This file is indexed.

/usr/include/CGAL/Alpha_shape_2.h is in libcgal-dev 4.11-2build1.

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
 929
 930
 931
 932
 933
 934
 935
 936
 937
 938
 939
 940
 941
 942
 943
 944
 945
 946
 947
 948
 949
 950
 951
 952
 953
 954
 955
 956
 957
 958
 959
 960
 961
 962
 963
 964
 965
 966
 967
 968
 969
 970
 971
 972
 973
 974
 975
 976
 977
 978
 979
 980
 981
 982
 983
 984
 985
 986
 987
 988
 989
 990
 991
 992
 993
 994
 995
 996
 997
 998
 999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
// Copyright (c) 1997, 2012  INRIA Sophia-Antipolis (France).
// 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)     : Tran Kai Frank DA
//                 Andreas Fabri <Andreas.Fabri@geometryfactory.com>

#ifndef CGAL_ALPHA_SHAPE_2_H
#define CGAL_ALPHA_SHAPE_2_H

#include <CGAL/license/Alpha_shapes_2.h>


#include <CGAL/basic.h>

#include <list>
#include <set>
#include <map>
#include <vector>
#include <algorithm>
#include <utility>
#include <iostream>

#include <CGAL/utility.h>
#include <CGAL/Unique_hash_map.h>
#include <CGAL/Triangulation_vertex_base_2.h>
#include <CGAL/Triangulation_face_base_2.h>
#include <CGAL/Alpha_shape_vertex_base_2.h>
#include <CGAL/Alpha_shape_face_base_2.h>
#include <CGAL/internal/Lazy_alpha_nt_2.h>



namespace CGAL {

template < class Dt,class ExactAlphaComparisonTag = Tag_false>
class Alpha_shape_2 : public Dt 
{
  // DEFINITION The class Alpha_shape_2<Dt> represents the family
  // of alpha-shapes of points in a plane for all positive alpha. It
  // maintains the underlying Delaunay triangulation which represents
  // connectivity and order among its faces. Each k-dimensional face of the
  // Delaunay triangulation is associated with an interval that specifies
  // for which values of alpha the face belongs to the alpha-shape (sorted
  // linear arrays resp. multimaps or interval trees). There are links
  // between the intervals and the k-dimensional faces of the Delaunay
  // triangulation (multimaps resp. std::hashtables).
  //

  //------------------------- TYPES ------------------------------------

public:
  typedef Dt Triangulation;
  typedef typename Dt::Geom_traits Gt;
  typedef typename Dt::Triangulation_data_structure Tds;

  typedef typename internal::Alpha_nt_selector_2<
    Gt, ExactAlphaComparisonTag, typename Dt::Weighted_tag>::Type_of_alpha  Type_of_alpha;
  typedef typename internal::Alpha_nt_selector_2<
    Gt, ExactAlphaComparisonTag, typename Dt::Weighted_tag>::Compute_squared_radius_2 Compute_squared_radius_2;
  typedef typename internal::Alpha_nt_selector_2<
    Gt, ExactAlphaComparisonTag, typename Dt::Weighted_tag>::Side_of_bounded_circle_2 Side_of_bounded_circle_2;

  typedef Type_of_alpha               NT;
  typedef Type_of_alpha               FT;

  //check simplices are correctly instantiated
  CGAL_static_assertion( (boost::is_same<NT, typename Dt::Face::NT>::value) );
  CGAL_static_assertion( (boost::is_same<NT, typename Dt::Vertex::NT>::value) );

  typedef typename Dt::Point Point;

  typedef typename Gt::Point_2 Point_2;
  typedef typename Gt::Segment_2 Segment;
  typedef typename Gt::Line_2 Line;

  typedef typename Dt::Face_handle Face_handle;
  typedef typename Dt::Vertex_handle Vertex_handle;
  typedef typename Dt::Edge Edge;

  typedef typename Dt::Face_circulator Face_circulator;
  typedef typename Dt::Edge_circulator Edge_circulator;
  typedef typename Dt::Vertex_circulator Vertex_circulator;

  typedef typename Dt::Finite_faces_iterator Finite_faces_iterator;
  typedef typename Dt::Edge_iterator Edge_iterator;
  typedef typename Dt::Finite_vertices_iterator Finite_vertices_iterator;

  typedef typename Dt::Locate_type Locate_type;
  typedef typename Dt::size_type size_type;

  using Dt::finite_vertices_begin;
  using Dt::finite_vertices_end;
  using Dt::faces_begin;
  using Dt::faces_end;
  using Dt::edges_begin;
  using Dt::edges_end;
  using Dt::number_of_vertices;
  using Dt::cw;
  using Dt::ccw;
  using Dt::VERTEX;
  using Dt::EDGE;
  using Dt::FACE;
  using Dt::OUTSIDE_CONVEX_HULL;
  using Dt::OUTSIDE_AFFINE_HULL;
  using Dt::dimension;
  using Dt::is_infinite;

  // for backward compatibility
  typedef Finite_vertices_iterator Vertex_iterator;
  typedef Finite_faces_iterator Face_iterator;

private:

  typedef std::multimap< Type_of_alpha, Face_handle >  Interval_face_map;
  typedef typename Interval_face_map::value_type    Interval_face;

  typedef typename Tds::Face::Interval_3            Interval3;
  
  typedef std::multimap< Interval3, Edge >          Interval_edge_map;
  typedef typename Interval_edge_map::value_type    Interval_edge;

  typedef std::pair< Type_of_alpha, Type_of_alpha > Interval2;
  typedef std::multimap< Interval2, Vertex_handle > Interval_vertex_map;
  typedef typename Interval_vertex_map::value_type  Interval_vertex;

  typedef Face_handle const const_void;
  typedef std::pair<const_void, int> const_Edge;

  typedef std::vector< Type_of_alpha > Alpha_spectrum;
  
  typedef std::vector< Segment > Vect_seg;

  typedef Unique_hash_map< Face_handle, bool > Marked_face_set;

public:

  typedef typename std::list< Vertex_handle >::iterator 
  Alpha_shape_vertices_iterator;
  typedef typename std::list< Edge >::iterator Alpha_shape_edges_iterator;

  typedef typename Alpha_spectrum::const_iterator Alpha_iterator;
  // An iterator that allow to traverse the sorted sequence of
  // different alpha-values. The iterator is bidirectional and
  // non-mutable. Its value-type is Type_of_alpha

  enum Classification_type {EXTERIOR, SINGULAR, REGULAR, INTERIOR};
  // Distinguishes the different cases for classifying a
  // k-dimensional face of the underlying Delaunay triangulation of
  // the alpha-shape.
  // 
  // `EXTERIOR' if the face does not belong to the alpha-complex.
  // 
  // `SINGULAR' if the face belongs to the boundary of the
  // alpha-shape, but is not incident to any higher-dimensional
  // face of the alpha-complex
  // 
  // `REGULAR' if face belongs to the boundary of the alpha-shape
  // and is incident to a higher-dimensional face of the
  // alpha-complex
  // 
  // `INTERIOR' if the face belongs to the alpha-complex, but does
  // not belong to the boundary of the alpha-shape

  enum Mode {GENERAL, REGULARIZED};
  // In general, an alpha shape is a non-connected, mixed-dimension
  // polygon. Its regularized version is formed by the set of
  // regular edges and their vertices

