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
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
/* -*- Mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
 * This file is part of the LibreOffice project.
 *
 * This Source Code Form is subject to the terms of the Mozilla Public
 * License, v. 2.0. If a copy of the MPL was not distributed with this
 * file, You can obtain one at http://mozilla.org/MPL/2.0/.
 *
 * This file incorporates work covered by the following license notice:
 *
 *   Licensed to the Apache Software Foundation (ASF) under one or more
 *   contributor license agreements. See the NOTICE file distributed
 *   with this work for additional information regarding copyright
 *   ownership. The ASF licenses this file to you under the Apache
 *   License, Version 2.0 (the "License"); you may not use this file
 *   except in compliance with the License. You may obtain a copy of
 *   the License at http://www.apache.org/licenses/LICENSE-2.0 .
 */

#include <algorithm>
#include <iterator>
#include <vector>
#include <utility>

#include <math.h>

#include <bezierclip.hxx>
#include <gauss.hxx>

// what to test
#define WITH_ASSERTIONS
//#define WITH_CONVEXHULL_TEST
//#define WITH_MULTISUBDIVIDE_TEST
//#define WITH_FATLINE_TEST
//#define WITH_CALCFOCUS_TEST
//#define WITH_SAFEPARAMBASE_TEST
//#define WITH_SAFEPARAMS_TEST
//#define WITH_SAFEPARAM_DETAILED_TEST
//#define WITH_SAFEFOCUSPARAM_CALCFOCUS
//#define WITH_SAFEFOCUSPARAM_TEST
//#define WITH_SAFEFOCUSPARAM_DETAILED_TEST
#define WITH_BEZIERCLIP_TEST

/* Implementation of the so-called 'Fat-Line Bezier Clipping Algorithm' by Sederberg et al.
 *
 * Actual reference is: T. W. Sederberg and T Nishita: Curve
 * intersection using Bezier clipping. In Computer Aided Design, 22
 * (9), 1990, pp. 538--549
 */

/* Misc helper
 * ===========
 */
int fallFac( int n, int k )
{
#ifdef WITH_ASSERTIONS
    assert(n>=k); // "For factorials, n must be greater or equal k"
    assert(n>=0); // "For factorials, n must be positive"
    assert(k>=0); // "For factorials, k must be positive"
#endif

    int res( 1 );

    while( k-- && n ) res *= n--;

    return res;
}

int fac( int n )
{
    return fallFac(n, n);
}

/* Bezier fat line clipping part
 * =============================
 */

void Impl_calcFatLine( FatLine& line, const Bezier& c )
{
    // Prepare normalized implicit line
    // ================================

    // calculate vector orthogonal to p1-p4:
    line.a = -(c.p0.y - c.p3.y);
    line.b = (c.p0.x - c.p3.x);

    // normalize
    const double len( sqrt( line.a*line.a + line.b*line.b ) );
    if( !tolZero(len) )
    {
        line.a /= len;
        line.b /= len;
    }

    line.c = -(line.a*c.p0.x + line.b*c.p0.y);

    // Determine bounding fat line from it
    // ===================================

    // calc control point distances
    const double dP2( calcLineDistance(line.a, line.b, line.c, c.p1.x, c.p1.y ) );
    const double dP3( calcLineDistance(line.a, line.b, line.c, c.p2.x, c.p2.y ) );

    // calc approximate bounding lines to curve (tight bounds are
    // possible here, but more expensive to calculate and thus not
    // worth the overhead)
    if( dP2 * dP3 > 0.0 )
    {
        line.dMin = 3.0/4.0 * std::min(0.0, std::min(dP2, dP3));
        line.dMax = 3.0/4.0 * std::max(0.0, std::max(dP2, dP3));
    }
    else
    {
        line.dMin = 4.0/9.0 * std::min(0.0, std::min(dP2, dP3));
        line.dMax = 4.0/9.0 * std::max(0.0, std::max(dP2, dP3));
    }
}

void Impl_calcBounds( Point2D&          leftTop,
                      Point2D&          rightBottom,
                      const Bezier&     c1          )
{
    leftTop.x = std::min( c1.p0.x, std::min( c1.p1.x, std::min( c1.p2.x, c1.p3.x ) ) );
    leftTop.y = std::min( c1.p0.y, std::min( c1.p1.y, std::min( c1.p2.y, c1.p3.y ) ) );
    rightBottom.x = std::max( c1.p0.x, std::max( c1.p1.x, std::max( c1.p2.x, c1.p3.x ) ) );
    rightBottom.y = std::max( c1.p0.y, std::max( c1.p1.y, std::max( c1.p2.y, c1.p3.y ) ) );
}

bool Impl_doBBoxIntersect( const Bezier& c1,
                           const Bezier& c2 )
{
    // calc rectangular boxes from c1 and c2
    Point2D lt1;
    Point2D rb1;
    Point2D lt2;
    Point2D rb2;

    Impl_calcBounds( lt1, rb1, c1 );
    Impl_calcBounds( lt2, rb2, c2 );

    if( std::min(rb1.x, rb2.x) < std::max(lt1.x, lt2.x) ||
        std::min(rb1.y, rb2.y) < std::max(lt1.y, lt2.y) )
    {
        return false;
    }
    else
    {
        return true;
    }
}

/* calculates two t's for the given bernstein control polygon: the first is
 * the intersection of the min value line with the convex hull from
 * the left, the second is the intersection of the max value line with
 * the convex hull from the right.
 */
bool Impl_calcSafeParams( double&           t1,
                          double&           t2,
                          const Polygon2D&  rPoly,
                          double            lowerYBound,
                          double            upperYBound )
{
    // need the convex hull of the control polygon, as this is
    // guaranteed to completely bound the curve
    Polygon2D convHull( convexHull(rPoly) );

    // init min and max buffers
    t1 = 0.0 ;
    double currLowerT( 1.0 );

    t2 = 1.0;
    double currHigherT( 0.0 );

    if( convHull.size() <= 1 )
        return false; // only one point? Then we're done with clipping

    /* now, clip against lower and higher bounds */
    Point2D p0;
    Point2D p1;

    bool bIntersection( false );

    for( Polygon2D::size_type i=0; i<convHull.size(); ++i )
    {
        // have to check against convHull.size() segments, as the
        // convex hull is, by definition, closed. Thus, for the
        // last point, we take the first point as partner.
        if( i+1 == convHull.size() )
        {
            // close the polygon
            p0 = convHull[i];
            p1 = convHull[0];
        }
        else
        {
            p0 = convHull[i];
            p1 = convHull[i+1];
        }

        // is the segment in question within or crossing the
        // horizontal band spanned by lowerYBound and upperYBound? If
        // not, we've got no intersection. If yes, we maybe don't have
        // an intersection, but we've got to update the permissible
        // range, nevertheless. This is because inside lying segments
        // leads to full range forbidden.
        if( (tolLessEqual(p0.y, upperYBound) || tolLessEqual(p1.y, upperYBound)) &&
            (tolGreaterEqual(p0.y, lowerYBound) || tolGreaterEqual(p1.y, lowerYBound)) )
        {
            // calc intersection of convex hull segment with
            // one of the horizontal bounds lines
            // to optimize a bit, r_x is calculated only in else case
            const double r_y( p1.y - p0.y );

            if( tolZero(r_y) )
            {
                // r_y is virtually zero, thus we've got a horizontal
                // line. Now check whether we maybe coincide with lower or
                // upper horizontal bound line.
                if( tolEqual(p0.y, lowerYBound) ||
                    tolEqual(p0.y, upperYBound) )
                {
                    // yes, simulate intersection then
                    currLowerT = std::min(currLowerT, std::min(p0.x, p1.x));
                    currHigherT = std::max(currHigherT, std::max(p0.x, p1.x));
                }
            }
            else
            {
                // check against lower and higher bounds
                // =====================================
                const double r_x( p1.x - p0.x );

                // calc intersection with horizontal dMin line
                const double currTLow( (lowerYBound - p0.y) * r_x / r_y + p0.x );

                // calc intersection with horizontal dMax line
                const double currTHigh( (upperYBound - p0.y) * r_x / r_y + p0.x );

                currLowerT = std::min(currLowerT, std::min(currTLow, currTHigh));
                currHigherT = std::max(currHigherT, std::max(currTLow, currTHigh));
            }

            // set flag that at least one segment is contained or
            // intersects given horizontal band.
            bIntersection = true;
        }
    }

#ifndef WITH_SAFEPARAMBASE_TEST
    // limit intersections found to permissible t parameter range
    t1 = std::max(0.0, currLowerT);
    t2 = std::min(1.0, currHigherT);
#endif

    return bIntersection;
}

/* calculates two t's for the given bernstein polynomial: the first is
 * the intersection of the min value line with the convex hull from
 * the left, the second is the intersection of the max value line with
 * the convex hull from the right.
 *
 * The polynomial coefficients c0 to c3 given to this method
 * must correspond to t values of 0, 1/3, 2/3 and 1, respectively.
 */
bool Impl_calcSafeParams_clip( double&          t1,
                               double&          t2,
                               const FatLine&   bounds,
                               double           c0,
                               double           c1,
                               double           c2,
                               double           c3 )
{
    /* first of all, determine convex hull of c0-c3 */
    Polygon2D poly(4);
    poly[0] = Point2D(0,        c0);
    poly[1] = Point2D(1.0/3.0,  c1);
    poly[2] = Point2D(2.0/3.0,  c2);
    poly[3] = Point2D(1,        c3);

#ifndef WITH_SAFEPARAM_DETAILED_TEST

    return Impl_calcSafeParams( t1, t2, poly, bounds.dMin, bounds.dMax );

#else
    bool bRet( Impl_calcSafeParams( t1, t2, poly, bounds.dMin, bounds.dMax ) );

