file is indexed.

/usr/include/CGAL/convex_hull_3.h is in libcgal-dev 4.2-5ubuntu1.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
// Copyright (c) 2001,2011  Max-Planck-Institute Saarbruecken (Germany).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
// You can redistribute it and/or modify it under the terms of the GNU
// General Public License as published by the Free Software Foundation,
// either version 3 of the License, or (at your option) any later version.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL$
// $Id$
// 
//
// Author(s)     : Susan Hert <hert@mpi-sb.mpg.de>
//               : Amol Prakash <prakash@mpi-sb.mpg.de>
//               : Andreas Fabri

#ifndef CGAL_CONVEX_HULL_3_H
#define CGAL_CONVEX_HULL_3_H
#include <CGAL/basic.h>
#include <CGAL/algorithm.h> 
#include <CGAL/convex_hull_2.h>
#include <CGAL/Polyhedron_incremental_builder_3.h>
#include <CGAL/Projection_traits_xy_3.h>
#include <CGAL/Projection_traits_xz_3.h>
#include <CGAL/Projection_traits_yz_3.h>
#include <CGAL/Convex_hull_traits_3.h>
#include <CGAL/Convex_hull_2/ch_assertions.h>
#include <CGAL/Triangulation_data_structure_2.h>
#include <CGAL/Triangulation_vertex_base_with_info_2.h>
#include <CGAL/Cartesian_converter.h>
#include <CGAL/Simple_cartesian.h>
#include <iostream>
#include <algorithm>
#include <utility>
#include <list>
#include <map>
#include <vector>
#include <boost/bind.hpp>
#include <boost/next_prior.hpp>
#include <boost/type_traits/is_floating_point.hpp>
#include <CGAL/internal/Exact_type_selector.h>


#ifndef CGAL_CH_NO_POSTCONDITIONS
#include <CGAL/convexity_check_3.h>
#endif // CGAL_CH_NO_POSTCONDITIONS


namespace CGAL {

namespace internal{  namespace Convex_hull_3{

//struct to select the default traits class for computing convex hull
template< class Point_3,
          class Is_floating_point=typename boost::is_floating_point<typename Kernel_traits<Point_3>::Kernel::FT>::type,
          class Has_filtered_predicates_tag=typename Kernel_traits<Point_3>::Kernel::Has_filtered_predicates_tag >
struct Default_traits_for_Chull_3{
  typedef typename Kernel_traits<Point_3>::Kernel type;
};

//FT is a floating point type and Kernel is a filtered kernel
template <class Point_3>
struct Default_traits_for_Chull_3<Point_3,boost::true_type,Tag_true>{
  typedef Convex_hull_traits_3< typename Kernel_traits<Point_3>::Kernel > type;
};

template <class Traits>
struct Default_polyhedron_for_Chull_3{
  typedef CGAL::Polyhedron_3<Traits> type;
};

template <class K>
struct Default_polyhedron_for_Chull_3<Convex_hull_traits_3<K> >{
  typedef typename  Convex_hull_traits_3<K>::Polyhedron_3 type;
};
 
//utility class to select the right version of internal predicate Is_on_positive_side_of_plane_3
template <class Traits,
          class Is_floating_point=
            typename boost::is_floating_point<typename Kernel_traits<typename Traits::Point_3>::Kernel::FT>::type,
          class Has_filtered_predicates_tag=typename Kernel_traits<typename Traits::Point_3>::Kernel::Has_filtered_predicates_tag,
          class Has_cartesian_tag=typename Kernel_traits<typename Traits::Point_3>::Kernel::Kernel_tag >
struct Use_advanced_filtering{
  typedef CGAL::Tag_false type;
};

template <class Traits>
struct Use_advanced_filtering<Traits,boost::true_type,Tag_true,Cartesian_tag>{
  typedef typename Kernel_traits<typename Traits::Point_3>::Kernel K;
  typedef CGAL::Boolean_tag<K::Has_static_filters> type;
};

//Predicates internally used
template <class Traits,class Tag_use_advanced_filtering=typename Use_advanced_filtering<Traits>::type >
class Is_on_positive_side_of_plane_3{
  typedef typename Traits::Point_3 Point_3;
  typename Traits::Plane_3 plane;
  typename Traits::Has_on_positive_side_3 has_on_positive_side;
public:
  typedef Protect_FPU_rounding<false> Protector;

