This file is indexed.

/usr/share/perl5/Math/PlanePath/DragonCurve.pm is in libmath-planepath-perl 117-1.

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
# Copyright 2011, 2012, 2013, 2014 Kevin Ryde

# This file is part of Math-PlanePath.
#
# Math-PlanePath is free software; 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, or (at your option) any later
# version.
#
# Math-PlanePath is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
# or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
# for more details.
#
# You should have received a copy of the GNU General Public License along
# with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.


# math-image --path=DragonCurve --lines --scale=20
# math-image --path=DragonCurve --all --scale=10
# math-image --path=DragonCurve --output=numbers_dash
#
# math-image --wx --path=DragonCurve,arms=4 --expression='i<16384?i:0'
#
# cf A088431 run lengths of dragon turns
#    A007400 cont frac 1/2^1 + 1/2^2 + 1/2^4 + 1/2^8 + ... 1/2^(2^n)
#            = 0.8164215090218931...
#    2,4,6 values
#    a(0)=0,
#    a(1)=1,
#    a(2)=4,
#    a(8n) = a(8n+3) = 2,
#    a(8n+4) = a(8n+7) = a(16n+5) = a(16n+14) = 4,
#    a(16n+6) = a(16n+13) = 6,
#    a(8n+1) = a(4n+1),
#    a(8n+2) = a(4n+2)
#
#    A060833 not adding to 2^k+2,
#            superset of positions of left turns ...
#
#    A166242 double or half according to dragon turn

#
# connection points
#   N = 26 = 11010
#     = 27 = 11011
#   N = 51 = 110011 = (3*16^k-1)/15 -> 1/5
#     = 52 = 110100
#   N = 101 = 1100101
#     = 102 = 1100110 = 2*(3*16^k-1)/15 -> 2/5
#   N


# Martin Gardner, "The Dragon Curve and Other Problems (Mathematical
# Games)", Scientific American, March 1967 (addenda from readers April and
# July).
#
# Reprinted in "Mathematical Magic Show", 1978.

# Chandler Davis and Donald Knuth,
# "Number Representations and Dragon Curves - I", C. Davis & D. E. Knuth,
# Journal Recreational Math., volume 3, number 2 (April 1970), pages 66-81.
# 16 pages
#
# Chandler Davis and Donald Knuth,
# "Number Representations and Dragon Curves - II", C. Davis & D. E. Knuth,
# Journal Recreational Math., volume 3, number 3 (July 1970), pages 133-149.
# 17 pages
#
# Revised in "Selected Papers on Fun and Games", 2010, pages 571-603 with
# addendum pages 603-614.
# http://trove.nla.gov.au/version/50039930
# 32+12=44 pages

# Sze-Man Ngai and Nhu Nguyen, "The Heighway Dragon Revisited", Discrete and
# Computational Geometry, May 2003 volume 29, issue 4, pages 603-623
# http://www.math.nmsu.edu/~nnguyen/23paper.ps

package Math::PlanePath::DragonCurve;
use 5.004;
use strict;
use List::Util 'min'; # 'max'
*max = \&Math::PlanePath::_max;

use vars '$VERSION', '@ISA';
$VERSION = 117;
use Math::PlanePath;
use Math::PlanePath::Base::NSEW;
@ISA = ('Math::PlanePath::Base::NSEW',
        'Math::PlanePath');

use Math::PlanePath::Base::Generic
  'is_infinite',
  'round_nearest';
use Math::PlanePath::Base::Digits
  'round_down_pow',
  'bit_split_lowtohigh',
  'digit_split_lowtohigh';
*_divrem_mutate = \&Math::PlanePath::_divrem_mutate;

use Math::PlanePath::DragonMidpoint;

# uncomment this to run the ### lines
# use Smart::Comments;



use constant n_start => 0;

use constant parameter_info_array => [ { name      => 'arms',
                                         share_key => 'arms_4',
                                         display   => 'Arms',
                                         type      => 'integer',
                                         minimum   => 1,
                                         maximum   => 4,
                                         default   => 1,
                                         width     => 1,
                                         description => 'Arms',
                                       } ];

{
  my @x_negative_at_n = (undef, 5,5,5,6);
  sub x_negative_at_n {
    my ($self) = @_;
    return $x_negative_at_n[$self->{'arms'}];
  }
}
{
  my @y_negative_at_n = (undef, 14,11,8,7);
  sub y_negative_at_n {
    my ($self) = @_;
    return $y_negative_at_n[$self->{'arms'}];
  }
}
{
  my @_UNDOCUMENTED__dxdy_list_at_n = (undef, 5, 5, 5, 3);
  sub _UNDOCUMENTED__dxdy_list_at_n {
    my ($self) = @_;
    return $_UNDOCUMENTED__dxdy_list_at_n[$self->{'arms'}];
  }
}