    Polygon2D convHull( convexHull( poly ) );

    cout << "# convex hull testing" << endl
         << "plot [t=0:1] ";
    cout << " bez("
         << poly[0].x << ","
         << poly[1].x << ","
         << poly[2].x << ","
         << poly[3].x << ",t),bez("
         << poly[0].y << ","
         << poly[1].y << ","
         << poly[2].y << ","
         << poly[3].y << ",t), "
         << "t, " << bounds.dMin << ", "
         << "t, " << bounds.dMax << ", "
         << t1 << ", t, "
         << t2 << ", t, "
         << "'-' using ($1):($2) title \"control polygon\" with lp, "
         << "'-' using ($1):($2) title \"convex hull\" with lp" << endl;

    unsigned int k;
    for( k=0; k<poly.size(); ++k )
    {
        cout << poly[k].x << " " << poly[k].y << endl;
    }
    cout << poly[0].x << " " << poly[0].y << endl;
    cout << "e" << endl;

    for( k=0; k<convHull.size(); ++k )
    {
        cout << convHull[k].x << " " << convHull[k].y << endl;
    }
    cout << convHull[0].x << " " << convHull[0].y << endl;
    cout << "e" << endl;

    return bRet;
#endif
}

void Impl_deCasteljauAt( Bezier&        part1,
                         Bezier&        part2,
                         const Bezier&  input,
                         double         t        )
{
    // deCasteljau bezier arc, scheme is:

    // First row is    C_0^n,C_1^n,...,C_n^n
    // Second row is         P_1^n,...,P_n^n
    // etc.
    // with P_k^r = (1 - x_s)P_{k-1}^{r-1} + x_s P_k{r-1}

    // this results in:

    // P1  P2  P3  P4
    // L1  P2  P3  R4
    //     L2  H   R3
    //         L3  R2
    //             L4/R1
    if( tolZero(t) )
    {
        // t is zero -> part2 is input curve, part1 is empty (input.p0, that is)
        part1.p0.x = part1.p1.x = part1.p2.x = part1.p3.x = input.p0.x;
        part1.p0.y = part1.p1.y = part1.p2.y = part1.p3.y = input.p0.y;
        part2 = input;
    }
    else if( tolEqual(t, 1.0) )
    {
        // t is one -> part1 is input curve, part2 is empty (input.p3, that is)
        part1 = input;
        part2.p0.x = part2.p1.x = part2.p2.x = part2.p3.x = input.p3.x;
        part2.p0.y = part2.p1.y = part2.p2.y = part2.p3.y = input.p3.y;
    }
    else
    {
        part1.p0.x = input.p0.x;                                    part1.p0.y = input.p0.y;
        part1.p1.x = (1.0 - t)*part1.p0.x + t*input.p1.x;           part1.p1.y = (1.0 - t)*part1.p0.y + t*input.p1.y;
        const double Hx ( (1.0 - t)*input.p1.x + t*input.p2.x ),    Hy ( (1.0 - t)*input.p1.y + t*input.p2.y );
        part1.p2.x = (1.0 - t)*part1.p1.x + t*Hx;                   part1.p2.y = (1.0 - t)*part1.p1.y + t*Hy;
        part2.p3.x = input.p3.x;                                    part2.p3.y = input.p3.y;
        part2.p2.x = (1.0 - t)*input.p2.x + t*input.p3.x;           part2.p2.y = (1.0 - t)*input.p2.y + t*input.p3.y;
        part2.p1.x = (1.0 - t)*Hx + t*part2.p2.x;                   part2.p1.y = (1.0 - t)*Hy + t*part2.p2.y;
        part2.p0.x = (1.0 - t)*part1.p2.x + t*part2.p1.x;           part2.p0.y = (1.0 - t)*part1.p2.y + t*part2.p1.y;
        part1.p3.x = part2.p0.x;                                    part1.p3.y = part2.p0.y;
    }
}

void printCurvesWithSafeRange( const Bezier& c1, const Bezier& c2, double t1_c1, double t2_c1,<--- The function 'printCurvesWithSafeRange' is never used.
                               const Bezier& c2_part, const FatLine& bounds_c2 )
{
    static int offset = 0;

    cout << "# safe param range testing" << endl
         << "plot [t=0.0:1.0] ";

    // clip safe ranges off c1
    Bezier c1_part1;
    Bezier c1_part2;
    Bezier c1_part3;

    // subdivide at t1_c1
    Impl_deCasteljauAt( c1_part1, c1_part2, c1, t1_c1 );
    // subdivide at t2_c1
    Impl_deCasteljauAt( c1_part1, c1_part3, c1_part2, t2_c1 );

    // output remaining segment (c1_part1)

    cout << "bez("
         << c1.p0.x+offset << ","
         << c1.p1.x+offset << ","
         << c1.p2.x+offset << ","
         << c1.p3.x+offset << ",t),bez("
         << c1.p0.y << ","
         << c1.p1.y << ","
         << c1.p2.y << ","
         << c1.p3.y << ",t), bez("
         << c2.p0.x+offset << ","
         << c2.p1.x+offset << ","
         << c2.p2.x+offset << ","
         << c2.p3.x+offset << ",t),bez("
         << c2.p0.y << ","
         << c2.p1.y << ","
         << c2.p2.y << ","
         << c2.p3.y << ",t), "
#if 1
         << "bez("
         << c1_part1.p0.x+offset << ","
         << c1_part1.p1.x+offset << ","
         << c1_part1.p2.x+offset << ","
         << c1_part1.p3.x+offset << ",t),bez("
         << c1_part1.p0.y << ","
         << c1_part1.p1.y << ","
         << c1_part1.p2.y << ","
         << c1_part1.p3.y << ",t), "
#endif
#if 1
         << "bez("
         << c2_part.p0.x+offset << ","
         << c2_part.p1.x+offset << ","
         << c2_part.p2.x+offset << ","
         << c2_part.p3.x+offset << ",t),bez("
         << c2_part.p0.y << ","
         << c2_part.p1.y << ","
         << c2_part.p2.y << ","
         << c2_part.p3.y << ",t), "
#endif
         << "linex("
         << bounds_c2.a << ","
         << bounds_c2.b << ","
         << bounds_c2.c << ",t)+" << offset << ", liney("
         << bounds_c2.a << ","
         << bounds_c2.b << ","
         << bounds_c2.c << ",t) title \"fat line (center)\", linex("
         << bounds_c2.a << ","
         << bounds_c2.b << ","
         << bounds_c2.c-bounds_c2.dMin << ",t)+" << offset << ", liney("
         << bounds_c2.a << ","
         << bounds_c2.b << ","
         << bounds_c2.c-bounds_c2.dMin << ",t) title \"fat line (min) \", linex("
         << bounds_c2.a << ","
         << bounds_c2.b << ","
         << bounds_c2.c-bounds_c2.dMax << ",t)+" << offset << ", liney("
         << bounds_c2.a << ","
         << bounds_c2.b << ","
         << bounds_c2.c-bounds_c2.dMax << ",t) title \"fat line (max) \"" << endl;

    offset += 1;
}

void printResultWithFinalCurves( const Bezier& c1, const Bezier& c1_part,
                                 const Bezier& c2, const Bezier& c2_part,
                                 double t1_c1, double t2_c1 )
{
    static int offset = 0;

    cout << "# final result" << endl
         << "plot [t=0.0:1.0] ";

    cout << "bez("
         << c1.p0.x+offset << ","
         << c1.p1.x+offset << ","
         << c1.p2.x+offset << ","
         << c1.p3.x+offset << ",t),bez("
         << c1.p0.y << ","
         << c1.p1.y << ","
         << c1.p2.y << ","
         << c1.p3.y << ",t), bez("
         << c1_part.p0.x+offset << ","
         << c1_part.p1.x+offset << ","
         << c1_part.p2.x+offset << ","
         << c1_part.p3.x+offset << ",t),bez("
         << c1_part.p0.y << ","
         << c1_part.p1.y << ","
         << c1_part.p2.y << ","
         << c1_part.p3.y << ",t), "
         << " pointmarkx(bez("
         << c1.p0.x+offset << ","
         << c1.p1.x+offset << ","
         << c1.p2.x+offset << ","
         << c1.p3.x+offset << ","
         << t1_c1 << "),t), "
         << " pointmarky(bez("
         << c1.p0.y << ","
         << c1.p1.y << ","
         << c1.p2.y << ","
         << c1.p3.y << ","
         << t1_c1 << "),t), "
         << " pointmarkx(bez("
         << c1.p0.x+offset << ","
         << c1.p1.x+offset << ","
         << c1.p2.x+offset << ","
         << c1.p3.x+offset << ","
         << t2_c1 << "),t), "
         << " pointmarky(bez("
         << c1.p0.y << ","
         << c1.p1.y << ","
         << c1.p2.y << ","
         << c1.p3.y << ","
         << t2_c1 << "),t), "

         << "bez("
         << c2.p0.x+offset << ","
         << c2.p1.x+offset << ","
         << c2.p2.x+offset << ","
         << c2.p3.x+offset << ",t),bez("
         << c2.p0.y << ","
         << c2.p1.y << ","
         << c2.p2.y << ","
         << c2.p3.y << ",t), "
         << "bez("
         << c2_part.p0.x+offset << ","
         << c2_part.p1.x+offset << ","
         << c2_part.p2.x+offset << ","
         << c2_part.p3.x+offset << ",t),bez("
         << c2_part.p0.y << ","
         << c2_part.p1.y << ","
         << c2_part.p2.y << ","
         << c2_part.p3.y << ",t)" << endl;

    offset += 1;
}

/** determine parameter ranges [0,t1) and (t2,1] on c1, where c1 is guaranteed to lie outside c2.
      Returns false, if the two curves don't even intersect.