  Is_on_positive_side_of_plane_3(const Traits& traits,const Point_3& p,const Point_3& q,const Point_3& r)
  :plane(traits.construct_plane_3_object()(p,q,r)),has_on_positive_side(traits.has_on_positive_side_3_object()) {}
    
  bool operator() (const Point_3& s) const 
  {
    return has_on_positive_side(plane,s);
  }
};
  

//This predicate uses copy of the code from the statically filtered version of
//Orientation_3. The rational is that the plane is a member of the functor
//so optimization are done to avoid doing several time operations on the plane.
//The main operator() first tries the static version of the predicate, then uses
//interval arithmetic (the protector must be created before using this predicate)
//and in case of failure, exact arithmetic is used.
template <class Kernel>
class Is_on_positive_side_of_plane_3<Convex_hull_traits_3<Kernel>,Tag_true>{
  typedef Simple_cartesian<CGAL::internal::Exact_type_selector<double>::Type>         PK;
  typedef Simple_cartesian<Interval_nt_advanced >                               CK;  
  typedef Convex_hull_traits_3<Kernel>                                          Traits;
  typedef typename Traits::Point_3                                              Point_3;
  
  Cartesian_converter<Kernel,CK>                        to_CK;
  Cartesian_converter<Kernel,PK>                        to_PK;

  const Point_3& p,q,r;
  mutable typename CK::Plane_3* ck_plane;
  mutable typename PK::Plane_3* pk_plane;
 
  double m10,m20,m21,Maxx,Maxy,Maxz;
  
  static const int STATIC_FILTER_FAILURE = 555;
  
  //this function is a made from the statically filtered version of Orientation_3
  int static_filtered(double psx,double psy, double psz) const{

    // Then semi-static filter.
    double apsx = CGAL::abs(psx);
    double apsy = CGAL::abs(psy);
    double apsz = CGAL::abs(psz);

    double maxx = (Maxx < apsx)? apsx : Maxx;
    double maxy = (Maxy < apsy)? apsy : Maxy;
    double maxz = (Maxz < apsz)? apsz : Maxz;

    double det =  psx*m10 - m20*psy + m21*psz;
    
    // Sort maxx < maxy < maxz.
    if (maxx > maxz)
        std::swap(maxx, maxz);
    if (maxy > maxz)
        std::swap(maxy, maxz);
    else if (maxy < maxx)
        std::swap(maxx, maxy);

    // Protect against underflow in the computation of eps.
    if (maxx < 1e-97) /* cbrt(min_double/eps) */ {
      if (maxx == 0)
        return 0;
    }
    // Protect against overflow in the computation of det.
    else if (maxz < 1e102) /* cbrt(max_double [hadamard]/4) */ {
      double eps = 5.1107127829973299e-15 * maxx * maxy * maxz;
      if (det > eps)  return 1;
      if (det < -eps) return -1;
    }
    return STATIC_FILTER_FAILURE;
  }
  
public:
  typedef typename Interval_nt_advanced::Protector           Protector;

  Is_on_positive_side_of_plane_3(const Traits&,const Point_3& p_,const Point_3& q_,const Point_3& r_)
  :p(p_),q(q_),r(r_),ck_plane(NULL),pk_plane(NULL)
  {
    double pqx = q.x() - p.x();
    double pqy = q.y() - p.y();
    double pqz = q.z() - p.z();
    double prx = r.x() - p.x();
    double pry = r.y() - p.y();
    double prz = r.z() - p.z();   


    m10 = pqy*prz - pry*pqz;
    m20 = pqx*prz - prx*pqz;
    m21 = pqx*pry - prx*pqy;
    
    double aprx = CGAL::abs(prx);
    double apry = CGAL::abs(pry);
    double aprz = CGAL::abs(prz);

    Maxx = CGAL::abs(pqx);
    if (Maxx < aprx) Maxx = aprx;
    Maxy = CGAL::abs(pqy);
    if (Maxy < apry) Maxy = apry;
    Maxz = CGAL::abs(pqz);
    if (Maxz < aprz) Maxz = aprz;
  }