  //------------------------ private VARIABLES -------------------------

private:

  // only finite edges and faces are inserted into the maps 
  Interval_face_map _interval_face_map;
  Interval_edge_map _interval_edge_map;
  Interval_vertex_map _interval_vertex_map;

  Alpha_spectrum _alpha_spectrum;
 
  Type_of_alpha _alpha;
  Mode _mode;
  // should be constants
  Type_of_alpha Infinity;
  Type_of_alpha UNDEFINED;
  
  mutable std::list< Vertex_handle > Alpha_shape_vertices_list;
  mutable std::list< Edge > Alpha_shape_edges_list;

  mutable bool use_vertex_cache;
  mutable bool use_edge_cache;
public:

  //------------------------- CONSTRUCTORS ------------------------------
 
  // Introduces an empty alpha-shape `A' for a positive
  // alpha-value `alpha'. Precondition: `alpha' >= 0.
  Alpha_shape_2(Type_of_alpha alpha = Type_of_alpha(0), 
		Mode m = GENERAL)
    : _alpha(alpha), _mode(m), Infinity(-1), UNDEFINED(-2),
      use_vertex_cache(false), use_edge_cache(false)
    {}
 
  // Introduces an alpha-shape `A' for a positive alpha-value
  // `alpha' that is initialized with the points in the range
  // from first to last
  template <class InputIterator>
  Alpha_shape_2(const InputIterator& first,  
		const InputIterator& last,  
		const Type_of_alpha& alpha = Type_of_alpha(0),
		Mode m = GENERAL)
    : _alpha(alpha), _mode(m), Infinity(-1), UNDEFINED(-2) ,
      use_vertex_cache(false), use_edge_cache(false)
    {
      Dt::insert(first, last);
      if (dimension() == 2)
	{
	  // Compute the associated _interval_face_map
	  initialize_interval_face_map();

	  // Compute the associated _interval_edge_map
	  initialize_interval_edge_map();
   
	  // Compute the associated _interval_vertex_map
	  initialize_interval_vertex_map();

	  // merge the two maps
	  initialize_alpha_spectrum();
	}
    }

  // Introduces an alpha-shape `A' for a positive alpha-value
  // `alpha' that is initialized with the triangulation 
  Alpha_shape_2(Dt& dt,
		const Type_of_alpha& alpha = Type_of_alpha(0),
		Mode m = GENERAL)
    : _alpha(alpha), _mode(m), Infinity(-1), UNDEFINED(-2) ,
      use_vertex_cache(false), use_edge_cache(false)
    {
      Dt::swap(dt);
      if (dimension() == 2)
	{
	  // Compute the associated _interval_face_map
	  initialize_interval_face_map();

	  // Compute the associated _interval_edge_map
	  initialize_interval_edge_map();
   
	  // Compute the associated _interval_vertex_map
	  initialize_interval_vertex_map();

	  // merge the two maps
	  initialize_alpha_spectrum();
	}
    }
 
public:

  //----------- OUTPUT POINTS CONNECTED BY PAIRS ----------------------

  std::list<Point_2> Output();
 
  std::ostream& op_ostream(std::ostream& os) const;

 
  //----------------------- OPERATIONS ---------------------------------

  // Introduces an alpha-shape `A' for a positive alpha-value
  // `alpha' that is initialized with the points in the range
  // from first to last

  template < class InputIterator >  
  std::ptrdiff_t make_alpha_shape(const InputIterator& first,  
		       const InputIterator& last) 
    {

      clear();

      size_type n =  Dt::insert(first, last);
 
      if (dimension() == 2)
	{   
	  // Compute the associated _interval_face_map
	  initialize_interval_face_map();

	  // Compute the associated _interval_edge_map
	  initialize_interval_edge_map();

	  // Compute the associated _interval_vertex_map
	  initialize_interval_vertex_map();
  
	  // merge the two maps
	  initialize_alpha_spectrum();
	}
      return n;
    }

private :

  //--------------------- INITIALIZATION OF PRIVATE MEMBERS -----------

  void initialize_interval_face_map();

  void initialize_interval_edge_map();

  void initialize_interval_vertex_map();

  void initialize_alpha_spectrum();

  //---------------------------------------------------------------------

public:

  void clear() 
    {
      // clears the structure
      Dt::clear();

      _interval_face_map.clear();
      _interval_edge_map.clear();
      _interval_vertex_map.clear();
    
      _alpha_spectrum.clear();

      Alpha_shape_vertices_list.clear();
      Alpha_shape_edges_list.clear();
    
      set_alpha(Type_of_alpha(0)); 
      set_mode(GENERAL);

    }

  //----------------------- PRIVATE MEMBERS --------------------------

private:
   
  struct Less 
  {
    bool operator()(const Interval_edge& ie, 
		    const Type_of_alpha& alpha) 
      {
	return ie.first.first < alpha; 
      }

    bool operator()( const Type_of_alpha& alpha, 
		     const Interval_edge& ie) 
      {
	return alpha < ie.first.first; 
      }

    // Needed for STL implementations of upper_bound which in debug mode 
    // check sortedness of range
    bool operator()(const Interval_edge& ie, 
		    const Interval_edge& ie2) const
      {
	return ie < ie2;
      }
  };

  
  //----------------------- ACCESS TO PRIVATE MEMBERS -----------------

private:
   
  
  Type_of_alpha find_interval(const Face_handle& f) const 
    {
      return f->get_alpha();
      // return the value Alpha f the face f
    }
 
  Interval3 find_interval(const_Edge e)  const 
    { // corriger le parametrage
      return (e.first)->get_ranges(e.second);
      // return the Interval3 for the edge n
    }


  //---------------------------------------------------------------------

public:

  Type_of_alpha set_alpha(const Type_of_alpha& alpha) 
    {
      // Sets the alpha-value to `alpha'. Precondition: `alpha' >= 0.
      // Returns the previous alpha
      Type_of_alpha previous_alpha = _alpha;
      _alpha = alpha;
      use_vertex_cache = false;
      use_edge_cache = false;
      return previous_alpha;
    }

  const Type_of_alpha&  get_alpha() const 
    {
      // Returns the current alpha-value.
      return _alpha;
    }
  
  const Type_of_alpha&  get_nth_alpha(size_type n) const 
    {
      // Returns the n-th alpha-value.
      // n < size()
      if (! _alpha_spectrum.empty())
	return _alpha_spectrum[n];
      else
	return UNDEFINED;
    }
  
  size_type number_of_alphas() const 
    {
      // Returns the number of not necessary different alpha-values
      return _alpha_spectrum.size();
    }

  //---------------------------------------------------------------------

private:

  // the dynamic version is not yet implemented
  // desactivate the triangulation member functions
  Vertex_handle insert(const Point& p);
  // Inserts point `p' in the alpha shape and returns the
  // corresponding vertex of the underlying Delaunay triangulation.
  // If point `p' coincides with an already existing vertex, this
  // vertex is returned and the alpha shape remains unchanged.
  // Otherwise, the vertex is inserted in the underlying Delaunay
  // triangulation and the associated intervals are updated.

  void remove(Vertex_handle v);
  // Removes the vertex from the underlying Delaunay triangulation.
  // The created hole is retriangulated and the associated intervals
  // are updated.