    @param t1
    Range [0,t1) on c1 is guaranteed to lie outside c2

    @param t2
    Range (t2,1] on c1 is guaranteed to lie outside c2

    @param c1_orig
    Original curve c1

    @param c1_part
    Subdivided current part of c1

    @param c2_orig
    Original curve c2

    @param c2_part
    Subdivided current part of c2
 */
bool Impl_calcClipRange( double&        t1,
                         double&        t2,
                         const Bezier&  c1_orig,
                         const Bezier&  c1_part,
                         const Bezier&  c2_orig,
                         const Bezier&  c2_part )
{
    // TODO: Maybe also check fat line orthogonal to P0P3, having P0
    //       and P3 as the extremal points

    if( Impl_doBBoxIntersect(c1_part, c2_part) )
    {
        // Calculate fat lines around c1
        FatLine bounds_c2;

        // must use the subdivided version of c2, since the fat line
        // algorithm works implicitly with the convex hull bounding
        // box.
        Impl_calcFatLine(bounds_c2, c2_part);

        // determine clip positions on c2. Can use original c1 (which
        // is necessary anyway, to get the t's on the original curve),
        // since the distance calculations work directly in the
        // Bernstein polynomial parameter domain.
        if( Impl_calcSafeParams_clip( t1, t2, bounds_c2,
                                      calcLineDistance( bounds_c2.a,
                                                        bounds_c2.b,
                                                        bounds_c2.c,
                                                        c1_orig.p0.x,
                                                        c1_orig.p0.y    ),
                                      calcLineDistance( bounds_c2.a,
                                                        bounds_c2.b,
                                                        bounds_c2.c,
                                                        c1_orig.p1.x,
                                                        c1_orig.p1.y    ),
                                      calcLineDistance( bounds_c2.a,
                                                        bounds_c2.b,
                                                        bounds_c2.c,
                                                        c1_orig.p2.x,
                                                        c1_orig.p2.y    ),
                                      calcLineDistance( bounds_c2.a,
                                                        bounds_c2.b,
                                                        bounds_c2.c,
                                                        c1_orig.p3.x,
                                                        c1_orig.p3.y    ) ) )
        {
            //printCurvesWithSafeRange(c1_orig, c2_orig, t1, t2, c2_part, bounds_c2);

            // they do intersect
            return true;
        }
    }

    // they don't intersect: nothing to do
    return false;
}

/* Tangent intersection part
 * =========================
 */

void Impl_calcFocus( Bezier& res, const Bezier& c )
{
    // arbitrary small value, for now
    // TODO: find meaningful value
    const double minPivotValue( 1.0e-20 );

    Point2D::value_type fMatrix[6];
    Point2D::value_type fRes[2];

    // calc new curve from hodograph, c and linear blend

    // Coefficients for derivative of c are (C_i=n(C_{i+1} - C_i)):

    // 3(P1 - P0), 3(P2 - P1), 3(P3 - P2) (bezier curve of degree 2)

    // The hodograph is then (bezier curve of 2nd degree is P0(1-t)^2 + 2P1(1-t)t + P2t^2):

    // 3(P1 - P0)(1-t)^2 + 6(P2 - P1)(1-t)t + 3(P3 - P2)t^2

    // rotate by 90 degrees: x=-y, y=x and you get the normal vector function N(t):

    // x(t) = -(3(P1.y - P0.y)(1-t)^2 + 6(P2.y - P1.y)(1-t)t + 3(P3.y - P2.y)t^2)
    // y(t) =   3(P1.x - P0.x)(1-t)^2 + 6(P2.x - P1.x)(1-t)t + 3(P3.x - P2.x)t^2

    // Now, the focus curve is defined to be F(t)=P(t) + c(t)N(t),
    // where P(t) is the original curve, and c(t)=c0(1-t) + c1 t

    // This results in the following expression for F(t):

    // x(t) =  P0.x (1-t)^3 + 3 P1.x (1-t)^2t + 3 P2.x (1.t)t^2 + P3.x t^3 -
    //          (c0(1-t) + c1 t)(3(P1.y - P0.y)(1-t)^2 + 6(P2.y - P1.y)(1-t)t + 3(P3.y - P2.y)t^2)

    // y(t) =  P0.y (1-t)^3 + 3 P1.y (1-t)^2t + 3 P2.y (1.t)t^2 + P3.y t^3 +
    //          (c0(1-t) + c1 t)(3(P1.x - P0.x)(1-t)^2 + 6(P2.x - P1.x)(1-t)t + 3(P3.x - P2.x)t^2)

    // As a heuristic, we set F(0)=F(1) (thus, the curve is closed and _tends_ to be small):

    // For F(0), the following results:

    // x(0) = P0.x - c0 3(P1.y - P0.y)
    // y(0) = P0.y + c0 3(P1.x - P0.x)

    // For F(1), the following results:

    // x(1) = P3.x - c1 3(P3.y - P2.y)
    // y(1) = P3.y + c1 3(P3.x - P2.x)

    // Reorder, collect and substitute into F(0)=F(1):

    // P0.x - c0 3(P1.y - P0.y) = P3.x - c1 3(P3.y - P2.y)
    // P0.y + c0 3(P1.x - P0.x) = P3.y + c1 3(P3.x - P2.x)

    // which yields

    // (P0.y - P1.y)c0 + (P3.y - P2.y)c1 = (P3.x - P0.x)/3
    // (P1.x - P0.x)c0 + (P2.x - P3.x)c1 = (P3.y - P0.y)/3

    // so, this is what we calculate here (determine c0 and c1):
    fMatrix[0] = c.p1.x - c.p0.x;
    fMatrix[1] = c.p2.x - c.p3.x;
    fMatrix[2] = (c.p3.y - c.p0.y)/3.0;
    fMatrix[3] = c.p0.y - c.p1.y;
    fMatrix[4] = c.p3.y - c.p2.y;
    fMatrix[5] = (c.p3.x - c.p0.x)/3.0;

    // TODO: determine meaningful value for
    if( !solve(fMatrix, 2, 3, fRes, minPivotValue) )
    {
        // TODO: generate meaningful values here
        // singular or nearly singular system -- use arbitrary
        // values for res
        fRes[0] = 0.0;
        fRes[1] = 1.0;

        cerr << "Matrix singular!" << endl;
    }

    // now, the reordered and per-coefficient collected focus curve is
    // the following third degree bezier curve F(t):

    // x(t) =  P0.x (1-t)^3 + 3 P1.x (1-t)^2t + 3 P2.x (1.t)t^2 + P3.x t^3 -
    //          (c0(1-t) + c1 t)(3(P1.y - P0.y)(1-t)^2 + 6(P2.y - P1.y)(1-t)t + 3(P3.y - P2.y)t^2)
    //      =  P0.x (1-t)^3 + 3 P1.x (1-t)^2t + 3 P2.x (1.t)t^2 + P3.x t^3 -
    //         (3c0P1.y(1-t)^3 - 3c0P0.y(1-t)^3 + 6c0P2.y(1-t)^2t - 6c0P1.y(1-t)^2t +
    //          3c0P3.y(1-t)t^2 - 3c0P2.y(1-t)t^2 +
    //          3c1P1.y(1-t)^2t - 3c1P0.y(1-t)^2t + 6c1P2.y(1-t)t^2 - 6c1P1.y(1-t)t^2 +
    //          3c1P3.yt^3 - 3c1P2.yt^3)
    //      =  (P0.x - 3 c0 P1.y + 3 c0 P0.y)(1-t)^3 +
    //         3(P1.x - c1 P1.y + c1 P0.y - 2 c0 P2.y + 2 c0 P1.y)(1-t)^2t +
    //         3(P2.x - 2 c1 P2.y + 2 c1 P1.y - c0 P3.y + c0 P2.y)(1-t)t^2 +
    //         (P3.x - 3 c1 P3.y + 3 c1 P2.y)t^3
    //      =  (P0.x - 3 c0(P1.y - P0.y))(1-t)^3 +
    //         3(P1.x - c1(P1.y - P0.y) - 2c0(P2.y - P1.y))(1-t)^2t +
    //         3(P2.x - 2 c1(P2.y - P1.y) - c0(P3.y - P2.y))(1-t)t^2 +
    //         (P3.x - 3 c1(P3.y - P2.y))t^3

    // y(t) =  P0.y (1-t)^3 + 3 P1.y (1-t)^2t + 3 P2.y (1-t)t^2 + P3.y t^3 +
    //          (c0(1-t) + c1 t)(3(P1.x - P0.x)(1-t)^2 + 6(P2.x - P1.x)(1-t)t + 3(P3.x - P2.x)t^2)
    //      =  P0.y (1-t)^3 + 3 P1.y (1-t)^2t + 3 P2.y (1-t)t^2 + P3.y t^3 +
    //         3c0(P1.x - P0.x)(1-t)^3 + 6c0(P2.x - P1.x)(1-t)^2t + 3c0(P3.x - P2.x)(1-t)t^2 +
    //         3c1(P1.x - P0.x)(1-t)^2t + 6c1(P2.x - P1.x)(1-t)t^2 + 3c1(P3.x - P2.x)t^3
    //      =  (P0.y + 3 c0 (P1.x - P0.x))(1-t)^3 +
    //         3(P1.y + 2 c0 (P2.x - P1.x) + c1 (P1.x - P0.x))(1-t)^2t +
    //         3(P2.y + c0 (P3.x - P2.x) + 2 c1 (P2.x - P1.x))(1-t)t^2 +
    //         (P3.y + 3 c1 (P3.x - P2.x))t^3

    // Therefore, the coefficients F0 to F3 of the focus curve are:

    // F0.x = (P0.x - 3 c0(P1.y - P0.y))                    F0.y = (P0.y + 3 c0 (P1.x - P0.x))
    // F1.x = (P1.x - c1(P1.y - P0.y) - 2c0(P2.y - P1.y))   F1.y = (P1.y + 2 c0 (P2.x - P1.x) + c1 (P1.x - P0.x))
    // F2.x = (P2.x - 2 c1(P2.y - P1.y) - c0(P3.y - P2.y))  F2.y = (P2.y + c0 (P3.x - P2.x) + 2 c1 (P2.x - P1.x))
    // F3.x = (P3.x - 3 c1(P3.y - P2.y))                    F3.y = (P3.y + 3 c1 (P3.x - P2.x))

    res.p0.x = c.p0.x - 3*fRes[0]*(c.p1.y - c.p0.y);
    res.p1.x = c.p1.x - fRes[1]*(c.p1.y - c.p0.y) - 2*fRes[0]*(c.p2.y - c.p1.y);
    res.p2.x = c.p2.x - 2*fRes[1]*(c.p2.y - c.p1.y) - fRes[0]*(c.p3.y - c.p2.y);
    res.p3.x = c.p3.x - 3*fRes[1]*(c.p3.y - c.p2.y);

    res.p0.y = c.p0.y + 3*fRes[0]*(c.p1.x - c.p0.x);
    res.p1.y = c.p1.y + 2*fRes[0]*(c.p2.x - c.p1.x) + fRes[1]*(c.p1.x - c.p0.x);
    res.p2.y = c.p2.y + fRes[0]*(c.p3.x - c.p2.x) + 2*fRes[1]*(c.p2.x - c.p1.x);
    res.p3.y = c.p3.y + 3*fRes[1]*(c.p3.x - c.p2.x);
}

bool Impl_calcSafeParams_focus( double&         t1,
                                double&         t2,
                                const Bezier&   curve,
                                const Bezier&   focus )
{
    // now, we want to determine which normals of the original curve
    // P(t) intersect with the focus curve F(t). The condition for
    // this statement is P'(t)(P(t) - F) = 0, i.e. hodograph P'(t) and
    // line through P(t) and F are perpendicular.
    // If you expand this equation, you end up with something like

    // (\sum_{i=0}^n (P_i - F)B_i^n(t))^T (\sum_{j=0}^{n-1} n(P_{j+1} - P_j)B_j^{n-1}(t))

    // Multiplying that out (as the scalar product is linear, we can
    // extract some terms) yields:

    // (P_i - F)^T n(P_{j+1} - P_j) B_i^n(t)B_j^{n-1}(t) + ...