  ~Is_on_positive_side_of_plane_3(){
    if (ck_plane!=NULL) delete ck_plane;
    if (pk_plane!=NULL) delete pk_plane;
  }
  
  bool operator() (const Point_3& s) const 
  {
    double psx = s.x() - p.x();
    double psy = s.y() - p.y();
    double psz = s.z() - p.z(); 
    
    int static_res = static_filtered(psx,psy,psz);
    if (static_res != STATIC_FILTER_FAILURE)
      return static_res == 1;
    
    try{
      if (ck_plane==NULL)
        ck_plane=new typename CK::Plane_3(to_CK(p),to_CK(q),to_CK(r));
      return ck_plane->has_on_positive_side(to_CK(s));
    }
    catch (Uncertain_conversion_exception){
      if (pk_plane==NULL)
        pk_plane=new typename PK::Plane_3(to_PK(p),to_PK(q),to_PK(r));
      return pk_plane->has_on_positive_side(to_PK(s));
    }
  }
};


template<class HDS, class ForwardIterator>
class Build_coplanar_poly : public Modifier_base<HDS> {
 public:
  Build_coplanar_poly(ForwardIterator i, ForwardIterator j) 
    {
      start = i;
      end = j;
    }
  void operator()( HDS& hds) {
    Polyhedron_incremental_builder_3<HDS> B(hds,true);
    ForwardIterator iter = start;
    int count = 0;
    while (iter != end)
      {
	count++;
	iter++;
      }
    B.begin_surface(count, 1, 2*count);
    iter = start;
    while (iter != end)
      {
	B.add_vertex(*iter);
	iter++;
      }
    iter = start;
    B.begin_facet();
    int p = 0;
    while (p < count)
      {
	B.add_vertex_to_facet(p);
	p++;
      }
    B.end_facet();
    B.end_surface();
  }
 private:
  ForwardIterator start;
  ForwardIterator end;    
};

template <class InputIterator, class Plane_3, class Polyhedron_3, class Traits>
void coplanar_3_hull(InputIterator first, InputIterator beyond,
                     Plane_3 plane, Polyhedron_3& P, const Traits& traits)
{
  typedef typename Traits::Point_3               Point_3;
  typedef typename Kernel_traits<Point_3>::Kernel R;
  typedef typename Traits::Vector_3              Vector_3;
  typedef Max_coordinate_3<Vector_3>             Max_coordinate_3;
  typedef Polyhedron_3                           Polyhedron;
  
  std::list<Point_3> CH_2;
  typedef typename std::list<Point_3>::iterator  CH_2_iterator;
  typedef typename Traits::Construct_orthogonal_vector_3
                                                   Construct_normal_vec;
  Max_coordinate_3 max_coordinate;

  Construct_normal_vec c_normal = 
                          traits.construct_orthogonal_vector_3_object();
  Vector_3 normal = c_normal(plane);
  int max_coord = max_coordinate(normal);
  switch (max_coord)
  {
     case 0:
     {
       convex_hull_points_2(first, beyond, std::back_inserter(CH_2),
            Projection_traits_yz_3<R>());
       break;
     }
     case 1:
     {
       convex_hull_points_2(first, beyond, std::back_inserter(CH_2),
            Projection_traits_xz_3<R>());
       break;
     }
     case 2:
     {
       convex_hull_points_2(first, beyond, std::back_inserter(CH_2),
            Projection_traits_xy_3<R>());
       break;
     }
     default:
       break;
  }
  typedef typename Polyhedron::Halfedge_data_structure HDS;

  Build_coplanar_poly<HDS,CH_2_iterator> poly(CH_2.begin(),CH_2.end());
  P.delegate(poly);
}


//
// visible is the set of facets visible from point  and reachable from
// start_facet.
//
template <class TDS_2, class Traits>
void
find_visible_set(TDS_2& tds, 
                 const typename Traits::Point_3& point, 
                 typename TDS_2::Face_handle start,
                 std::list<typename TDS_2::Face_handle>& visible,
                 std::map<typename TDS_2::Vertex_handle, typename TDS_2::Edge>& outside,
                 const Traits& traits)
{
   typedef typename Traits::Plane_3                   Plane_3;
   typedef typename TDS_2::Face_handle Face_handle;
   typedef typename TDS_2::Vertex_handle Vertex_handle;
   typename Traits::Has_on_positive_side_3 has_on_positive_side =
            traits.has_on_positive_side_3_object();

   std::vector<Vertex_handle> vertices;
   vertices.reserve(10);
   int VISITED=1, BORDER=2;
   visible.clear();
   typename std::list<Face_handle>::iterator  vis_it;
   visible.push_back(start);
   start->info() = VISITED;
   vertices.push_back(start->vertex(0));
   vertices.push_back(start->vertex(1));
   vertices.push_back(start->vertex(2));
   start->vertex(0)->info() = start->vertex(1)->info() = start->vertex(2)->info() = VISITED;
 
   for (vis_it = visible.begin(); vis_it != visible.end(); vis_it++)
   {
      // check all the neighbors of the current face to see if they have 
      // already been visited or not and if not whether they are visible 
      // or not.