  //---------------------------------------------------------------------

public:

  Mode set_mode(Mode mode = GENERAL ) 
    {
      // Sets `A' to its general or regularized version. Returns the
      // previous mode.

      Mode previous_mode = _mode;
      _mode = mode;
      return previous_mode;
    }

  Mode get_mode() const 
    {
      // Returns whether `A' is general or regularized.
      return _mode;
    }

  //---------------------------------------------------------------------

private:


void
update_alpha_shape_vertex_list()const;

  //---------------------------------------------------------------------

  void
  update_alpha_shape_edges_list() const;
  

  //---------------------------------------------------------------------

public:

  Alpha_shape_vertices_iterator alpha_shape_vertices_begin() const
    { 
    if(!use_vertex_cache){
      update_alpha_shape_vertex_list();
    }
      return Alpha_shape_vertices_list.begin();
    }



  // for backward compatibility
  Alpha_shape_vertices_iterator Alpha_shape_vertices_begin()
    {
      return alpha_shape_vertices_begin();
    }
  //---------------------------------------------------------------------

  Alpha_shape_vertices_iterator alpha_shape_vertices_end() const
    {
      return Alpha_shape_vertices_list.end();
    }

  Alpha_shape_vertices_iterator Alpha_shape_vertices_end() const
    {
      return Alpha_shape_vertices_list.end();
    }
  //---------------------------------------------------------------------

  Alpha_shape_edges_iterator alpha_shape_edges_begin() const
    {
      if(!use_edge_cache){
      update_alpha_shape_edges_list();
    }
      return Alpha_shape_edges_list.begin();
    }

  Alpha_shape_edges_iterator Alpha_shape_edges_begin() const
  {
    return alpha_shape_edges_begin();
  }
  //---------------------------------------------------------------------

  Alpha_shape_edges_iterator alpha_shape_edges_end() const
    {
      return Alpha_shape_edges_list.end();
    }

  Alpha_shape_edges_iterator Alpha_shape_edges_end() const
    {
      return Alpha_shape_edges_list.end();
    }
public: 
  
  // Traversal of the alpha-Values
  // 
  // The alpha shape class defines an iterator that allows to
  // visit the sorted sequence of alpha-values. This iterator is
  // non-mutable and bidirectional. Its value type is Type_of_alpha.

  Alpha_iterator alpha_begin() const 
    {
      // Returns an iterator that allows to traverse the sorted sequence
      // of alpha-values of `A'.
      return _alpha_spectrum.begin(); 
    }

  Alpha_iterator alpha_end() const 
    {
      // Returns the corresponding past-the-end iterator.
      return _alpha_spectrum.end(); 
    }
  
  Alpha_iterator alpha_find(const Type_of_alpha& alpha) const 
    {
      // Returns an iterator pointing to an element with alpha-value
      // `alpha', or the corresponding past-the-end iterator if such an
      // element is not found.
      return std::find(_alpha_spectrum.begin(),
		       _alpha_spectrum.end(),
		       alpha);
    }

  Alpha_iterator alpha_lower_bound(const Type_of_alpha& alpha) const 
    {
      // Returns an iterator pointing to the first element with
      // alpha-value not less than `alpha'.
      return std::lower_bound(_alpha_spectrum.begin(),
			      _alpha_spectrum.end(),
			      alpha);
    }

  Alpha_iterator alpha_upper_bound(const Type_of_alpha& alpha) const 
    {
      // Returns an iterator pointing to the first element with
      // alpha-value greater than `alpha'.
      return std::upper_bound(_alpha_spectrum.begin(),
			      _alpha_spectrum.end(),
			      alpha);
    }

  //--------------------- PREDICATES -----------------------------------

  // the classification predicates take 
  //      amortized const time if STL_STD::HASH_TABLES
  //      O(log #alpha_shape ) otherwise

  Classification_type  classify(const Point& p ) const 
    {
      return classify( p, get_alpha());
    }

  
  Classification_type  classify(const Point& p,   
				const Type_of_alpha& alpha) const 
    {
      // Classifies a point `p' with respect to `A'.
      Locate_type type;
      int i;
      Face_handle pFace = this->locate(p, type, i);
      switch (type) 
	{
	case VERTEX            : return classify(pFace->vertex(i), alpha);
	case EDGE              : return classify(pFace, i, alpha);
	case FACE              : return classify(pFace, alpha);
	case OUTSIDE_CONVEX_HULL : 
	case OUTSIDE_AFFINE_HULL : return EXTERIOR;
	default                : return EXTERIOR;
	}
    }

  //---------------------------------------------------------------------

  Classification_type  classify(const Face_handle& f) const 
    {
      // Classifies the face `f' of the underlying Delaunay
      // triangulation with respect to `A'.
      return classify(f, get_alpha());
    }
  
  Classification_type  classify(const Face_handle& f, 
				const Type_of_alpha& alpha) const 
    {
      // Classifies the face `f' of the underlying Delaunay
      // triangulation with respect to `A'.
      // we consider close circles :
      // f->radius < alpha <=> f exterior
      // problem the operator [] is non-const

      if (is_infinite(f)) return EXTERIOR;
    
      // the version that computes the squared radius seems to be 
      // much faster
    
      return (find_interval(f) <= alpha) ? 
	INTERIOR :
	EXTERIOR;

    }

  //---------------------------------------------------------------------
  
  Classification_type  classify(const Edge& edge) const
    {  
      return classify(edge.first, edge.second, get_alpha());
    }

  Classification_type  classify(const Face_handle& f, 
				int i) const 
    {  
      return classify(f, i, get_alpha());
    }

  Classification_type  classify(const Edge& edge,
				const Type_of_alpha& alpha) const 
    {  
      return classify(edge.first, edge.second, alpha);
    }

  Classification_type  classify(const Face_handle& f, 
				int i,
				const Type_of_alpha& alpha) const;

  
  //---------------------------------------------------------------------

  Classification_type  classify(const Vertex_handle& v) const 
    {
      return classify(v, get_alpha());
    }

  Classification_type  classify(const Vertex_handle& v,
				const Type_of_alpha& alpha) const;

  //--------------------- NB COMPONENTS ---------------------------------
  int
  number_solid_components() const
    {
      return number_of_solid_components(get_alpha());
    }

  int
  number_of_solid_components() const
    {
      return number_of_solid_components(get_alpha());
    }

  size_type 
  number_solid_components(const Type_of_alpha& /* alpha */) const
    {
      return number_of_solid_components(get_alpha());
    }

  size_type
  number_of_solid_components(const Type_of_alpha& alpha) const;

private:

  void traverse(const Face_handle& pFace,
		Marked_face_set& marked_face_set, 
		const Type_of_alpha alpha) const;

//  class Line_face_circulator;

  //----------------------------------------------------------------------

public:

  Alpha_iterator find_optimal_alpha(size_type nb_components);  	

private:

  Type_of_alpha find_alpha_solid() const;