    // If we combine the B_i^n(t)B_j^{n-1}(t) product, we arrive at a
    // Bernstein polynomial of degree 2n-1, as

    // \binom{n}{i}(1-t)^{n-i}t^i) \binom{n-1}{j}(1-t)^{n-1-j}t^j) =
    // \binom{n}{i}\binom{n-1}{j}(1-t)^{2n-1-i-j}t^{i+j}

    // Thus, with the defining equation for a 2n-1 degree Bernstein
    // polynomial

    // \sum_{i=0}^{2n-1} d_i B_i^{2n-1}(t)

    // the d_i are calculated as follows:

    // d_i = \sum_{j+k=i, j\in\{0,...,n\}, k\in\{0,...,n-1\}} \frac{\binom{n}{j}\binom{n-1}{k}}{\binom{2n-1}{i}} n (P_{k+1} - P_k)^T(P_j - F)

    // Okay, but F is now not a single point, but itself a curve
    // F(u). Thus, for every value of u, we get a different 2n-1
    // bezier curve from the above equation. Therefore, we have a
    // tensor product bezier patch, with the following defining
    // equation:

    // d(t,u) = \sum_{i=0}^{2n-1} \sum_{j=0}^m B_i^{2n-1}(t) B_j^{m}(u) d_{ij}, where
    // d_{ij} = \sum_{k+l=i, l\in\{0,...,n\}, k\in\{0,...,n-1\}} \frac{\binom{n}{l}\binom{n-1}{k}}{\binom{2n-1}{i}} n (P_{k+1} - P_k)^T(P_l - F_j)

    // as above, only that now F is one of the focus' control points.

    // Note the difference in the binomial coefficients to the
    // reference paper, these formulas most probably contained a typo.

    // To determine, where D(t,u) is _not_ zero (these are the parts
    // of the curve that don't share normals with the focus and can
    // thus be safely clipped away), we project D(u,t) onto the
    // (d(t,u), t) plane, determine the convex hull there and proceed
    // as for the curve intersection part (projection is orthogonal to
    // u axis, thus simply throw away u coordinate).

    // \fallfac are so-called falling factorials (see Concrete
    // Mathematics, p. 47 for a definition).

    // now, for tensor product bezier curves, the convex hull property
    // holds, too. Thus, we simply project the control points (t_{ij},
    // u_{ij}, d_{ij}) onto the (t,d) plane and calculate the
    // intersections of the convex hull with the t axis, as for the
    // bezier clipping case.

    // calc polygon of control points (t_{ij}, d_{ij}):

    const int n( 3 ); // cubic bezier curves, as a matter of fact
    const int i_card( 2*n );
    const int j_card( n + 1 );
    const int k_max( n-1 );
    Polygon2D controlPolygon( i_card*j_card ); // vector of (t_{ij}, d_{ij}) in row-major order

    int i, j, k, l; // variable notation from formulas above and Sederberg article
    Point2D::value_type d;
    for( i=0; i<i_card; ++i )
    {
        for( j=0; j<j_card; ++j )
        {
            // calc single d_{ij} sum:
            for( d=0.0, k=std::max(0,i-n); k<=k_max && k<=i; ++k )
            {
                l = i - k; // invariant: k + l = i
                assert(k>=0 && k<=n-1); // k \in {0,...,n-1}
                assert(l>=0 && l<=n);   // l \in {0,...,n}

                // TODO: find, document and assert proper limits for n and int's max_val.
                // This becomes important should anybody wants to use
                // this code for higher-than-cubic beziers
                d += static_cast<double>(fallFac(n,l)*fallFac(n-1,k)*fac(i)) /
                    static_cast<double>(fac(l)*fac(k) * fallFac(2*n-1,i)) * n *
                    ( (curve[k+1].x - curve[k].x)*(curve[l].x - focus[j].x) +   // dot product here
                      (curve[k+1].y - curve[k].y)*(curve[l].y - focus[j].y) );
            }

            // Note that the t_{ij} values are evenly spaced on the
            // [0,1] interval, thus t_{ij}=i/(2n-1)
            controlPolygon[ i*j_card + j ] = Point2D( i/(2.0*n-1.0), d );
        }
    }

#ifndef WITH_SAFEFOCUSPARAM_DETAILED_TEST

    // calc safe parameter range, to determine [0,t1] and [t2,1] where
    // no zero crossing is guaranteed.
    return Impl_calcSafeParams( t1, t2, controlPolygon, 0.0, 0.0 );

#else
    bool bRet( Impl_calcSafeParams( t1, t2, controlPolygon, 0.0, 0.0 ) );

    Polygon2D convHull( convexHull( controlPolygon ) );

    cout << "# convex hull testing (focus)" << endl
         << "plot [t=0:1] ";
    cout << "'-' using ($1):($2) title \"control polygon\" with lp, "
         << "'-' using ($1):($2) title \"convex hull\" with lp" << endl;

    unsigned int count;
    for( count=0; count<controlPolygon.size(); ++count )
    {
        cout << controlPolygon[count].x << " " << controlPolygon[count].y << endl;
    }
    cout << controlPolygon[0].x << " " << controlPolygon[0].y << endl;
    cout << "e" << endl;

    for( count=0; count<convHull.size(); ++count )
    {
        cout << convHull[count].x << " " << convHull[count].y << endl;
    }
    cout << convHull[0].x << " " << convHull[0].y << endl;
    cout << "e" << endl;

    return bRet;
#endif
}

/** Calc all values t_i on c1, for which safeRanges functor does not
    give a safe range on c1 and c2.

    This method is the workhorse of the bezier clipping. Because c1
    and c2 must be alternatingly tested against each other (first
    determine safe parameter interval on c1 with regard to c2, then
    the other way around), we call this method recursively with c1 and
    c2 swapped.

    @param result
    Output iterator where the final t values are added to. If curves
    don't intersect, nothing is added.

    @param delta
    Maximal allowed distance to true critical point (measured in the
    original curve's coordinate system)

    @param safeRangeFunctor
    Functor object, that must provide the following operator():
    bool safeRangeFunctor( double& t1,
                           double& t2,
                           const Bezier& c1_orig,
                           const Bezier& c1_part,
                           const Bezier& c2_orig,
                           const Bezier& c2_part );
    This functor must calculate the safe ranges [0,t1] and [t2,1] on
    c1_orig, where c1_orig is 'safe' from c2_part. If the whole
    c1_orig is safe, false must be returned, true otherwise.
 */
template <class Functor> void Impl_applySafeRanges_rec( std::back_insert_iterator< std::vector< std::pair<double, double> > >&    result,
                                                        double                                                                          delta,
                                                        const Functor&                                                                  safeRangeFunctor,
                                                        int                                                                             recursionLevel,
                                                        const Bezier&                                                                   c1_orig,
                                                        const Bezier&                                                                   c1_part,
                                                        double                                                                          last_t1_c1,
                                                        double                                                                          last_t2_c1,
                                                        const Bezier&                                                                   c2_orig,
                                                        const Bezier&                                                                   c2_part,
                                                        double                                                                          last_t1_c2,
                                                        double                                                                          last_t2_c2  )
{
    // check end condition
    // ===================

    // TODO: tidy up recursion handling. maybe put everything in a
    // struct and swap that here at method entry

    // TODO: Implement limit on recursion depth. Should that limit be
    // reached, chances are that we're on a higher-order tangency. For
    // this case, AW proposed to take the middle of the current
    // interval, and to correct both curve's tangents at that new
    // endpoint to be equal. That virtually generates a first-order
    // tangency, and justifies to return a single intersection
    // point. Otherwise, inside/outside test might fail here.

    for( int i=0; i<recursionLevel; ++i ) cerr << " ";
    if( recursionLevel % 2 )
    {
        cerr << "level: " << recursionLevel
             << " t: "
             << last_t1_c2 + (last_t2_c2 - last_t1_c2)/2.0
             << ", c1: " << last_t1_c2 << " " << last_t2_c2
             << ", c2: " << last_t1_c1 << " " << last_t2_c1
             << endl;
    }
    else
    {
        cerr << "level: " << recursionLevel
             << " t: "
             << last_t1_c1 + (last_t2_c1 - last_t1_c1)/2.0
             << ", c1: " << last_t1_c1 << " " << last_t2_c1
             << ", c2: " << last_t1_c2 << " " << last_t2_c2
             << endl;
    }

    // refine solution
    // ===============

    double t1_c1, t2_c1;

    // Note: we first perform the clipping and only test for precision
    // sufficiency afterwards, since we want to exploit the fact that
    // Impl_calcClipRange returns false if the curves don't
    // intersect. We would have to check that separately for the end
    // condition, otherwise.