      for(int i=0; i < 3; i++) {
        // the facet on the other side of the current halfedge
        Face_handle f = (*vis_it)->neighbor(i);
        // if haven't already seen this facet
        if (f->info() == 0) {
          f->info() = VISITED;
          Plane_3 plane(f->vertex(0)->point(),f->vertex(1)->point(),f->vertex(2)->point());
          int ind = f->index(*vis_it);
          if ( has_on_positive_side(plane, point) ){  // is visible
            visible.push_back(f);
            Vertex_handle vh = f->vertex(ind);
            if(vh->info() == 0){ vertices.push_back(vh); vh->info() = VISITED;}
          } else {
            f->info() = BORDER;
            f->vertex(TDS_2::cw(ind))->info() = BORDER;            
            f->vertex(TDS_2::ccw(ind))->info() = BORDER;
            outside.insert(std::make_pair(f->vertex(TDS_2::cw(ind)),
                                          typename TDS_2::Edge(f,ind)));
          }
        } else if(f->info() == BORDER) {
          int ind = f->index(*vis_it);
          f->vertex(TDS_2::cw(ind))->info() = BORDER;            
          f->vertex(TDS_2::ccw(ind))->info() = BORDER;
          outside.insert(std::make_pair(f->vertex(TDS_2::cw(ind)),
                                        typename TDS_2::Edge(f,ind)));
        }
      }
   }
 
   for(typename std::vector<Vertex_handle>::iterator vit =  vertices.begin();
       vit != vertices.end();
       ++vit){
     if((*vit)->info() != BORDER){
       tds.delete_vertex(*vit);
     } else {
       (*vit)->info() = 0;
     }
   }

}

// using a third template parameter for the point instead of getting it from
// the traits class as it should be is required by M$VC6
template <class Face_handle, class Traits, class Point>
typename std::list<Point>::iterator
farthest_outside_point(Face_handle f, std::list<Point>& outside_set,
                       const Traits& traits)
{

   typedef typename std::list<Point>::iterator Outside_set_iterator;
   CGAL_ch_assertion(!outside_set.empty());

   typename Traits::Plane_3 plane(f->vertex(0)->point(),f->vertex(1)->point(),f->vertex(2)->point());

   typename Traits::Less_signed_distance_to_plane_3 less_dist_to_plane =
            traits.less_signed_distance_to_plane_3_object();
   Outside_set_iterator farthest_it =
          std::max_element(outside_set.begin(),
                           outside_set.end(), 
                           boost::bind(less_dist_to_plane, plane, _1, _2));
   return farthest_it;
}

template <class Face_handle, class Traits, class Point>
void     
partition_outside_sets(const std::list<Face_handle>& new_facets,
                       std::list<Point>& vis_outside_set, 
                       std::list<Face_handle>& pending_facets,
                       const Traits& traits)
{
  typename std::list<Face_handle>::const_iterator        f_list_it;
  typename std::list<Point>::iterator  point_it, to_splice;
   
  // walk through all the new facets and check each unassigned outside point
  // to see if it belongs to the outside set of this new facet.
  for (f_list_it = new_facets.begin(); (f_list_it != new_facets.end()) && (! vis_outside_set.empty());
        ++f_list_it)
  {
    Face_handle f = *f_list_it;
    Is_on_positive_side_of_plane_3<Traits> is_on_positive_side(
      traits,f->vertex(0)->point(),f->vertex(1)->point(),f->vertex(2)->point());
    std::list<Point>& point_list = f->points;

    for (point_it = vis_outside_set.begin();point_it != vis_outside_set.end();){
      if( is_on_positive_side(*point_it) ) {
        to_splice = point_it;
        ++point_it;
        point_list.splice(point_list.end(), vis_outside_set, to_splice);
      } else {
         ++point_it;
      }
    }
    if(! point_list.empty()){
      pending_facets.push_back(f);
      f->it = boost::prior(pending_facets.end());
    } else {
      f->it = pending_facets.end();
    }
  }
   
   
   for (; f_list_it != new_facets.end();++f_list_it)
    (*f_list_it)->it = pending_facets.end();
}