  //---------------------- PREDICATES ------------------------------------

private:

  bool is_attached(const Face_handle& f, int i) const
  {
    Bounded_side b = Side_of_bounded_circle_2()(*this)(f->vertex(cw(i))->point(),
                                                       f->vertex(ccw(i))->point(),
                                                       f->vertex(i)->point());

    return (b == ON_BOUNDED_SIDE) ? true : false;
  }

  //-------------------- GEOMETRIC PRIMITIVES ----------------------------

  Type_of_alpha squared_radius(const Face_handle& f) const
  {
    return Compute_squared_radius_2()(*this)(f->vertex(0)->point(),
                                             f->vertex(1)->point(),
                                             f->vertex(2)->point());
  }

  Type_of_alpha squared_radius(const Face_handle& f, int i) const
  {
    return Compute_squared_radius_2()(*this)(f->vertex(ccw(i))->point(),
                                             f->vertex(cw(i))->point());
  }

  //---------------------------------------------------------------------

private:
  // prevent default copy constructor and default assigment
  
  Alpha_shape_2(const Alpha_shape_2& A);

  Alpha_shape_2& operator=(const Alpha_shape_2& A);


public:
  // to debug
  void print_edge_map();

};


//---------------------------------------------------------------------
//----------------------- MEMBER FUNCTIONS -----------------------------
//---------------------------------------------------------------------


template < class Dt, class EACT >
void 
Alpha_shape_2<Dt,EACT>::initialize_interval_face_map() 
{
  Type_of_alpha alpha_f;

  // only finite faces
  for(Finite_faces_iterator face_it = faces_begin(); face_it != faces_end(); ++face_it)
    {
      alpha_f = squared_radius(face_it);
      _interval_face_map.insert(Interval_face(alpha_f, face_it));

      // cross references
      face_it->set_alpha(alpha_f);
    } 
}

//-------------------------------------------------------------------------

template < class Dt, class EACT >
void 
Alpha_shape_2<Dt,EACT>::initialize_interval_edge_map() 
{  
  Edge_iterator edge_it;
  Edge edge;

  // only finite faces
  for( edge_it = edges_begin(); 
       edge_it != edges_end(); 
       ++edge_it) 
    { 
      Interval3 interval;
      edge = (*edge_it);

      Face_handle pFace = edge.first;
      int i = edge.second;
      Face_handle pNeighbor = pFace->neighbor(i);
      int Neigh_i = pNeighbor->index(pFace);
      
      // not on the convex hull
      
      if(!is_infinite(pFace) && !is_infinite(pNeighbor)) 
	{ 
	  Type_of_alpha squared_radius_Face = 
	    find_interval(pFace);
	  Type_of_alpha squared_radius_Neighbor = 
	    find_interval(pNeighbor);
	  if (squared_radius_Neighbor < squared_radius_Face) 
	    {	    
	      edge =  Edge(pNeighbor, Neigh_i);
	      Type_of_alpha coord_tmp = squared_radius_Face;
	      squared_radius_Face = squared_radius_Neighbor;
	      squared_radius_Neighbor = coord_tmp;
	    }
	  interval = (is_attached(pFace, i) || 
		      is_attached(pNeighbor, Neigh_i)) ?
	    make_triple(UNDEFINED,
			squared_radius_Face, 
			squared_radius_Neighbor):
	    make_triple(squared_radius(pFace, i),
			squared_radius_Face, 
			squared_radius_Neighbor);
	}
      else 
	{ // on the convex hull

	  if(is_infinite(pFace)) 
	    {
	      if (!is_infinite(pNeighbor)) 
		{
		  interval =  (is_attached(pNeighbor,
					   Neigh_i)) ?
		    make_triple(UNDEFINED,
				find_interval(pNeighbor), 
				Infinity):
		    make_triple(squared_radius(pNeighbor, 
					       Neigh_i), 
				find_interval(pNeighbor), 
				Infinity); 
		  edge =  Edge(pNeighbor, Neigh_i);
		}
	      else 
		{
		  // both faces are infinite by definition unattached
		  // the edge is finite by construction
		  CGAL_triangulation_precondition((is_infinite(pNeighbor) 
						   && is_infinite(pFace)));
		  interval = make_triple(squared_radius(pFace, i), 
					 Infinity,
					 Infinity);
		}
	    }
	  else 
	    { // is_infinite(pNeighbor)
	   
	      CGAL_triangulation_precondition((is_infinite(pNeighbor) 
					       && !is_infinite(pFace)));
	      if (is_attached(pFace, i))
		interval = make_triple(UNDEFINED,
				       find_interval(pFace),
				       Infinity);
	      else
		interval = make_triple(squared_radius(pFace, i),
				       find_interval(pFace),
				       Infinity);

	    }
	}
      
      _interval_edge_map.insert(Interval_edge(interval, edge));
      
      // cross-links
      (edge.first)->set_ranges(edge.second,interval);
      // MY : to fix a bug I store the interval of the edge in both faces 
      Face_handle neighbor = (edge.first)->neighbor(edge.second);
      int ni = neighbor->index(edge.first);
      neighbor->set_ranges( ni, interval);

    }

  // Remark:
  // The interval_edge_map will be sorted as follows
  // first the attached edges on the convex hull
  // second                   not on the convex hull
  // third the un-attached edges on the convex hull
  // finally                     not on the convex hull
  // 
  // if we are in regularized mode we should sort differently
  // by the second third first Key
}

//-------------------------------------------------------------------------

template < class Dt, class EACT >
void 
Alpha_shape_2<Dt,EACT>::initialize_interval_vertex_map() 
{
  Type_of_alpha alpha_mid_v;
  Type_of_alpha alpha_max_v;
  Type_of_alpha alpha_f;

  Finite_vertices_iterator vertex_it;

  for( vertex_it = finite_vertices_begin(); 
       vertex_it != finite_vertices_end(); 
       ++vertex_it) 
    {
      Vertex_handle v = vertex_it;
      Face_handle f;

      alpha_max_v = Type_of_alpha(0);    
      alpha_mid_v = (!_interval_face_map.empty() ?
		     (--_interval_face_map.end())->first :
		     Type_of_alpha(0));

      //----------------- examine incident edges --------------------------
      
// 	// if we used Edelsbrunner and Muecke's definition
// 	// singular means not incident to any higher-dimensional face
// 	// regular means incident to a higher-dimensional face
// 	Edge_circulator edge_circ = this->incident_edges(v),
// 	edge_done(edge_circ);

// 	do 
      
// 	{ 
// 	f = (*edge_circ).first;
// 	int i = (*edge_circ).second;

// 	if (is_infinite(f, i))
      
// 	{
// 	alpha_max_v = Infinity;
// 	}
// 	else
      
// 	{
// 	Interval3 interval3 = find_interval(const_Edge(f, i));
				
	
// 	alpha_mid_v = (interval3.first != UNDEFINED) ?
// 	(CGAL::min)(alpha_mid_v, interval3.first): 
// 	(CGAL::min)(alpha_mid_v, interval3.second); 
			
// 	if (alpha_max_v != Infinity)
      
// 	{
// 	alpha_max_v = (interval3.third != Infinity) ?
// 	(CGAL::max)(alpha_max_v, interval3.third):
// 	Infinity;
// 	}
// 	}
// 	}
// 	while(++edge_circ != edge_done);
     

      //-------------- examine incident faces --------------------------
      
      // we use a different definition than Edelsbrunner and Muecke
      // singular means not incident to any 2-dimensional face
      // regular means incident to a 2-dimensional face
     