    // determine safe range on c1_orig
    if( safeRangeFunctor( t1_c1, t2_c1, c1_orig, c1_part, c2_orig, c2_part ) )
    {
        // now, t1 and t2 are calculated on the original curve
        // (but against a fat line calculated from the subdivided
        // c2, namely c2_part). If the [t1,t2] range is outside
        // our current [last_t1,last_t2] range, we're done in this
        // branch - the curves no longer intersect.
        if( tolLessEqual(t1_c1, last_t2_c1) && tolGreaterEqual(t2_c1, last_t1_c1) )
        {
            // As noted above, t1 and t2 are calculated on the
            // original curve, but against a fat line
            // calculated from the subdivided c2, namely
            // c2_part. Our domain to work on is
            // [last_t1,last_t2], on the other hand, so values
            // of [t1,t2] outside that range are irrelevant
            // here. Clip range appropriately.
            t1_c1 = std::max(t1_c1, last_t1_c1);
            t2_c1 = std::min(t2_c1, last_t2_c1);

            // TODO: respect delta
            // for now, end condition is just a fixed threshold on the t's

            // check end condition
            // ===================

#if 1
            if( fabs(last_t2_c1 - last_t1_c1) < 0.0001 &&
                fabs(last_t2_c2 - last_t1_c2) < 0.0001  )
#else
            if( fabs(last_t2_c1 - last_t1_c1) < 0.01 &&
                fabs(last_t2_c2 - last_t1_c2) < 0.01    )
#endif
            {
                // done. Add to result
                if( recursionLevel % 2 )
                {
                    // uneven level: have to swap the t's, since curves are swapped, too
                    *result++ = std::make_pair( last_t1_c2 + (last_t2_c2 - last_t1_c2)/2.0,
                                                  last_t1_c1 + (last_t2_c1 - last_t1_c1)/2.0 );
                }
                else
                {
                    *result++ = std::make_pair( last_t1_c1 + (last_t2_c1 - last_t1_c1)/2.0,
                                                  last_t1_c2 + (last_t2_c2 - last_t1_c2)/2.0 );
                }

#if 0
                //printResultWithFinalCurves( c1_orig, c1_part, c2_orig, c2_part, last_t1_c1, last_t2_c1 );
                printResultWithFinalCurves( c1_orig, c1_part, c2_orig, c2_part, t1_c1, t2_c1 );
#else
                // calc focus curve of c2
                Bezier focus;
                Impl_calcFocus(focus, c2_part); // need to use subdivided c2

                safeRangeFunctor( t1_c1, t2_c1, c1_orig, c1_part, c2_orig, c2_part );

                //printResultWithFinalCurves( c1_orig, c1_part, c2_orig, focus, t1_c1, t2_c1 );
                printResultWithFinalCurves( c1_orig, c1_part, c2_orig, focus, last_t1_c1, last_t2_c1 );
#endif
            }
            else
            {
                // heuristic: if parameter range is not reduced by at least
                // 20%, subdivide longest curve, and clip shortest against
                // both parts of longest
//                if( (last_t2_c1 - last_t1_c1 - t2_c1 + t1_c1) / (last_t2_c1 - last_t1_c1) < 0.2 )
                if( false )
                {
                    // subdivide and descend
                    // =====================

                    Bezier part1;
                    Bezier part2;

                    double intervalMiddle;

                    if( last_t2_c1 - last_t1_c1 > last_t2_c2 - last_t1_c2 )
                    {
                        // subdivide c1
                        // ============

                        intervalMiddle = last_t1_c1 + (last_t2_c1 - last_t1_c1)/2.0;

                        // subdivide at the middle of the interval (as
                        // we're not subdividing on the original
                        // curve, this simply amounts to subdivision
                        // at 0.5)
                        Impl_deCasteljauAt( part1, part2, c1_part, 0.5 );

                        // and descend recursively with swapped curves
                        Impl_applySafeRanges_rec( result, delta, safeRangeFunctor, recursionLevel+1,
                                                  c2_orig, c2_part, last_t1_c2, last_t2_c2,
                                                  c1_orig, part1, last_t1_c1, intervalMiddle );

                        Impl_applySafeRanges_rec( result, delta, safeRangeFunctor, recursionLevel+1,
                                                  c2_orig, c2_part, last_t1_c2, last_t2_c2,
                                                  c1_orig, part2, intervalMiddle, last_t2_c1 );
                    }
                    else
                    {
                        // subdivide c2
                        // ============

                        intervalMiddle = last_t1_c2 + (last_t2_c2 - last_t1_c2)/2.0;

                        // subdivide at the middle of the interval (as
                        // we're not subdividing on the original
                        // curve, this simply amounts to subdivision
                        // at 0.5)
                        Impl_deCasteljauAt( part1, part2, c2_part, 0.5 );

                        // and descend recursively with swapped curves
                        Impl_applySafeRanges_rec( result, delta, safeRangeFunctor, recursionLevel+1,
                                                  c2_orig, part1, last_t1_c2, intervalMiddle,
                                                  c1_orig, c1_part, last_t1_c1, last_t2_c1 );

                        Impl_applySafeRanges_rec( result, delta, safeRangeFunctor, recursionLevel+1,
                                                  c2_orig, part2, intervalMiddle, last_t2_c2,
                                                  c1_orig, c1_part, last_t1_c1, last_t2_c1 );
                    }
                }
                else
                {
                    // apply calculated clip
                    // =====================

                    // clip safe ranges off c1_orig
                    Bezier c1_part1;
                    Bezier c1_part2;
                    Bezier c1_part3;

                    // subdivide at t1_c1
                    Impl_deCasteljauAt( c1_part1, c1_part2, c1_orig, t1_c1 );

                    // subdivide at t2_c1. As we're working on
                    // c1_part2 now, we have to adapt t2_c1 since
                    // we're no longer in the original parameter
                    // interval. This is based on the following
                    // assumption: t2_new = (t2-t1)/(1-t1), which
                    // relates the t2 value into the new parameter
                    // range [0,1] of c1_part2.
                    Impl_deCasteljauAt( c1_part1, c1_part3, c1_part2, (t2_c1-t1_c1)/(1.0-t1_c1) );

                    // descend with swapped curves and c1_part1 as the
                    // remaining (middle) segment
                    Impl_applySafeRanges_rec( result, delta, safeRangeFunctor, recursionLevel+1,
                                              c2_orig, c2_part, last_t1_c2, last_t2_c2,
                                              c1_orig, c1_part1, t1_c1, t2_c1 );
                }
            }
        }
    }
}

struct ClipBezierFunctor
{
    bool operator()( double& t1_c1,
                     double& t2_c1,
                     const Bezier& c1_orig,
                     const Bezier& c1_part,
                     const Bezier& c2_orig,
                     const Bezier& c2_part ) const
    {
        return Impl_calcClipRange( t1_c1, t2_c1, c1_orig, c1_part, c2_orig, c2_part );
    }
};

struct BezierTangencyFunctor
{
    bool operator()( double& t1_c1,
                     double& t2_c1,
                     const Bezier& c1_orig,
                     const Bezier& c1_part,
                     const Bezier& c2_orig,
                     const Bezier& c2_part ) const
    {
        // calc focus curve of c2
        Bezier focus;
        Impl_calcFocus(focus, c2_part); // need to use subdivided c2
                                        // here, as the whole curve is
                                        // used for focus calculation

        // determine safe range on c1_orig
        bool bRet( Impl_calcSafeParams_focus( t1_c1, t2_c1,
                                              c1_orig, // use orig curve here, need t's on original curve
                                              focus ) );

        cerr << "range: " << t2_c1 - t1_c1 << ", ret: " << bRet << endl;

        return bRet;
    }
};

/** Perform a bezier clip (curve against curve)

    @param result
    Output iterator where the final t values are added to. This
    iterator will remain empty, if there are no intersections.

    @param delta
    Maximal allowed distance to true intersection (measured in the
    original curve's coordinate system)
 */
void clipBezier( std::back_insert_iterator< std::vector< std::pair<double, double> > >&   result,
                 double                                                                         delta,
                 const Bezier&                                                                  c1,
                 const Bezier&                                                                  c2        )
{
#if 0
    // first of all, determine list of collinear normals. Collinear
    // normals typically separate two intersections, thus, subdivide
    // at all collinear normal's t values beforehand. This will cater
    // for tangent intersections, where two or more intersections are
    // infinitesimally close together.

    // TODO: evaluate effects of higher-than-second-order
    // tangencies. Sederberg et al. state that collinear normal
    // algorithm then degrades quickly.

    std::vector< std::pair<double,double> > results;
    std::back_insert_iterator< std::vector< std::pair<double, double> > > ii(results);

    Impl_calcCollinearNormals( ii, delta, 0, c1, c1, 0.0, 1.0, c2, c2, 0.0, 1.0 );

    // As Sederberg's collinear normal theorem is only sufficient, not
    // necessary for two intersections left and right, we've to test
    // all segments generated by the collinear normal algorithm
    // against each other. In other words, if the two curves are both
    // divided in a left and a right part, the collinear normal
    // theorem does _not_ state that the left part of curve 1 does not
    // e.g. intersect with the right part of curve 2.

    // divide c1 and c2 at collinear normal intersection points
    std::vector< Bezier > c1_segments( results.size()+1 );
    std::vector< Bezier > c2_segments( results.size()+1 );
    Bezier c1_remainder( c1 );
    Bezier c2_remainder( c2 );
    unsigned int i;
    for( i=0; i<results.size(); ++i )
    {
        Bezier c1_part2;
        Impl_deCasteljauAt( c1_segments[i], c1_part2, c1_remainder, results[i].first );
        c1_remainder = c1_part2;

        Bezier c2_part2;
        Impl_deCasteljauAt( c2_segments[i], c2_part2, c2_remainder, results[i].second );
        c2_remainder = c2_part2;
    }
    c1_segments[i] = c1_remainder;
    c2_segments[i] = c2_remainder;

    // now, c1/c2_segments contain all segments, then
    // clip every resulting segment against every other
    unsigned int c1_curr, c2_curr;
    for( c1_curr=0; c1_curr<c1_segments.size(); ++c1_curr )
    {
        for( c2_curr=0; c2_curr<c2_segments.size(); ++c2_curr )
        {
            if( c1_curr != c2_curr )
            {
                Impl_clipBezier_rec(result, delta, 0,
                                    c1_segments[c1_curr], c1_segments[c1_curr],
                                    0.0, 1.0,
                                    c2_segments[c2_curr], c2_segments[c2_curr],
                                    0.0, 1.0);
            }
        }
    }
#else
    Impl_applySafeRanges_rec( result, delta, BezierTangencyFunctor(), 0, c1, c1, 0.0, 1.0, c2, c2, 0.0, 1.0 );
    //Impl_applySafeRanges_rec( result, delta, ClipBezierFunctor(), 0, c1, c1, 0.0, 1.0, c2, c2, 0.0, 1.0 );
#endif
    // that's it, boys'n'girls!
}

int main(int argc, const char *argv[])
{
    double curr_Offset( 0 );
    unsigned int i,j,k;<--- Shadowed declaration

    Bezier someCurves[] =
        {
//            {Point2D(0.0,0.0),Point2D(0.0,1.0),Point2D(1.0,1.0),Point2D(1.0,0.0)},
//            {Point2D(0.0,0.0),Point2D(0.0,1.0),Point2D(1.0,1.0),Point2D(1.0,0.5)},
//            {Point2D(1.0,0.0),Point2D(0.0,0.0),Point2D(0.0,1.0),Point2D(1.0,1.0)}
//            {Point2D(0.25+1,0.5),Point2D(0.25+1,0.708333),Point2D(0.423611+1,0.916667),Point2D(0.770833+1,0.980324)},
//            {Point2D(0.0+1,0.0),Point2D(0.0+1,1.0),Point2D(1.0+1,1.0),Point2D(1.0+1,0.5)}