template <class TDS_2, class Traits>
void
ch_quickhull_3_scan(TDS_2& tds,
                    std::list<typename TDS_2::Face_handle>& pending_facets,
                    const Traits& traits)
{
  typedef typename TDS_2::Edge                            Edge;
  typedef typename TDS_2::Face_handle                     Face_handle;
  typedef typename TDS_2::Vertex_handle                   Vertex_handle;
  typedef typename Traits::Point_3			  Point_3;
  typedef std::list<Point_3>                              Outside_set;
  typedef typename std::list<Point_3>::iterator           Outside_set_iterator;
  typedef std::map<typename TDS_2::Vertex_handle, typename TDS_2::Edge> Border_edges;

  std::list<Face_handle>                     visible_set;
  typename std::list<Face_handle>::iterator  vis_set_it;
  Outside_set                                vis_outside_set;
  Border_edges                               border;

  while (!pending_facets.empty())
  {
     vis_outside_set.clear();

     Face_handle f_handle = pending_facets.front();

     Outside_set_iterator farthest_pt_it = farthest_outside_point(f_handle, f_handle->points, traits);
     Point_3 farthest_pt = *farthest_pt_it;
     f_handle->points.erase(farthest_pt_it);
     find_visible_set(tds, farthest_pt, f_handle, visible_set, border, traits);

     // for each visible facet
     for (vis_set_it = visible_set.begin(); vis_set_it != visible_set.end();
          vis_set_it++)
     {
       
        //   add its outside set to the global outside set list
       std::list<Point_3>& point_list = (*vis_set_it)->points;
       if(! point_list.empty()){
         vis_outside_set.splice(vis_outside_set.end(), point_list, point_list.begin(), point_list.end());
       }

       if((*vis_set_it)->it != pending_facets.end()){
         pending_facets.erase((*vis_set_it)->it);
       }
       (*vis_set_it)->info() = 0;
     }

     std::vector<Edge> edges;
     edges.reserve(border.size());
     typename Border_edges::iterator it = border.begin();
     Edge e = it->second;
     e.first->info() = 0; 
     edges.push_back(e);
     border.erase(it);
     while(! border.empty()){
       it = border.find(e.first->vertex(TDS_2::ccw(e.second)));
       assert(it != border.end());
       e = it->second;
       e.first->info() = 0; 
       edges.push_back(e);
       border.erase(it);
     }

     // If we want to reuse the faces we must only pass |edges| many, and call delete_face for the others.
     // Also create facets if necessary
     std::ptrdiff_t diff = visible_set.size() - edges.size();
     if(diff < 0){
       for(int i = 0; i<-diff;i++){
         visible_set.push_back(tds.create_face());
       }
     } else {
       for(int i = 0; i<diff;i++){
         tds.delete_face(visible_set.back());
         visible_set.pop_back();
       }
     }
     Vertex_handle vh = tds.star_hole(edges.begin(), edges.end(), visible_set.begin(), visible_set.end());
     vh->point() = farthest_pt;
     vh->info() = 0;     
  
     // now partition the set of outside set points among the new facets.
   
     partition_outside_sets(visible_set, vis_outside_set, 
                            pending_facets, traits);

  }
}

template <class TDS_2, class Traits>
void non_coplanar_quickhull_3(std::list<typename Traits::Point_3>& points,
                              TDS_2& tds, const Traits& traits)
{
  typedef typename Traits::Point_3                        Point_3;

  typedef typename TDS_2::Face_handle                     Face_handle;
  typedef typename TDS_2::Face_iterator                     Face_iterator;
  typedef typename std::list<Point_3>::iterator           P3_iterator;

  std::list<Face_handle> pending_facets;

  typename Is_on_positive_side_of_plane_3<Traits>::Protector p;
  
  // for each facet, look at each unassigned point and decide if it belongs
  // to the outside set of this facet.
  for(Face_iterator fit = tds.faces_begin(); fit != tds.faces_end(); ++fit){
    Is_on_positive_side_of_plane_3<Traits> is_on_positive_side(
      traits,fit->vertex(0)->point(),fit->vertex(1)->point(),fit->vertex(2)->point() );
    for (P3_iterator point_it = points.begin() ; point_it != points.end(); )
    {
      if( is_on_positive_side(*point_it) ) {
        P3_iterator to_splice = point_it;
        ++point_it;
        fit->points.splice(fit->points.end(), points, to_splice);
      } else {
       ++point_it;
      }
    }
  }
  // add all the facets with non-empty outside sets to the set of facets for
  // further consideration
  for(Face_iterator fit = tds.faces_begin(); fit != tds.faces_end(); ++fit){
    if (! fit->points.empty()){
      pending_facets.push_back(fit);
      fit->it = boost::prior(pending_facets.end());
        } else {
      fit->it =  pending_facets.end();
    }
  }


  ch_quickhull_3_scan(tds, pending_facets, traits);