#------------------------------------------------------------------------------

sub new {
  my $self = shift->SUPER::new(@_);
  $self->{'arms'} = max(1, min(4, $self->{'arms'} || 1));
  return $self;
}

{
  # sub state_string {
  #   my ($state) = @_;
  #   my $digit = $state & 3;  $state >>= 2;
  #   my $rot = $state & 3;  $state >>= 2;
  #   my $rev = $state & 1;  $state >>= 1;
  #   return "rot=$rot rev=$rev (digit=$digit)";
  # }

  # generated by tools/dragon-curve-table.pl
  # next_state length 32
  my @next_state = (12,16, 4,16,  0,20, 8,20,  4,24,12,24,  8,28, 0,28,
                    0,20, 0,28,  4,24, 4,16,  8,28, 8,20, 12,16,12,24);
  my @digit_to_x = ( 0, 0, 1, 1,  0, 1, 1, 0,  0, 0,-1,-1,  0,-1,-1, 0,
                     0, 1, 1, 2,  0, 0,-1,-1,  0,-1,-1,-2,  0, 0, 1, 1);
  my @digit_to_y = ( 0,-1,-1, 0,  0, 0, 1, 1,  0, 1, 1, 0,  0, 0,-1,-1,
                     0, 0, 1, 1,  0, 1, 1, 2,  0, 0,-1,-1,  0,-1,-1,-2);
  my @digit_to_dxdy = ( 1, 0,undef,undef,  0, 1,undef,undef, -1, 0,undef,undef,  0,-1,undef,undef,
                        1, 0,undef,undef,  0, 1,undef,undef, -1, 0,undef,undef,  0,-1);

  sub n_to_xy {
    my ($self, $n) = @_;
    ### DragonCurve n_to_xy(): $n

    if ($n < 0) { return; }
    if (is_infinite($n)) { return ($n, $n); }

    my $int = int($n);      # integer part
    $n -= $int;             # $n = fraction part
    my $zero = ($int * 0);  # inherit bignum 0

    my $arm = _divrem_mutate ($int, $self->{'arms'});
    my @digits = digit_split_lowtohigh($int,4);
    ### @digits

    # initial state from rotation by arm and number of digits
    my $state = ((scalar(@digits) + $arm) & 3) << 2;

    my $len = (2+$zero) ** $#digits;
    my $x = $zero;
    my $y = $zero;
    foreach my $digit (reverse @digits) {  # high to low
      ### at: "x=$x,y=$y  len=$len digit=$digit state=$state"
      # ### state is: state_string($state)

      $state += $digit;
      $x += $len * $digit_to_x[$state];
      $y += $len * $digit_to_y[$state];
      $state = $next_state[$state];
      $len /= 2;
    }

    ### final: "x=$x y=$y  state=$state"
    # ### state is: state_string($state)
    ### final: "frac dx=$digit_to_dxdy[$state], dy=$digit_to_dxdy[$state+1]"

    return ($n * $digit_to_dxdy[$state] + $x,
            $n * $digit_to_dxdy[$state+1] + $y);
  }
}


{
  # generated by tools/dragon-curve-dxdy.pl
  # next_state length 32
  my @next_state = ( 0, 6,20, 2,  4,10,24, 6,  8,14,28,10, 12, 2,16,14,
                     0,22,20,18,  4,26,24,22,  8,30,28,26, 12,18,16,30);
  my @state_to_dxdy = ( 1, 0,-1, 1,  0, 1,-1,-1, -1, 0, 1,-1,  0,-1, 1, 1,
                        1, 0,-1,-1,  0, 1, 1,-1, -1, 0, 1, 1,  0,-1,-1, 1);

  sub n_to_dxdy {
    my ($self, $n) = @_;
    ### n_to_dxdy(): $n

    my $int = int($n);
    $n -= $int;  # $n fraction part
    ### $int
    ### $n

    my $state = 4 * _divrem_mutate ($int, $self->{'arms'});
    ### arm as initial state: $state

    foreach my $bit (reverse bit_split_lowtohigh($int)) {  # high to low
      $state = $next_state[$state + $bit];
    }
    $state &= 0x1C;  # mask out "prevbit" from state, leaving state==0 mod 4