      Face_circulator face_circ = this->incident_faces(v),
	done = face_circ;
   
      if (!face_circ.is_empty()) 
	{
	  do 
	    {
	      f = face_circ;
	      if (is_infinite(f)) 
		{
		  alpha_max_v = Infinity;
		  // continue;
		}
	      else 
		{
		  alpha_f = find_interval(f);
		  // if we define singular as not incident to a 2-dimensional
		  // face
		  alpha_mid_v = (CGAL::min)(alpha_mid_v, alpha_f);
		    
		  if (alpha_max_v != Infinity)
		    alpha_max_v = (CGAL::max)(alpha_max_v, alpha_f);
			    
		}
	    }
	  while(++face_circ != done);
	}
	
 
      Interval2 interval = std::make_pair(alpha_mid_v, alpha_max_v);
      _interval_vertex_map.insert(Interval_vertex(interval, vertex_it));

      // cross references
      vertex_it->set_range(interval);
    }
}

//-------------------------------------------------------------------------

template < class Dt, class EACT >
void 
Alpha_shape_2<Dt,EACT>::initialize_alpha_spectrum() 
{

  // skip the attached edges 
  // <=> _interval_edge_map.first.first == UNDEFINED
  typename Interval_edge_map::iterator 
    edge_it = std::upper_bound(_interval_edge_map.begin(),
			       _interval_edge_map.end(),
			       UNDEFINED,
			       Less());

  // merge the maps which is sorted and contains the alpha-values
  // of the unattached edges and the triangles.
  // eliminate duplicate values due to for example attached edges
  // merge and copy from STL since assignment should be function object
	
  typename Interval_face_map::iterator
    face_it = _interval_face_map.begin();

  _alpha_spectrum.reserve(_interval_face_map.size() +
			  _interval_edge_map.size()/ 2 );
  // should be only the number of unattached edges
  // size_type nb_unattached_edges;
  // distance(edge_it, _interval_edge_map.end(), nb_unattached_edges);
  // however the distance function is expensive

  while (edge_it != _interval_edge_map.end() ||
	 face_it != _interval_face_map.end()) 
    {
      if (face_it != _interval_face_map.end() &&
	  (edge_it == _interval_edge_map.end() ||
	   ((*face_it).first < (*edge_it).first.first))) 
	{
	  if (((_alpha_spectrum.empty() || 
		_alpha_spectrum.back() < (*face_it).first)) && 
	      ((*face_it).first > Type_of_alpha(0)))
	    _alpha_spectrum.push_back((*face_it).first);
	  face_it++;
	}
      else
	{
	  if (((_alpha_spectrum.empty() || 
		_alpha_spectrum.back() < (*edge_it).first.first)) &&
	      (((*edge_it).first.first) > Type_of_alpha(0)))
	    _alpha_spectrum.push_back((*edge_it).first.first);
	  edge_it++;
	}
    }
    
  while (edge_it != _interval_edge_map.end()) 
    {
      if (((_alpha_spectrum.empty() || 
	    _alpha_spectrum.back() < (*edge_it).first.first))&&
	  (((*edge_it).first.first) > Type_of_alpha(0)))
	_alpha_spectrum.push_back((*edge_it).first.first);
      edge_it++;
    }

  while (face_it != _interval_face_map.end()) 
    { 
      if (((_alpha_spectrum.empty() || 
	    _alpha_spectrum.back() < (*face_it).first))&&
	  ((*face_it).first > Type_of_alpha(0)))
	_alpha_spectrum.push_back((*face_it).first);
      face_it++;
    }

}


//-------------------------------------------------------------------------



template < class Dt, class EACT >
void
Alpha_shape_2<Dt,EACT>::update_alpha_shape_vertex_list()const {
	//typedef typename Alpha_shape_2<Dt,EACT>::Interval_vertex_map 
	//                                              Interval_vertex_map;
	typename Interval_vertex_map::const_iterator vertex_alpha_it;

	//const typename Alpha_shape_2<Dt,EACT>::Interval2* pInterval2;
	const Interval2* pInterval2;
	Vertex_handle v;
	Alpha_shape_vertices_list.clear();
	// write the regular vertices

	for (vertex_alpha_it = _interval_vertex_map.begin(); 
		vertex_alpha_it != _interval_vertex_map.end() &&
		(*vertex_alpha_it).first.first <= get_alpha();
		++vertex_alpha_it) 
		{
		pInterval2 = &(*vertex_alpha_it).first;

		if((pInterval2->second > get_alpha()
		|| pInterval2->second == Infinity)) 
		{
		// alpha must be larger than the min boundary
		// and alpha is smaller than the upper boundary
		// which might be infinity 
		// write the vertex
		v = (*vertex_alpha_it).second;
		CGAL_triangulation_assertion((classify(v) == REGULAR));
		Alpha_shape_vertices_list.push_back(v);
		}
		}
	 
	if (get_mode() == Alpha_shape_2<Dt,EACT>::GENERAL) 
		{
		// write the singular vertices
		for (; 
		vertex_alpha_it != _interval_vertex_map.end();
		++vertex_alpha_it) 
		{
		v = (*vertex_alpha_it).second;
		CGAL_triangulation_assertion((classify(v) == SINGULAR));

		Alpha_shape_vertices_list.push_back(v);
		}
		}
	use_vertex_cache = true;
  }

//-------------------------------------------------------------------------

template < class Dt, class EACT >
void
Alpha_shape_2<Dt,EACT>::update_alpha_shape_edges_list() const 
{

  // Writes the edges of the alpha shape `A' for the current $\alpha$-value
  // to the container where 'out' refers to. Returns an output iterator 
  // which is the end of the constructed range.
  //typedef  typename Alpha_shape_2<Dt,EACT>::Interval_edge_map Interval_edge_map;
  typename Interval_edge_map::const_iterator edge_alpha_it;

  //const typename Alpha_shape_2<Dt,EACT>::Interval3* pInterval;
  const Interval3* pInterval;
  Alpha_shape_edges_list.clear();
  if (get_mode() == REGULARIZED) 
    {
      // it is much faster looking at the sorted intervals 
      // than looking at all sorted faces
      // alpha must be larger than the mid boundary
      // and alpha is smaller than the upper boundary
      for (edge_alpha_it = _interval_edge_map.begin(); 
	   edge_alpha_it != _interval_edge_map.end() &&
	     (*edge_alpha_it).first.first <= get_alpha();
	   ++edge_alpha_it) 
	{
	  pInterval = &(*edge_alpha_it).first;
      