  //std::cout << "|V(tds)| = " << tds.number_of_vertices() << std::endl;
//  CGAL_ch_expensive_postcondition(all_points_inside(points.begin(),
//                                                    points.end(),P,traits));
//  CGAL_ch_postcondition(is_strongly_convex_3(P, traits));
}


namespace internal{
  
template <class HDS,class TDS>
class Build_convex_hull_from_TDS_2 : public CGAL::Modifier_base<HDS> {
  typedef std::map<typename TDS::Vertex_handle,unsigned> Vertex_map;
  
  const TDS& t;
  template <class Builder>
  static unsigned get_vertex_index( Vertex_map& vertex_map,
                                    typename TDS::Vertex_handle vh,
                                    Builder& builder,
                                    unsigned& vindex)
  {
    std::pair<typename Vertex_map::iterator,bool>
      res=vertex_map.insert(std::make_pair(vh,vindex));
    if (res.second){
      builder.add_vertex(vh->point());
      ++vindex;
    }
    return res.first->second;
  }
  
public:
  Build_convex_hull_from_TDS_2(const TDS& t_):t(t_) 
  {
    CGAL_assertion(t.dimension()==2);
  }
  void operator()( HDS& hds) {
    // Postcondition: `hds' is a valid polyhedral surface.
    
    CGAL::Polyhedron_incremental_builder_3<HDS> B( hds, true);
    Vertex_map vertex_map;
    //start the surface
    B.begin_surface( t.number_of_vertices(), t.number_of_faces());
    unsigned vindex=0;
    for (typename TDS::Face_iterator it=t.faces_begin();it!=t.faces_end();++it)
    {
      unsigned i0=get_vertex_index(vertex_map,it->vertex(0),B,vindex);
      unsigned i1=get_vertex_index(vertex_map,it->vertex(1),B,vindex);
      unsigned i2=get_vertex_index(vertex_map,it->vertex(2),B,vindex);
      B.begin_facet();
      B.add_vertex_to_facet( i0 );
      B.add_vertex_to_facet( i1 );
      B.add_vertex_to_facet( i2 );
      B.end_facet();      
    }
    B.end_surface();
  }
};
  
} //namespace internal

template <class InputIterator, class Polyhedron_3, class Traits>
void
ch_quickhull_polyhedron_3(std::list<typename Traits::Point_3>& points,
                          InputIterator point1_it, InputIterator point2_it,
                          InputIterator point3_it, Polyhedron_3& P,
                          const Traits& traits)
{
  typedef typename Traits::Point_3	  		  Point_3;  
  typedef typename Traits::Plane_3		      	  Plane_3;
  typedef typename std::list<Point_3>::iterator           P3_iterator;

  typedef typename Kernel_traits<typename Traits::Point_3>::Kernel R;
  typedef Triangulation_data_structure_2<
    Triangulation_vertex_base_with_info_2<int, GT3_for_CH3<R> >,
    Convex_hull_face_base_2<int, R> >                           Tds;  
  typedef typename Tds::Vertex_handle                     Vertex_handle;
  typedef typename Tds::Face_handle                     Face_handle;

  // found three points that are not collinear, so construct the plane defined
  // by these points and then find a point that has maximum distance from this
  // plane.   
  typename Traits::Construct_plane_3 construct_plane =
         traits.construct_plane_3_object();
  Plane_3 plane = construct_plane(*point3_it, *point2_it, *point1_it);
  typedef typename Traits::Less_signed_distance_to_plane_3      Dist_compare; 
  Dist_compare compare_dist = traits.less_signed_distance_to_plane_3_object();
  
  typename Traits::Coplanar_3  coplanar = traits.coplanar_3_object(); 
  // find both min and max here since using signed distance.  If all points
  // are on the negative side of the plane, the max element will be on the
  // plane.
  std::pair<P3_iterator, P3_iterator> min_max;
  min_max = CGAL::min_max_element(points.begin(), points.end(), 
                                  boost::bind(compare_dist, plane, _1, _2),
                                  boost::bind(compare_dist, plane, _1, _2));
  P3_iterator max_it;
  if (coplanar(*point1_it, *point2_it, *point3_it, *min_max.second))
  {
     max_it = min_max.first;
     // want the orientation of the points defining the plane to be positive
     // so have to reorder these points if all points were on negative side
     // of plane
     std::swap(*point1_it, *point3_it);
  }
  else
     max_it = min_max.second;