    ### final state: $state
    ### dx: $state_to_dxdy[$state]
    ### dy: $state_to_dxdy[$state+1],
    ### frac dx: $state_to_dxdy[$state+2],
    ### frac dy: $state_to_dxdy[$state+3],

    return ($state_to_dxdy[$state]   + $n * $state_to_dxdy[$state+2],
            $state_to_dxdy[$state+1] + $n * $state_to_dxdy[$state+3]);
  }
}

# point N=2^(2k) at XorY=+/-2^k  radius 2^k
#       N=2^(2k-1) at X=Y=+/-2^(k-1) radius sqrt(2)*2^(k-1)
# radius = sqrt(2^level)
# R(l)-R(l-1) = sqrt(2^level) - sqrt(2^(level-1))
#             = sqrt(2^level) * (1 - 1/sqrt(2))
# about 0.29289
#
my @try_dx = (0,0,-1,-1);
my @try_dy = (0,1,1,0);

sub xy_to_n {
  return scalar((shift->xy_to_n_list(@_))[0]);
}
sub xy_to_n_list {
  my ($self, $x, $y) = @_;
  ### DragonCurve xy_to_n(): "$x, $y"

  $x = round_nearest($x);
  $y = round_nearest($y);

  if (is_infinite($x)) {
    return $x;  # infinity
  }
  if (is_infinite($y)) {
    return $y;  # infinity
  }

  if ($x == 0 && $y == 0) {
    return (0 .. $self->arms_count - 1);
  }

  my @n_list;
  my $xm = $x+$y;  # rotate -45 and mul sqrt(2)
  my $ym = $y-$x;
  foreach my $dx (0,-1) {
    foreach my $dy (0,1) {
      my $t = $self->Math::PlanePath::DragonMidpoint::xy_to_n
        ($xm+$dx, $ym+$dy);
      next unless defined $t;

      my ($tx,$ty) = $self->n_to_xy($t)
        or next;

      if ($tx == $x && $ty == $y) {
        ### found: $t
        if (@n_list && $t < $n_list[0]) {
          unshift @n_list, $t;
        } else {
          push @n_list, $t;
        }
        if (@n_list == 2) {
          return @n_list;
        }
      }
    }
  }
  return @n_list;
}

#------------------------------------------------------------------------------

sub xy_is_visited {
  my ($self, $x, $y) = @_;

  my $arms_count = $self->{'arms'};
  if ($arms_count == 4) {
    # yes, whole plane visited
    return 1;
  }

  my $xm = $x+$y;
  my $ym = $y-$x;
  {
    my $arm = Math::PlanePath::DragonMidpoint::_xy_to_arm($xm,$ym);
    if ($arm < $arms_count) {
      # yes, segment $xm,$ym is on the desired arms
      return 1;
    }
    if ($arm == 2 && $arms_count == 1) {
      # no, segment $xm,$ym is on arm 2, which means its opposite is only on
      # arm 1,2,3 not arm 0 so arms_count==1 cannot be visited
      return 0;
    }
  }
  return (Math::PlanePath::DragonMidpoint::_xy_to_arm($xm-1,$ym+1)
          < $arms_count);
}


#------------------------------------------------------------------------------

# f = (1 - 1/sqrt(2) = .292
# 1/f = 3.41
# N = 2^level
# Rend = sqrt(2)^level
# Rmin = Rend / 2  maybe
# Rmin^2 = (2^level)/4
# N = 4 * Rmin^2
#
# not exact
sub rect_to_n_range {
  my ($self, $x1,$y1, $x2,$y2) = @_;
  ### DragonCurve rect_to_n_range(): "$x1,$y1  $x2,$y2"
  my $xmax = int(max(abs($x1),abs($x2)));
  my $ymax = int(max(abs($y1),abs($y2)));
  return (0,
          $self->{'arms'} * ($xmax*$xmax + $ymax*$ymax + 1) * 7);
}