// tangency1
//            {Point2D(0.627124+1,0.828427),Point2D(0.763048+1,0.828507),Point2D(0.885547+1,0.77312),Point2D(0.950692+1,0.67325)},
//            {Point2D(0.0,1.0),Point2D(0.1,1.0),Point2D(0.4,1.0),Point2D(0.5,1.0)}

//            {Point2D(0.0,0.0),Point2D(0.0,1.0),Point2D(1.0,1.0),Point2D(1.0,0.5)},
//            {Point2D(0.60114,0.933091),Point2D(0.69461,0.969419),Point2D(0.80676,0.992976),Point2D(0.93756,0.998663)}
//            {Point2D(1.0,0.0),Point2D(0.0,0.0),Point2D(0.0,1.0),Point2D(1.0,1.0)},
//            {Point2D(0.62712,0.828427),Point2D(0.76305,0.828507),Point2D(0.88555,0.77312),Point2D(0.95069,0.67325)}

// clipping1
//            {Point2D(0.0,0.0),Point2D(0.0,3.5),Point2D(1.0,-2.5),Point2D(1.0,1.0)},
//            {Point2D(0.0,1.0),Point2D(3.5,1.0),Point2D(-2.5,0.0),Point2D(1.0,0.0)}

// tangency2
//            {Point2D(0.0,1.0),Point2D(3.5,1.0),Point2D(-2.5,0.0),Point2D(1.0,0.0)},
//            {Point2D(15.3621,0.00986464),Point2D(15.3683,0.0109389),Point2D(15.3682,0.0109315),Point2D(15.3621,0.00986464)}

// tangency3
//            {Point2D(1.0,0.0),Point2D(0.0,0.0),Point2D(0.0,1.0),Point2D(1.0,1.0)},
//            {Point2D(-0.5,0.0),Point2D(0.5,0.0),Point2D(0.5,1.0),Point2D(-0.5,1.0)}

// tangency4
//            {Point2D(-0.5,0.0),Point2D(0.5,0.0),Point2D(0.5,1.0),Point2D(-0.5,1.0)},
//            {Point2D(0.26,0.4),Point2D(0.25,0.5),Point2D(0.25,0.5),Point2D(0.26,0.6)}

// tangency5
//            {Point2D(0.0,0.0),Point2D(0.0,3.5),Point2D(1.0,-2.5),Point2D(1.0,1.0)},
//            {Point2D(15.3621,0.00986464),Point2D(15.3683,0.0109389),Point2D(15.3682,0.0109315),Point2D(15.3621,0.00986464)}

// tangency6
//            {Point2D(0.0,0.0),Point2D(0.0,3.5),Point2D(1.0,-2.5),Point2D(1.0,1.0)},
//            {Point2D(15.3621,10.00986464),Point2D(15.3683,10.0109389),Point2D(15.3682,10.0109315),Point2D(15.3621,10.00986464)}

// tangency7
//            {Point2D(2.505,0.0),Point2D(2.505+4.915,4.300),Point2D(2.505+3.213,10.019),Point2D(2.505-2.505,10.255)},
//            {Point2D(15.3621,10.00986464),Point2D(15.3683,10.0109389),Point2D(15.3682,10.0109315),Point2D(15.3621,10.00986464)}

// tangency Sederberg example
            {Point2D(2.505,0.0),Point2D(2.505+4.915,4.300),Point2D(2.505+3.213,10.019),Point2D(2.505-2.505,10.255)},
            {Point2D(5.33+9.311,0.0),Point2D(5.33+9.311-13.279,4.205),Point2D(5.33+9.311-10.681,9.119),Point2D(5.33+9.311-2.603,10.254)}

// clipping2
//            {Point2D(-0.5,0.0),Point2D(0.5,0.0),Point2D(0.5,1.0),Point2D(-0.5,1.0)},
//            {Point2D(0.2575,0.4),Point2D(0.2475,0.5),Point2D(0.2475,0.5),Point2D(0.2575,0.6)}

//            {Point2D(0.0,0.1),Point2D(0.2,3.5),Point2D(1.0,-2.5),Point2D(1.1,1.2)},
//            {Point2D(0.0,1.0),Point2D(3.5,0.9),Point2D(-2.5,0.1),Point2D(1.1,0.2)}
//            {Point2D(0.0,0.1),Point2D(0.2,3.0),Point2D(1.0,-2.0),Point2D(1.1,1.2)},
//            {Point2D(0.627124+1,0.828427),Point2D(0.763048+1,0.828507),Point2D(0.885547+1,0.77312),Point2D(0.950692+1,0.67325)}
//            {Point2D(0.0,1.0),Point2D(3.0,0.9),Point2D(-2.0,0.1),Point2D(1.1,0.2)}
//            {Point2D(0.0,4.0),Point2D(0.1,5.0),Point2D(0.9,5.0),Point2D(1.0,4.0)},
//            {Point2D(0.0,0.0),Point2D(0.1,0.5),Point2D(0.9,0.5),Point2D(1.0,0.0)},
//            {Point2D(0.0,0.1),Point2D(0.1,1.5),Point2D(0.9,1.5),Point2D(1.0,0.1)},
//            {Point2D(0.0,-4.0),Point2D(0.1,-5.0),Point2D(0.9,-5.0),Point2D(1.0,-4.0)}
        };

    // output gnuplot setup
    cout << "#!/usr/bin/gnuplot -persist" << endl
         << "#" << endl
         << "# automatically generated by bezierclip, don't change!" << endl
         << "#" << endl
         << "set parametric" << endl
         << "bez(p,q,r,s,t) = p*(1-t)**3+q*3*(1-t)**2*t+r*3*(1-t)*t**2+s*t**3" << endl
         << "bezd(p,q,r,s,t) = 3*(q-p)*(1-t)**2+6*(r-q)*(1-t)*t+3*(s-r)*t**2" << endl
         << "pointmarkx(c,t) = c-0.03*t" << endl
         << "pointmarky(c,t) = c+0.03*t" << endl
         << "linex(a,b,c,t) = a*-c + t*-b" << endl
         << "liney(a,b,c,t) = b*-c + t*a" << endl << endl
         << "# end of setup" << endl << endl;

#ifdef WITH_CONVEXHULL_TEST
    // test convex hull algorithm
    const double convHull_xOffset( curr_Offset );
    curr_Offset += 20;
    cout << "# convex hull testing" << endl
         << "plot [t=0:1] ";
    for( i=0; i<sizeof(someCurves)/sizeof(Bezier); ++i )
    {
        Polygon2D aTestPoly(4);
        aTestPoly[0] = someCurves[i].p0;
        aTestPoly[1] = someCurves[i].p1;
        aTestPoly[2] = someCurves[i].p2;
        aTestPoly[3] = someCurves[i].p3;

        aTestPoly[0].x += convHull_xOffset;
        aTestPoly[1].x += convHull_xOffset;
        aTestPoly[2].x += convHull_xOffset;
        aTestPoly[3].x += convHull_xOffset;

        cout << " bez("
             << aTestPoly[0].x << ","
             << aTestPoly[1].x << ","
             << aTestPoly[2].x << ","
             << aTestPoly[3].x << ",t),bez("
             << aTestPoly[0].y << ","
             << aTestPoly[1].y << ","
             << aTestPoly[2].y << ","
             << aTestPoly[3].y << ",t), '-' using ($1):($2) title \"convex hull " << i << "\" with lp";

        if( i+1<sizeof(someCurves)/sizeof(Bezier) )
            cout << ",\\" << endl;
        else
            cout << endl;
    }
    for( i=0; i<sizeof(someCurves)/sizeof(Bezier); ++i )
    {
        Polygon2D aTestPoly(4);
        aTestPoly[0] = someCurves[i].p0;
        aTestPoly[1] = someCurves[i].p1;
        aTestPoly[2] = someCurves[i].p2;
        aTestPoly[3] = someCurves[i].p3;

        aTestPoly[0].x += convHull_xOffset;
        aTestPoly[1].x += convHull_xOffset;
        aTestPoly[2].x += convHull_xOffset;
        aTestPoly[3].x += convHull_xOffset;

        Polygon2D convHull( convexHull(aTestPoly) );

        for( k=0; k<convHull.size(); ++k )
        {
            cout << convHull[k].x << " " << convHull[k].y << endl;
        }
        cout << convHull[0].x << " " << convHull[0].y << endl;
        cout << "e" << endl;
    }
#endif

#ifdef WITH_MULTISUBDIVIDE_TEST
    // test convex hull algorithm
    const double multiSubdivide_xOffset( curr_Offset );
    curr_Offset += 20;
    cout << "# multi subdivide testing" << endl
         << "plot [t=0:1] ";
    for( i=0; i<sizeof(someCurves)/sizeof(Bezier); ++i )
    {
        Bezier c( someCurves[i] );
        Bezier c1_part1;
        Bezier c1_part2;
        Bezier c1_part3;

        c.p0.x += multiSubdivide_xOffset;
        c.p1.x += multiSubdivide_xOffset;
        c.p2.x += multiSubdivide_xOffset;
        c.p3.x += multiSubdivide_xOffset;

        const double t1( 0.1+i/(3.0*sizeof(someCurves)/sizeof(Bezier)) );
        const double t2( 0.9-i/(3.0*sizeof(someCurves)/sizeof(Bezier)) );

        // subdivide at t1
        Impl_deCasteljauAt( c1_part1, c1_part2, c, t1 );

        // subdivide at t2_c1. As we're working on
        // c1_part2 now, we have to adapt t2_c1 since
        // we're no longer in the original parameter
        // interval. This is based on the following
        // assumption: t2_new = (t2-t1)/(1-t1), which
        // relates the t2 value into the new parameter
        // range [0,1] of c1_part2.
        Impl_deCasteljauAt( c1_part1, c1_part3, c1_part2, (t2-t1)/(1.0-t1) );