  // if the maximum distance point is on the plane then all are coplanar
  if (coplanar(*point1_it, *point2_it, *point3_it, *max_it)) {
     coplanar_3_hull(points.begin(), points.end(), plane, P, traits);
  } else {  
    Tds tds;
    Vertex_handle v0 = tds.create_vertex(); v0->set_point(*point1_it);
    Vertex_handle v1 = tds.create_vertex(); v1->set_point(*point2_it);
    Vertex_handle v2 = tds.create_vertex(); v2->set_point(*point3_it);
    Vertex_handle v3 = tds.create_vertex(); v3->set_point(*max_it);

    v0->info() = v1->info() = v2->info() = v3->info() = 0;
    Face_handle f0 = tds.create_face(v0,v1,v2);
    Face_handle f1 = tds.create_face(v3,v1,v0);
    Face_handle f2 = tds.create_face(v3,v2,v1);
    Face_handle f3 = tds.create_face(v3,v0,v2);
    tds.set_dimension(2);
    f0->set_neighbors(f2, f3, f1);
    f1->set_neighbors(f0, f3, f2);
    f2->set_neighbors(f0, f1, f3);
    f3->set_neighbors(f0, f2, f1);

    points.erase(point1_it);
    points.erase(point2_it);
    points.erase(point3_it);
    points.erase(max_it);
    if (!points.empty()){
      non_coplanar_quickhull_3(points, tds, traits);
      internal::Build_convex_hull_from_TDS_2<typename Polyhedron_3::HalfedgeDS,Tds> builder(tds);
      P.delegate(builder);
    }
    else
      P.make_tetrahedron(v0->point(),v1->point(),v2->point(),v3->point());
  }
  
}

} } //namespace internal::Convex_hull_3

template <class InputIterator, class Traits>
void
convex_hull_3(InputIterator first, InputIterator beyond, 
              Object& ch_object, const Traits& traits)
{  
  typedef typename Traits::Point_3	  		  Point_3;  
  typedef std::list<Point_3>                              Point_3_list;
  typedef typename Point_3_list::iterator                 P3_iterator;
  typedef std::pair<P3_iterator,P3_iterator>              P3_iterator_pair;

  if (first == beyond)    // No point
    return;

  // If the first and last point are equal the collinearity test some lines below will always be true.
  Point_3_list points(first, beyond);
  std::size_t size = points.size();
  while((size > 1) && (points.front() == points.back())){
    points.pop_back();
    --size;
  }

  if ( size == 1 )                // 1 point 
  {
      ch_object = make_object(*points.begin());
      return;
  }
  else if ( size == 2 )           // 2 points 
  {
      typedef typename Traits::Segment_3                 Segment_3;  
      typename Traits::Construct_segment_3 construct_segment =
             traits.construct_segment_3_object();
      Segment_3 seg = construct_segment(*points.begin(), *(++points.begin()));
      ch_object = make_object(seg);
      return;
  }
  else if ( size == 3 )           // 3 points 
  {
      typedef typename Traits::Triangle_3                Triangle_3;  
      typename Traits::Construct_triangle_3 construct_triangle =
             traits.construct_triangle_3_object();
      Triangle_3 tri = construct_triangle(*(points.begin()), 
                                          *(++points.begin()),
                                          *(--points.end()));
      ch_object = make_object(tri);
      return;
  }

  // at least 4 points 
  typename Traits::Collinear_3 collinear = traits.collinear_3_object();
  
  P3_iterator point1_it = points.begin();
  P3_iterator point2_it = points.begin();
  point2_it++;
  P3_iterator point3_it = points.end();
  point3_it--;

  // find three that are not collinear
  while (point2_it != points.end() && 
         collinear(*point1_it,*point2_it,*point3_it))
    point2_it++;
  

  // all are collinear, so the answer is a segment
  if (point2_it == points.end())
  {
     typedef typename Traits::Less_distance_to_point_3      Less_dist; 