	  CGAL_triangulation_assertion(pInterval->second != Infinity);
	  // since this happens only for convex hull of dimension 2
	  // thus singular

	  if(pInterval->second <= get_alpha() &&
	     (pInterval->third > get_alpha()
	      || pInterval->third == Infinity)) 
	    {
	      // alpha must be larger than the mid boundary
	      // and alpha is smaller than the upper boundary
	      // which might be infinity 
	      // visualize the boundary
 CGAL_triangulation_assertion((classify((*edge_alpha_it).second.first,
					(*edge_alpha_it).second.second)
			       == REGULAR));
	      Alpha_shape_edges_list.push_back(Edge((*edge_alpha_it).second.first,
						    (*edge_alpha_it).second.second));
	    }
	}
    }
  else 
    {  // get_mode() == GENERAL -------------------------------------------
      // draw the edges
      for (edge_alpha_it = _interval_edge_map.begin(); 
	   edge_alpha_it != _interval_edge_map.end() &&
	     (*edge_alpha_it).first.first <= get_alpha();
	   ++edge_alpha_it) 
	{
	  pInterval = &(*edge_alpha_it).first;
      
	  if (pInterval->first == UNDEFINED) 
	    {
	      CGAL_triangulation_assertion(pInterval->second != Infinity);
	      // since this happens only for convex hull of dimension 2
	      // thus singular
	
	      if(pInterval->second <= get_alpha() &&
		 (pInterval->third > get_alpha()
		  || pInterval->third == Infinity)) 
		{
		  // alpha must be larger than the mid boundary
		  // and alpha is smaller than the upper boundary
		  // which might be infinity 
		  // visualize the boundary
 CGAL_triangulation_assertion((classify((*edge_alpha_it).second.first,
					(*edge_alpha_it).second.second) 
			       == REGULAR));
		  Alpha_shape_edges_list.push_back(Edge((*edge_alpha_it).second.first,
							(*edge_alpha_it).second.second));
		}
	    }
	  else 
	    {
	
	      if(pInterval->third > get_alpha()
		 || pInterval->third == Infinity) 
		{
		  // if alpha is smaller than the upper boundary
		  // which might be infinity 
		  // visualize the boundary
 CGAL_triangulation_assertion(((classify((*edge_alpha_it).second.first,
					 (*edge_alpha_it).second.second) 
				== REGULAR)
			       || (classify((*edge_alpha_it).second.first,
					    (*edge_alpha_it).second.second) 
				   == SINGULAR)));
		  Alpha_shape_edges_list.push_back(Edge((*edge_alpha_it).second.first,
							(*edge_alpha_it).second.second));
		}
	    }
      
	}
    
    }
  use_edge_cache = true;
}

//-------------------------------------------------------------------------

template < class Dt, class EACT >
typename Alpha_shape_2<Dt,EACT>::Classification_type  
Alpha_shape_2<Dt,EACT>::classify(const Face_handle& f, int i, 
			    const Type_of_alpha& alpha) const
{
  // Classifies the edge `e' of the underlying Delaunay
  // triangulation with respect to `A'.

  // the version that uses a simplified version without crosslinks
  // is much slower
  if (is_infinite(f, i))
    return EXTERIOR;

  // we store only finite edges in _edge_interval_map
  Interval3 interval = find_interval(const_Edge(f, i));
  //  (*(_edge_interval_map.find(const_Edge(f, i)))).second;
 
  if (alpha < interval.second) 
    {
      if (get_mode() == REGULARIZED ||
	  interval.first == UNDEFINED ||
	  alpha < interval.first)
	return EXTERIOR;
      else // alpha >= interval.first
	return SINGULAR;
    }
  else 
    {   // alpha >= interval.second
      if (interval.third == Infinity ||
	  alpha < interval.third)
	return REGULAR;
      else // alpha >= interval.third
	return INTERIOR;
    }
   
}

//-------------------------------------------------------------------------

template < class Dt, class EACT >
typename Alpha_shape_2<Dt,EACT>::Classification_type  
Alpha_shape_2<Dt,EACT>::classify(const Vertex_handle& v,
			    const Type_of_alpha& alpha) const 
{
  // Classifies the vertex `v' of the underlying Delaunay
  // triangulation with respect to `A'.
  Interval2 interval = v->get_range();
 
  if (alpha < interval.first) 
    {
      if (get_mode() == REGULARIZED) 
	return EXTERIOR;
      else // general => vertices are never exterior
	return SINGULAR;
    }
  else 
    {   // alpha >= interval.first
      if (interval.second == Infinity ||
	  alpha < interval.second)
	return REGULAR;
      else // alpha >= interval.second
	return INTERIOR;
    }
}

//-------------------------------------------------------------------------

template < class Dt, class EACT >
typename Alpha_shape_2<Dt,EACT>::size_type
Alpha_shape_2<Dt,EACT>::number_of_solid_components(const Type_of_alpha& alpha) const
{
  // Determine the number of connected solid components 
  typedef typename Marked_face_set::Data Data;
  Marked_face_set marked_face_set(false);
  Finite_faces_iterator face_it;
  size_type nb_solid_components = 0;
  
  if (number_of_vertices()==0)
    return 0;
  
  // only finite faces
  for( face_it = faces_begin(); 
       face_it != faces_end(); 
       ++face_it) 
    {
      Face_handle pFace = face_it;
      CGAL_triangulation_postcondition( pFace != NULL);
      
      if (classify(pFace, alpha) == INTERIOR){
	Data& data = marked_face_set[pFace];
	if(data == false)
	  {
	    // we traverse only interior faces
	    traverse(pFace, marked_face_set, alpha);
	    nb_solid_components++;  
	  }
      }
    }
  return nb_solid_components;
}

//-------------------------------------------------------------------------

template < class Dt, class EACT >
void
Alpha_shape_2<Dt,EACT>::traverse(const Face_handle& pFace,
			    Marked_face_set& marked_face_set, 
			    const Type_of_alpha alpha) const 
{
  typedef typename Marked_face_set::Data Data;
  std::list<Face_handle> faces;
  faces.push_back(pFace);
  Face_handle pNeighbor, fh;

  while(! faces.empty()){
    fh = faces.front();
    faces.pop_front();
    for (int i=0; i<3; i++)
      {
	pNeighbor = fh->neighbor(i);
	 CGAL_triangulation_assertion(pNeighbor != NULL);
	if (classify(pNeighbor, alpha) == INTERIOR){
	  Data& data = marked_face_set[pNeighbor];
	  if(data == false){
	    data = true;
	    faces.push_back(pNeighbor);
	  }
	}
      }
  }
}

//-------------------------------------------------------------------------

template < class Dt, class EACT >
typename Alpha_shape_2<Dt,EACT>::Alpha_iterator
Alpha_shape_2<Dt,EACT>::find_optimal_alpha(size_type nb_components) 
{
  // find the minimum alpha that satisfies the properties
  // (1) nb_components solid components
  // (2) all data points on the boundary or in its interior
  Type_of_alpha alpha = find_alpha_solid();
  // from this alpha on the alpha_solid satisfies property (2)
  
  Alpha_iterator first = alpha_lower_bound(alpha);

  if (number_of_solid_components(alpha) == nb_components) 
    {
      if ((first+1) < alpha_end())
	return (first+1);
      else
	return first;
    }

  // do binary search on the alpha values
  // number_of_solid_components() is a monotone function 
  // if we start with find_alpha_solid
  
  Alpha_iterator last = alpha_end();
  Alpha_iterator middle;
  
  std::ptrdiff_t len = last - first - 1;
  std::ptrdiff_t half;

  while (len > 0) 
    {
      half = len / 2;
      middle = first + half;

#ifdef CGAL_DEBUG_ALPHA_SHAPE_2
      std::cout << "first : " << *first << " last : " << *(first+len)
		<< " mid : " << *middle 
		<< " nb comps : " << number_of_solid_components(*middle) 
		<< std::endl;
#endif // CGAL_DEBUG_ALPHA_SHAPE_2

      if (number_of_solid_components(*middle) > nb_components) 
	{
	  first = middle + 1;
	  len = len - half - 1; 
	} 
      else 
	{ // number_of_solid_components(*middle) <= nb_components
	  len = half;
	}
    }
  if ((first+1) < alpha_end())
    return (first+1);
  else
    return first;
}