        // subdivide at t2
        Impl_deCasteljauAt( c1_part3, c1_part2, c, t2 );

        cout << " bez("
             << c1_part1.p0.x << ","
             << c1_part1.p1.x << ","
             << c1_part1.p2.x << ","
             << c1_part1.p3.x << ",t), bez("
             << c1_part1.p0.y+0.01 << ","
             << c1_part1.p1.y+0.01 << ","
             << c1_part1.p2.y+0.01 << ","
             << c1_part1.p3.y+0.01 << ",t) title \"middle " << i << "\", "
             << " bez("
             << c1_part2.p0.x << ","
             << c1_part2.p1.x << ","
             << c1_part2.p2.x << ","
             << c1_part2.p3.x << ",t), bez("
             << c1_part2.p0.y << ","
             << c1_part2.p1.y << ","
             << c1_part2.p2.y << ","
             << c1_part2.p3.y << ",t) title \"right " << i << "\", "
             << " bez("
             << c1_part3.p0.x << ","
             << c1_part3.p1.x << ","
             << c1_part3.p2.x << ","
             << c1_part3.p3.x << ",t), bez("
             << c1_part3.p0.y << ","
             << c1_part3.p1.y << ","
             << c1_part3.p2.y << ","
             << c1_part3.p3.y << ",t) title \"left " << i << "\"";

        if( i+1<sizeof(someCurves)/sizeof(Bezier) )
            cout << ",\\" << endl;
        else
            cout << endl;
    }
#endif

#ifdef WITH_FATLINE_TEST
    // test fatline algorithm
    const double fatLine_xOffset( curr_Offset );
    curr_Offset += 20;
    cout << "# fat line testing" << endl
         << "plot [t=0:1] ";
    for( i=0; i<sizeof(someCurves)/sizeof(Bezier); ++i )
    {
        Bezier c( someCurves[i] );

        c.p0.x += fatLine_xOffset;
        c.p1.x += fatLine_xOffset;
        c.p2.x += fatLine_xOffset;
        c.p3.x += fatLine_xOffset;

        FatLine line;

        Impl_calcFatLine(line, c);

        cout << " bez("
             << c.p0.x << ","
             << c.p1.x << ","
             << c.p2.x << ","
             << c.p3.x << ",t), bez("
             << c.p0.y << ","
             << c.p1.y << ","
             << c.p2.y << ","
             << c.p3.y << ",t) title \"bezier " << i << "\", linex("
             << line.a << ","
             << line.b << ","
             << line.c << ",t), liney("
             << line.a << ","
             << line.b << ","
             << line.c << ",t) title \"fat line (center) on " << i << "\", linex("
             << line.a << ","
             << line.b << ","
             << line.c-line.dMin << ",t), liney("
             << line.a << ","
             << line.b << ","
             << line.c-line.dMin << ",t) title \"fat line (min) on " << i << "\", linex("
             << line.a << ","
             << line.b << ","
             << line.c-line.dMax << ",t), liney("
             << line.a << ","
             << line.b << ","
             << line.c-line.dMax << ",t) title \"fat line (max) on " << i << "\"";

        if( i+1<sizeof(someCurves)/sizeof(Bezier) )
            cout << ",\\" << endl;
        else
            cout << endl;
    }
#endif

#ifdef WITH_CALCFOCUS_TEST
    // test focus curve algorithm
    const double focus_xOffset( curr_Offset );
    curr_Offset += 20;
    cout << "# focus line testing" << endl
         << "plot [t=0:1] ";
    for( i=0; i<sizeof(someCurves)/sizeof(Bezier); ++i )
    {
        Bezier c( someCurves[i] );

        c.p0.x += focus_xOffset;
        c.p1.x += focus_xOffset;
        c.p2.x += focus_xOffset;
        c.p3.x += focus_xOffset;

        // calc focus curve
        Bezier focus;
        Impl_calcFocus(focus, c);

        cout << " bez("
             << c.p0.x << ","
             << c.p1.x << ","
             << c.p2.x << ","
             << c.p3.x << ",t), bez("
             << c.p0.y << ","
             << c.p1.y << ","
             << c.p2.y << ","
             << c.p3.y << ",t) title \"bezier " << i << "\", bez("
             << focus.p0.x << ","
             << focus.p1.x << ","
             << focus.p2.x << ","
             << focus.p3.x << ",t), bez("
             << focus.p0.y << ","
             << focus.p1.y << ","
             << focus.p2.y << ","
             << focus.p3.y << ",t) title \"focus " << i << "\"";

        if( i+1<sizeof(someCurves)/sizeof(Bezier) )
            cout << ",\\" << endl;
        else
            cout << endl;
    }
#endif

#ifdef WITH_SAFEPARAMBASE_TEST
    // test safe params base method
    double safeParamsBase_xOffset( curr_Offset );
    cout << "# safe param base method testing" << endl
         << "plot [t=0:1] ";
    for( i=0; i<sizeof(someCurves)/sizeof(Bezier); ++i )
    {
        Bezier c( someCurves[i] );

        c.p0.x += safeParamsBase_xOffset;
        c.p1.x += safeParamsBase_xOffset;
        c.p2.x += safeParamsBase_xOffset;
        c.p3.x += safeParamsBase_xOffset;

        Polygon2D poly(4);
        poly[0] = c.p0;
        poly[1] = c.p1;
        poly[2] = c.p2;
        poly[3] = c.p3;

        double t1, t2;

        bool bRet( Impl_calcSafeParams( t1, t2, poly, 0, 1 ) );

        Polygon2D convHull( convexHull( poly ) );

        cout << " bez("
             << poly[0].x << ","
             << poly[1].x << ","
             << poly[2].x << ","
             << poly[3].x << ",t),bez("
             << poly[0].y << ","
             << poly[1].y << ","
             << poly[2].y << ","
             << poly[3].y << ",t), "
             << "t+" << safeParamsBase_xOffset << ", 0, "
             << "t+" << safeParamsBase_xOffset << ", 1, ";
        if( bRet )
        {
            cout << t1+safeParamsBase_xOffset << ", t, "
                 << t2+safeParamsBase_xOffset << ", t, ";
        }
        cout << "'-' using ($1):($2) title \"control polygon\" with lp, "
             << "'-' using ($1):($2) title \"convex hull\" with lp";

        if( i+1<sizeof(someCurves)/sizeof(Bezier) )
            cout << ",\\" << endl;
        else
            cout << endl;

        safeParamsBase_xOffset += 2;
    }

    safeParamsBase_xOffset = curr_Offset;
    for( i=0; i<sizeof(someCurves)/sizeof(Bezier); ++i )
    {
        Bezier c( someCurves[i] );

        c.p0.x += safeParamsBase_xOffset;
        c.p1.x += safeParamsBase_xOffset;
        c.p2.x += safeParamsBase_xOffset;
        c.p3.x += safeParamsBase_xOffset;

        Polygon2D poly(4);
        poly[0] = c.p0;
        poly[1] = c.p1;
        poly[2] = c.p2;
        poly[3] = c.p3;

        double t1, t2;

        Impl_calcSafeParams( t1, t2, poly, 0, 1 );

        Polygon2D convHull( convexHull( poly ) );

        unsigned int k;<--- Shadow variable
        for( k=0; k<poly.size(); ++k )
        {
            cout << poly[k].x << " " << poly[k].y << endl;
        }
        cout << poly[0].x << " " << poly[0].y << endl;
        cout << "e" << endl;

        for( k=0; k<convHull.size(); ++k )
        {
            cout << convHull[k].x << " " << convHull[k].y << endl;
        }
        cout << convHull[0].x << " " << convHull[0].y << endl;
        cout << "e" << endl;

        safeParamsBase_xOffset += 2;
    }
    curr_Offset += 20;
#endif

#ifdef WITH_SAFEPARAMS_TEST
    // test safe parameter range algorithm
    const double safeParams_xOffset( curr_Offset );
    curr_Offset += 20;
    cout << "# safe param range testing" << endl
         << "plot [t=0.0:1.0] ";
    for( i=0; i<sizeof(someCurves)/sizeof(Bezier); ++i )
    {
        for( j=i+1; j<sizeof(someCurves)/sizeof(Bezier); ++j )
        {
            Bezier c1( someCurves[i] );
            Bezier c2( someCurves[j] );

            c1.p0.x += safeParams_xOffset;
            c1.p1.x += safeParams_xOffset;
            c1.p2.x += safeParams_xOffset;
            c1.p3.x += safeParams_xOffset;
            c2.p0.x += safeParams_xOffset;
            c2.p1.x += safeParams_xOffset;
            c2.p2.x += safeParams_xOffset;
            c2.p3.x += safeParams_xOffset;

            double t1, t2;

            if( Impl_calcClipRange(t1, t2, c1, c1, c2, c2) )
            {
                // clip safe ranges off c1
                Bezier c1_part1;
                Bezier c1_part2;
                Bezier c1_part3;

                // subdivide at t1_c1
                Impl_deCasteljauAt( c1_part1, c1_part2, c1, t1 );
                // subdivide at t2_c1
                Impl_deCasteljauAt( c1_part1, c1_part3, c1_part2, (t2-t1)/(1.0-t1) );

                // output remaining segment (c1_part1)

                cout << " bez("
                     << c1.p0.x << ","
                     << c1.p1.x << ","
                     << c1.p2.x << ","
                     << c1.p3.x << ",t),bez("
                     << c1.p0.y << ","
                     << c1.p1.y << ","
                     << c1.p2.y << ","
                     << c1.p3.y << ",t), bez("
                     << c2.p0.x << ","
                     << c2.p1.x << ","
                     << c2.p2.x << ","
                     << c2.p3.x << ",t),bez("
                     << c2.p0.y << ","
                     << c2.p1.y << ","
                     << c2.p2.y << ","
                     << c2.p3.y << ",t), bez("
                     << c1_part1.p0.x << ","
                     << c1_part1.p1.x << ","
                     << c1_part1.p2.x << ","
                     << c1_part1.p3.x << ",t),bez("
                     << c1_part1.p0.y << ","
                     << c1_part1.p1.y << ","
                     << c1_part1.p2.y << ","
                     << c1_part1.p3.y << ",t)";

                if( i+2<sizeof(someCurves)/sizeof(Bezier) )
                    cout << ",\\" << endl;
                else
                    cout << endl;
            }
        }
    }
#endif