     Less_dist less_dist = traits.less_distance_to_point_3_object();
     P3_iterator_pair endpoints = 
      min_max_element(points.begin(), points.end(), 
                      boost::bind(less_dist, *points.begin(), _1, _2), 
                      boost::bind(less_dist, *points.begin(), _1, _2));

     typename Traits::Construct_segment_3 construct_segment =
            traits.construct_segment_3_object();
     typedef typename Traits::Segment_3                 Segment_3;  

     Segment_3 seg = construct_segment(*endpoints.first, *endpoints.second);
     ch_object = make_object(seg);
     return;
  }

  // result will be a polyhedron
  typename internal::Convex_hull_3::Default_polyhedron_for_Chull_3<Traits>::type P;

  P3_iterator minx, maxx, miny, it;
  minx = maxx = miny = it = points.begin();
  ++it;
  for(; it != points.end(); ++it){
    if(it->x() < minx->x()) minx = it;
    if(it->x() > maxx->x()) maxx = it;
    if(it->y() < miny->y()) miny = it;
  }
  if(! collinear(*minx, *maxx, *miny) ){  
    internal::Convex_hull_3::ch_quickhull_polyhedron_3(points, minx, maxx, miny, P, traits);
  } else {
    internal::Convex_hull_3::ch_quickhull_polyhedron_3(points, point1_it, point2_it, point3_it, P, traits);
  }
  CGAL_assertion(P.size_of_vertices()>=3);
  if (boost::next(P.vertices_begin(),3) == P.vertices_end()){
    typedef typename Traits::Triangle_3                Triangle_3;
    typename Traits::Construct_triangle_3 construct_triangle =
           traits.construct_triangle_3_object();
    Triangle_3 tri = construct_triangle(P.halfedges_begin()->vertex()->point(), 
                                        P.halfedges_begin()->next()->vertex()->point(),
                                        P.halfedges_begin()->opposite()->vertex()->point());
    ch_object = make_object(tri);
  }
  else
    ch_object = make_object(P);
}


template <class InputIterator>
void convex_hull_3(InputIterator first, InputIterator beyond, 
		   Object& ch_object)
{
   typedef typename std::iterator_traits<InputIterator>::value_type Point_3;
   typedef typename internal::Convex_hull_3::Default_traits_for_Chull_3<Point_3>::type Traits;
   convex_hull_3(first, beyond, ch_object, Traits());
}



template <class InputIterator, class Polyhedron_3, class Traits>
void convex_hull_3(InputIterator first, InputIterator beyond,
                   Polyhedron_3& polyhedron,  const Traits& traits)
{
  typedef typename Traits::Point_3                Point_3;  
  typedef std::list<Point_3>                      Point_3_list;
  typedef typename Point_3_list::iterator         P3_iterator;

  Point_3_list points(first, beyond);
  CGAL_ch_precondition(points.size() > 3);

  // at least 4 points 
  typename Traits::Collinear_3 collinear = traits.collinear_3_object();
  typename Traits::Equal_3 equal = traits.equal_3_object();

  P3_iterator point1_it = points.begin();
  P3_iterator point2_it = points.begin();
  point2_it++;

  // find three that are not collinear
  while (point2_it != points.end() && equal(*point1_it,*point2_it))
    ++point2_it;

  CGAL_ch_precondition_msg(point2_it != points.end(), 
        "All points are equal; cannot construct polyhedron.");
  
  P3_iterator point3_it = point2_it;
  ++point3_it;
  
  CGAL_ch_precondition_msg(point3_it != points.end(), 
        "Only two points with different coordinates; cannot construct polyhedron.");
  
  while (point3_it != points.end() && collinear(*point1_it,*point2_it,*point3_it))
    ++point3_it;
  
  CGAL_ch_precondition_msg(point3_it != points.end(), 
        "All points are collinear; cannot construct polyhedron.");
  
  polyhedron.clear();
  // result will be a polyhedron
  internal::Convex_hull_3::ch_quickhull_polyhedron_3(points, point1_it, point2_it, point3_it,
                                                     polyhedron, traits);

}


template <class InputIterator, class Polyhedron_3>
void convex_hull_3(InputIterator first, InputIterator beyond,
                   Polyhedron_3& polyhedron)
{
   typedef typename std::iterator_traits<InputIterator>::value_type Point_3;
   typedef typename internal::Convex_hull_3::Default_traits_for_Chull_3<Point_3>::type Traits;
   convex_hull_3(first, beyond, polyhedron, Traits());
}

} // namespace CGAL

#endif // CGAL_CONVEX_HULL_3_H