//-------------------------------------------------------------------------

template < class Dt, class EACT >
typename Alpha_shape_2<Dt,EACT>::Type_of_alpha 
Alpha_shape_2<Dt,EACT>::find_alpha_solid() const 
{
  // compute the minumum alpha such that all data points 
  // are either on the boundary or in the interior
  // not necessarily connected
  // starting point for searching 
  // takes O(#alpha_shape) time
  Type_of_alpha alpha_solid = 0;
  
  if (number_of_vertices()<3) return alpha_solid;

  Finite_vertices_iterator vertex_it;
  // only finite vertices
  for( vertex_it = finite_vertices_begin(); 
       vertex_it != finite_vertices_end();
       ++vertex_it) 
    {
      Type_of_alpha alpha_min_v = (--_interval_face_map.end())->first;

      Face_circulator face_circ = this->incident_faces(vertex_it);
      Face_circulator  done = face_circ;
      do 
	{
	  Face_handle f = face_circ;
	  if (! is_infinite(f))
	    alpha_min_v = (CGAL::min)(find_interval(f),
                                      alpha_min_v);
	}
      while (++face_circ != done);
      alpha_solid = (CGAL::max)(alpha_min_v, alpha_solid);

    }
  return alpha_solid;
}

//-------------------------------------------------------------------------

template < class Dt, class EACT >
std::ostream& 
Alpha_shape_2<Dt,EACT>::op_ostream(std::ostream& os) const
{
  typedef typename Alpha_shape_2<Dt,EACT>::Interval_vertex_map Interval_vertex_map ;
  typedef typename Alpha_shape_2<Dt,EACT>::Interval_edge_map Interval_edge_map;

  typename Interval_vertex_map::const_iterator vertex_alpha_it;
  const typename Alpha_shape_2<Dt,EACT>::Interval2* pInterval2;
  typename Interval_edge_map::const_iterator edge_alpha_it;
  const typename Alpha_shape_2<Dt,EACT>::Interval3* pInterval;

  Unique_hash_map< Vertex_handle, size_type > V;
  size_type number_of_vertices = 0;

  if (get_mode() == Alpha_shape_2<Dt,EACT>::REGULARIZED)
  {
    typename Alpha_shape_2<Dt,EACT>::Vertex_handle v;
    for (vertex_alpha_it = _interval_vertex_map.begin();
         vertex_alpha_it != _interval_vertex_map.end() &&
         (*vertex_alpha_it).first.first <= get_alpha();
         ++vertex_alpha_it)
    {
      pInterval2 = &(*vertex_alpha_it).first;

#ifdef CGAL_DEBUG_ALPHA_SHAPE_2
      typename Alpha_shape_2<Dt,EACT>::Type_of_alpha alpha =
          get_alpha();
      typename Alpha_shape_2<Dt,EACT>::Type_of_alpha alpha_mid =
          pInterval2->first;
      typename Alpha_shape_2<Dt,EACT>::Type_of_alpha alpha_max =
          pInterval2->second;
#endif // CGAL_DEBUG_ALPHA_SHAPE_2

      if((pInterval2->second > get_alpha()
          || pInterval2->second == Infinity))
      {
        // alpha must be larger than the min boundary
        // and alpha is smaller than the upper boundary
        // which might be infinity
        // write the vertex

        v = (*vertex_alpha_it).second;
        CGAL_triangulation_assertion((classify(v) ==
                                      Alpha_shape_2<Dt,EACT>::REGULAR));
        // if we used Edelsbrunner and Muecke's definition
        // regular means incident to a higher-dimensional face
        // we would write too many vertices

        V[v] = number_of_vertices++;
        os << v->point() << std::endl;
      }
    }
    // the vertices are oriented counterclockwise

    typename Alpha_shape_2<Dt,EACT>::Face_handle f;
    int i;

    for (edge_alpha_it = _interval_edge_map.begin();
         edge_alpha_it != _interval_edge_map.end() &&
         (*edge_alpha_it).first.first <= get_alpha();
         ++edge_alpha_it)
    {
      pInterval = &(*edge_alpha_it).first;

      CGAL_triangulation_assertion(pInterval->second != Infinity);
      // since this happens only for convex hull of dimension 1
      // thus singular

      if(pInterval->second <= get_alpha() &&
         (pInterval->third > get_alpha()
          || pInterval->third == Infinity))
      {
        // alpha must be larger than the mid boundary
        // and alpha is smaller than the upper boundary
        // which might be infinity
        // visualize the boundary


        f = (*edge_alpha_it).second.first;
        i = (*edge_alpha_it).second.second;

        // assure that all vertices are in ccw order
        if (classify(f) == Alpha_shape_2<Dt,EACT>::EXTERIOR)
        {
          // take the reverse face
          typename Alpha_shape_2<Dt,EACT>::Face_handle
              pNeighbor = f->neighbor(i);
          i = pNeighbor->index(f);
          f = pNeighbor;
        }

        CGAL_triangulation_assertion((classify(f) ==
                                      Alpha_shape_2<Dt,EACT>::INTERIOR));

        CGAL_triangulation_assertion((classify(f, i) ==
                                      Alpha_shape_2<Dt,EACT>::REGULAR));

        os << V[f->vertex(f->ccw(i))] << ' '
                                      << V[f->vertex(f->cw(i))] << std::endl;
      }
    }
  }
  else
  { // get_mode() == GENERAL -----------------------------------------
    typename Alpha_shape_2<Dt,EACT>::Vertex_handle v;

    // write the regular vertices
    for (vertex_alpha_it = _interval_vertex_map.begin();
         vertex_alpha_it != _interval_vertex_map.end() &&
         (*vertex_alpha_it).first.first <= get_alpha();
         ++vertex_alpha_it)
    {
      pInterval2 = &(*vertex_alpha_it).first;

      if((pInterval2->second > get_alpha()
          || pInterval2->second == Infinity))
      {
        // alpha must be larger than the min boundary
        // and alpha is smaller than the upper boundary
        // which might be infinity
        // write the vertex

        v = (*vertex_alpha_it).second;
        CGAL_triangulation_assertion((classify(v) ==
                                      Alpha_shape_2<Dt,EACT>::REGULAR));
        V[v] = number_of_vertices++;
        os << v->point() << std::endl;
      }
    }

    // write the singular vertices
    for (;
         vertex_alpha_it != _interval_vertex_map.end();
         ++vertex_alpha_it)
    {
      v = (*vertex_alpha_it).second;
      CGAL_triangulation_assertion((classify(v) ==
                                    Alpha_shape_2<Dt,EACT>::SINGULAR));

      V[v] = number_of_vertices++;
      os << v->point() << std::endl;
    }

    // the vertices are oriented counterclockwise

    typename Alpha_shape_2<Dt,EACT>::Face_handle f;
    int i;

    for (edge_alpha_it = _interval_edge_map.begin();
         edge_alpha_it != _interval_edge_map.end() &&
         (*edge_alpha_it).first.first <= get_alpha();
         ++edge_alpha_it)
    {
      pInterval = &(*edge_alpha_it).first;