#ifdef WITH_SAFEPARAM_DETAILED_TEST
    // test safe parameter range algorithm
    const double safeParams2_xOffset( curr_Offset );
    curr_Offset += 20;
    if( sizeof(someCurves)/sizeof(Bezier) > 1 )
    {
        Bezier c1( someCurves[0] );
        Bezier c2( someCurves[1] );

        c1.p0.x += safeParams2_xOffset;
        c1.p1.x += safeParams2_xOffset;
        c1.p2.x += safeParams2_xOffset;
        c1.p3.x += safeParams2_xOffset;
        c2.p0.x += safeParams2_xOffset;
        c2.p1.x += safeParams2_xOffset;
        c2.p2.x += safeParams2_xOffset;
        c2.p3.x += safeParams2_xOffset;

        double t1, t2;

        // output happens here
        Impl_calcClipRange(t1, t2, c1, c1, c2, c2);
    }
#endif

#ifdef WITH_SAFEFOCUSPARAM_TEST
    // test safe parameter range from focus algorithm
    const double safeParamsFocus_xOffset( curr_Offset );
    curr_Offset += 20;
    cout << "# safe param range from focus testing" << endl
         << "plot [t=0.0:1.0] ";
    for( i=0; i<sizeof(someCurves)/sizeof(Bezier); ++i )
    {
        for( j=i+1; j<sizeof(someCurves)/sizeof(Bezier); ++j )
        {
            Bezier c1( someCurves[i] );
            Bezier c2( someCurves[j] );

            c1.p0.x += safeParamsFocus_xOffset;
            c1.p1.x += safeParamsFocus_xOffset;
            c1.p2.x += safeParamsFocus_xOffset;
            c1.p3.x += safeParamsFocus_xOffset;
            c2.p0.x += safeParamsFocus_xOffset;
            c2.p1.x += safeParamsFocus_xOffset;
            c2.p2.x += safeParamsFocus_xOffset;
            c2.p3.x += safeParamsFocus_xOffset;

            double t1, t2;

            Bezier focus;
#ifdef WITH_SAFEFOCUSPARAM_CALCFOCUS
#if 0
            {
                // clip safe ranges off c1_orig
                Bezier c1_part1;
                Bezier c1_part2;
                Bezier c1_part3;

                // subdivide at t1_c1
                Impl_deCasteljauAt( c1_part1, c1_part2, c2, 0.30204 );

                // subdivide at t2_c1. As we're working on
                // c1_part2 now, we have to adapt t2_c1 since
                // we're no longer in the original parameter
                // interval. This is based on the following
                // assumption: t2_new = (t2-t1)/(1-t1), which
                // relates the t2 value into the new parameter
                // range [0,1] of c1_part2.
                Impl_deCasteljauAt( c1_part1, c1_part3, c1_part2, (0.57151-0.30204)/(1.0-0.30204) );

                c2 = c1_part1;
                Impl_calcFocus( focus, c2 );
            }
#else
            Impl_calcFocus( focus, c2 );
#endif
#else
            focus = c2;
#endif
            // determine safe range on c1
            bool bRet( Impl_calcSafeParams_focus( t1, t2,
                                                  c1, focus ) );

            cerr << "t1: " << t1 << ", t2: " << t2 << endl;

            // clip safe ranges off c1
            Bezier c1_part1;
            Bezier c1_part2;
            Bezier c1_part3;

            // subdivide at t1_c1
            Impl_deCasteljauAt( c1_part1, c1_part2, c1, t1 );
            // subdivide at t2_c1
            Impl_deCasteljauAt( c1_part1, c1_part3, c1_part2, (t2-t1)/(1.0-t1) );

            // output remaining segment (c1_part1)

            cout << " bez("
                 << c1.p0.x << ","
                 << c1.p1.x << ","
                 << c1.p2.x << ","
                 << c1.p3.x << ",t),bez("
                 << c1.p0.y << ","
                 << c1.p1.y << ","
                 << c1.p2.y << ","
                 << c1.p3.y << ",t) title \"c1\", "
#ifdef WITH_SAFEFOCUSPARAM_CALCFOCUS
                 << "bez("
                 << c2.p0.x << ","
                 << c2.p1.x << ","
                 << c2.p2.x << ","
                 << c2.p3.x << ",t),bez("
                 << c2.p0.y << ","
                 << c2.p1.y << ","
                 << c2.p2.y << ","
                 << c2.p3.y << ",t) title \"c2\", "
                 << "bez("
                 << focus.p0.x << ","
                 << focus.p1.x << ","
                 << focus.p2.x << ","
                 << focus.p3.x << ",t),bez("
                 << focus.p0.y << ","
                 << focus.p1.y << ","
                 << focus.p2.y << ","
                 << focus.p3.y << ",t) title \"focus\"";
#else
                 << "bez("
                 << c2.p0.x << ","
                 << c2.p1.x << ","
                 << c2.p2.x << ","
                 << c2.p3.x << ",t),bez("
                 << c2.p0.y << ","
                 << c2.p1.y << ","
                 << c2.p2.y << ","
                 << c2.p3.y << ",t) title \"focus\"";
#endif
            if( bRet )
            {
                cout << ", bez("
                     << c1_part1.p0.x << ","
                     << c1_part1.p1.x << ","
                     << c1_part1.p2.x << ","
                     << c1_part1.p3.x << ",t),bez("
                     << c1_part1.p0.y+0.01 << ","
                     << c1_part1.p1.y+0.01 << ","
                     << c1_part1.p2.y+0.01 << ","
                     << c1_part1.p3.y+0.01 << ",t) title \"part\"";
            }

            if( i+2<sizeof(someCurves)/sizeof(Bezier) )
                cout << ",\\" << endl;
            else
                cout << endl;
        }
    }
#endif

#ifdef WITH_SAFEFOCUSPARAM_DETAILED_TEST
    // test safe parameter range algorithm
    const double safeParams3_xOffset( curr_Offset );
    curr_Offset += 20;
    if( sizeof(someCurves)/sizeof(Bezier) > 1 )
    {
        Bezier c1( someCurves[0] );
        Bezier c2( someCurves[1] );

        c1.p0.x += safeParams3_xOffset;
        c1.p1.x += safeParams3_xOffset;
        c1.p2.x += safeParams3_xOffset;
        c1.p3.x += safeParams3_xOffset;
        c2.p0.x += safeParams3_xOffset;
        c2.p1.x += safeParams3_xOffset;
        c2.p2.x += safeParams3_xOffset;
        c2.p3.x += safeParams3_xOffset;

        double t1, t2;

        Bezier focus;
#ifdef WITH_SAFEFOCUSPARAM_CALCFOCUS
        Impl_calcFocus( focus, c2 );
#else
        focus = c2;
#endif

        // determine safe range on c1, output happens here
        Impl_calcSafeParams_focus( t1, t2,
                                   c1, focus );
    }
#endif

#ifdef WITH_BEZIERCLIP_TEST
    std::vector< std::pair<double, double> >                                result;
    std::back_insert_iterator< std::vector< std::pair<double, double> > > ii(result);

    // test full bezier clipping
    const double bezierClip_xOffset( curr_Offset );
    cout << endl << endl << "# bezier clip testing" << endl
         << "plot [t=0:1] ";
    for( i=0; i<sizeof(someCurves)/sizeof(Bezier); ++i )
    {
        for( j=i+1; j<sizeof(someCurves)/sizeof(Bezier); ++j )
        {
            Bezier c1( someCurves[i] );
            Bezier c2( someCurves[j] );

            c1.p0.x += bezierClip_xOffset;
            c1.p1.x += bezierClip_xOffset;
            c1.p2.x += bezierClip_xOffset;
            c1.p3.x += bezierClip_xOffset;
            c2.p0.x += bezierClip_xOffset;
            c2.p1.x += bezierClip_xOffset;
            c2.p2.x += bezierClip_xOffset;
            c2.p3.x += bezierClip_xOffset;

            cout << " bez("
                 << c1.p0.x << ","
                 << c1.p1.x << ","
                 << c1.p2.x << ","
                 << c1.p3.x << ",t),bez("
                 << c1.p0.y << ","
                 << c1.p1.y << ","
                 << c1.p2.y << ","
                 << c1.p3.y << ",t), bez("
                 << c2.p0.x << ","
                 << c2.p1.x << ","
                 << c2.p2.x << ","
                 << c2.p3.x << ",t),bez("
                 << c2.p0.y << ","
                 << c2.p1.y << ","
                 << c2.p2.y << ","
                 << c2.p3.y << ",t), '-' using (bez("
                 << c1.p0.x << ","
                 << c1.p1.x << ","
                 << c1.p2.x << ","
                 << c1.p3.x
                 << ",$1)):(bez("
                 << c1.p0.y << ","
                 << c1.p1.y << ","
                 << c1.p2.y << ","
                 << c1.p3.y << ",$1)) title \"bezier " << i << " clipped against " << j << " (t on " << i << ")\", "
                 << " '-' using (bez("
                 << c2.p0.x << ","
                 << c2.p1.x << ","
                 << c2.p2.x << ","
                 << c2.p3.x
                 << ",$1)):(bez("
                 << c2.p0.y << ","
                 << c2.p1.y << ","
                 << c2.p2.y << ","
                 << c2.p3.y << ",$1)) title \"bezier " << i << " clipped against " << j << " (t on " << j << ")\"";

            if( i+2<sizeof(someCurves)/sizeof(Bezier) )
                cout << ",\\" << endl;
            else
                cout << endl;
        }
    }
    for( i=0; i<sizeof(someCurves)/sizeof(Bezier); ++i )
    {
        for( j=i+1; j<sizeof(someCurves)/sizeof(Bezier); ++j )
        {
            result.clear();
            Bezier c1( someCurves[i] );
            Bezier c2( someCurves[j] );

            c1.p0.x += bezierClip_xOffset;
            c1.p1.x += bezierClip_xOffset;
            c1.p2.x += bezierClip_xOffset;
            c1.p3.x += bezierClip_xOffset;
            c2.p0.x += bezierClip_xOffset;
            c2.p1.x += bezierClip_xOffset;
            c2.p2.x += bezierClip_xOffset;
            c2.p3.x += bezierClip_xOffset;

            clipBezier( ii, 0.00001, c1, c2 );

            for( k=0; k<result.size(); ++k )
            {
                cout << result[k].first << endl;
            }
            cout << "e" << endl;

            for( k=0; k<result.size(); ++k )
            {
                cout << result[k].second << endl;
            }
            cout << "e" << endl;
        }
    }
#endif

    return 0;
}

/* vim:set shiftwidth=4 softtabstop=4 expandtab: */