#ifdef CGAL_DEBUG_ALPHA_SHAPE_2
      typename Alpha_shape_2<Dt,EACT>::Type_of_alpha alpha =
          get_alpha();
      typename Alpha_shape_2<Dt,EACT>::Type_of_alpha alpha_min =
          pInterval->first;
      typename Alpha_shape_2<Dt,EACT>::Type_of_alpha alpha_mid =
          pInterval->second;
      typename Alpha_shape_2<Dt,EACT>::Type_of_alpha alpha_max =
          pInterval->third;
#endif // CGAL_DEBUG_ALPHA_SHAPE_2

      if(pInterval->third > get_alpha()
         || pInterval->third == Infinity)
      {
        // if alpha is smaller than the upper boundary
        // which might be infinity
        // visualize the boundary

        f = (*edge_alpha_it).second.first;
        i = (*edge_alpha_it).second.second;

        // write the regular edges
        if (pInterval->second != Infinity &&
            pInterval->second <= get_alpha())
        {
          CGAL_triangulation_assertion((classify(f, i) ==
                                        Alpha_shape_2<Dt,EACT>::REGULAR));
          // assure that all vertices are in ccw order
          if (classify(f) == Alpha_shape_2<Dt,EACT>::EXTERIOR)
          {
            // take the reverse face
            typename Alpha_shape_2<Dt,EACT>::Face_handle
                pNeighbor = f->neighbor(i);

            i = pNeighbor->index(f);
            f = pNeighbor;
          }

          CGAL_triangulation_assertion((classify(f) ==
                                        Alpha_shape_2<Dt,EACT>::INTERIOR));

          os << V[f->vertex(f->ccw(i))] << ' '
                                        << V[f->vertex(f->cw(i))] << std::endl;
        }
        else
        { // pInterval->second == Infinity ||
          //                           pInterval->second >= get_alpha())
          // pInterval->second == Infinity happens only for convex hull
          // of dimension 1 thus singular

          // write the singular edges
          if (pInterval->first != UNDEFINED)
          {
            CGAL_triangulation_assertion((classify(f, i) ==
                                          Alpha_shape_2<Dt,EACT>::SINGULAR));
            os << V[f->vertex(f->ccw(i))] << ' '
                                          << V[f->vertex(f->cw(i))] << std::endl;
          }
        }
      }
    }
  }
  return os;
}

//-------------------------------------------------------------------

template < class Dt, class EACT >
std::ostream& 
operator<<(std::ostream& os, const Alpha_shape_2<Dt>& A)
{
  return A.op_ostream(os);
}

//-------------------------------------------------------------------

template < class Dt, class EACT >
std::list<typename Alpha_shape_2<Dt,EACT>::Point_2>
Alpha_shape_2<Dt,EACT>::Output () 
{
  typename Interval_edge_map::const_iterator edge_alpha_it;

  const Interval3* pInterval;
  std::list<Point_2> L;

  if (get_mode() == REGULARIZED)
  {
    // it is much faster looking at the sorted intervals
    // than looking at all sorted faces
    // alpha must be larger than the mid boundary
    // and alpha is smaller than the upper boundary
    for (edge_alpha_it = _interval_edge_map.begin();
         edge_alpha_it != _interval_edge_map.end() &&
         (*edge_alpha_it).first.first <= get_alpha();
         ++edge_alpha_it)
    {
      pInterval = &(*edge_alpha_it).first;

      if (pInterval->second != Infinity)
      {
        // since this happens only for convex hull of dimension 1
        // thus singular

        if(pInterval->second <= get_alpha() &&
           (pInterval->third > get_alpha()
            || pInterval->third == Infinity))
        {
          // alpha must be larger than the mid boundary
          // and alpha is smaller than the upper boundary
          // which might be infinity
          // visualize the boundary

          CGAL_triangulation_assertion((classify((*edge_alpha_it).second.first,
                                                 (*edge_alpha_it).second.second) ==
                                        REGULAR));

          // if we used Edelsbrunner and Muecke's definition
          // regular means incident to a higher-dimensional face
          // thus we would write to many vertices
          L.push_back((this->segment((*edge_alpha_it).second.first,
                                     (*edge_alpha_it).second.second))
                      .source());
          L.push_back((this->segment((*edge_alpha_it).second.first,
                                     (*edge_alpha_it).second.second))
                      .target());
        }
      }
    }
  }
  else
  {  // get_mode() == GENERAL
    // draw the edges
    for (edge_alpha_it = _interval_edge_map.begin();
         edge_alpha_it != _interval_edge_map.end() &&
         (*edge_alpha_it).first.first <= get_alpha();
         ++edge_alpha_it)
    {
      pInterval = &(*edge_alpha_it).first;

      if (pInterval->first == UNDEFINED)
      {

        CGAL_triangulation_assertion(pInterval->second != Infinity);
        // since this happens only for convex hull of dimension 1
        // thus singular

        if(pInterval->second <= get_alpha() &&
           (pInterval->third > get_alpha()
            || pInterval->third == Infinity))
        {
          // alpha must be larger than the mid boundary
          // and alpha is smaller than the upper boundary
          // which might be infinity
          // visualize the boundary

          CGAL_triangulation_assertion((classify((*edge_alpha_it).second.first,
                                                 (*edge_alpha_it).second.second) ==
                                        REGULAR));
          L.push_back((this->segment((*edge_alpha_it).second.first,
                                     (*edge_alpha_it).second.second))
                      .source());
          L.push_back((this->segment((*edge_alpha_it).second.first,
                                     (*edge_alpha_it).second.second))
                      .target());
        }
      }
      else
      {
        if(pInterval->third > get_alpha()
           || pInterval->third == Infinity)
        {
          // if alpha is smaller than the upper boundary
          // which might be infinity
          // visualize the boundary

          CGAL_triangulation_assertion(((classify((*edge_alpha_it).second.first,
                                                  (*edge_alpha_it).second.second) ==
                                         REGULAR) ||
                                        (classify((*edge_alpha_it).second.first,
                                                  (*edge_alpha_it).second.second) ==
                                         SINGULAR)));
          L.push_back((this->segment((*edge_alpha_it).second.first,
                                     (*edge_alpha_it).second.second))
                      .source());
          L.push_back((this->segment((*edge_alpha_it).second.first,
                                     (*edge_alpha_it).second.second))
                      .target());
        }
      }
    }
  }
  return L;
}

template < class Dt, class EACT >
void Alpha_shape_2<Dt,EACT>::print_edge_map()
{
  for (typename Interval_edge_map::iterator iemapit= _interval_edge_map.begin();
       iemapit != _interval_edge_map.end(); ++iemapit) {
    Interval3 interval = (*iemapit).first;
    Edge edge = (*iemapit).second;
    Point p1 = edge.first->vertex(cw(edge.second))->point();
    Point p2 = edge.first->vertex(ccw(edge.second))->point();
    std::cout << "[ (" <<	p1 << ") - (" << p2 << ") ] :            "
              << interval.first << " "
              << interval.second << " " << interval.third << std::endl;
  }
}

} //namespace CGAL


#endif //CGAL_ALPHA_SHAPE_2_H