# Not quite right yet ...
#
# sub rect_to_n_range {
#   my ($self, $x1,$y1, $x2,$y2) = @_;
#   ### DragonCurve rect_to_n_range(): "$x1,$y1  $x2,$y2"
#
#
#    my ($length, $level_limit) = round_down_pow
#      ((max(abs($x1),abs($x2))**2 + max(abs($y1),abs($y2))**2 + 1) * 7,
#       2);
#    $level_limit += 2;
#    ### $level_limit
#
#    if (is_infinite($level_limit)) {
#      return ($level_limit,$level_limit);
#    }
#
#    $x1 = round_nearest ($x1);
#    $y1 = round_nearest ($y1);
#    $x2 = round_nearest ($x2);
#    $y2 = round_nearest ($y2);
#    ($x1,$x2) = ($x2,$x1) if $x1 > $x2;
#    ($y1,$y2) = ($y2,$y1) if $y1 > $y2;
#    ### sorted range: "$x1,$y1  $x2,$y2"
#
#
#    my @xend = (0, 1);
#    my @yend = (0, 0);
#    my @xmin = (0, 0);
#    my @xmax = (0, 1);
#    my @ymin = (0, 0);
#    my @ymax = (0, 0);
#    my @sidemax = (0, 1);
#    my $extend = sub {
#      my ($i) = @_;
#      ### extend(): $i
#      while ($i >= $#xend) {
#        ### extend from: $#xend
#        my $xend = $xend[-1];
#        my $yend = $yend[-1];
#        ($xend,$yend) = ($xend-$yend,  # rotate +45
#                         $xend+$yend);
#        push @xend, $xend;
#        push @yend, $yend;
#        my $xmax = $xmax[-1];
#        my $xmin = $xmin[-1];
#        my $ymax = $ymax[-1];
#        my $ymin = $ymin[-1];
#        ### assert: $xmax >= $xmin
#        ### assert: $ymax >= $ymin
#
#        #    ### at: "end=$xend,$yend   $xmin..$xmax  $ymin..$ymax"
#        push @xmax, max($xmax, $xend + $ymax);
#        push @xmin, min($xmin, $xend + $ymin);
#
#        push @ymax, max($ymax, $yend - $xmin);
#        push @ymin, min($ymin, $yend - $xmax);
#
#        push @sidemax, max ($xmax[-1], -$xmin[-1],
#                             $ymax[-1], -$ymin[-1],
#                             abs($xend),
#                             abs($yend));
#      }
#      ### @sidemax
#    };
#
#    my $rect_dist = sub {
#      my ($x,$y) = @_;
#      my $xd = ($x < $x1 ? $x1 - $x
#                : $x > $x2 ? $x - $x2
#                : 0);
#      my $yd = ($y < $y1 ? $y1 - $y
#                : $y > $y2 ? $y - $y2
#                : 0);
#      return max($xd,$yd);
#    };
#
#    my $arms = $self->{'arms'};
#    ### $arms
#    my $n_lo;
#    {
#      my $top = 0;
#      for (;;) {
#      ARM_LO: foreach my $arm (0 .. $arms-1) {
#          my $i = 0;
#          my @digits;
#          if ($top > 0) {
#            @digits = ((0)x($top-1), 1);
#          } else {
#            @digits = (0);
#          }
#
#          for (;;) {
#            my $n = 0;
#            foreach my $digit (reverse @digits) { # high to low
#              $n = 2*$n + $digit;
#            }
#            $n = $n*$arms + $arm;
#            my ($nx,$ny) = $self->n_to_xy($n);
#            my $nh = &$rect_dist ($nx,$ny);
#
#            ### lo consider: "i=$i  digits=".join(',',reverse @digits)."  is n=$n xy=$nx,$ny nh=$nh"
#
#            if ($i == 0 && $nh == 0) {
#              ### lo found inside: $n
#              if (! defined $n_lo || $n < $n_lo) {
#                $n_lo = $n;
#              }
#              next ARM_LO;
#            }
#
#            if ($i == 0 || $nh > $sidemax[$i+2]) {
#              ### too far away: "nxy=$nx,$ny   nh=$nh vs ".$sidemax[$i+2]." at i=$i"
#
#              while (++$digits[$i] > 1) {
#                $digits[$i] = 0;
#                if (++$i <= $top) {
#                  ### backtrack up ...
#                } else {
#                  ### not found within this top and arm, next arm ...
#                  next ARM_LO;
#                }
#              }
#            } else {
#              ### lo descend ...
#              ### assert: $i > 0
#              $i--;
#              $digits[$i] = 0;
#            }
#          }
#        }
#
#        # if an $n_lo was found on any arm within this $top then done
#        if (defined $n_lo) {
#          last;
#        }
#
#        ### lo extend top ...
#        if (++$top > $level_limit) {
#          ### nothing below level limit ...
#          return (1,0);
#        }
#        &$extend($top+3);
#      }
#    }
#
#    my $n_hi = 0;
#   ARM_HI: foreach my $arm (reverse 0 .. $arms-1) {
#      &$extend($level_limit+2);
#      my @digits = ((1) x $level_limit);
#      my $i = $#digits;
#      for (;;) {
#        my $n = 0;
#        foreach my $digit (reverse @digits) { # high to low
#          $n = 2*$n + $digit;
#        }
#
#        $n = $n*$arms + $arm;
#        my ($nx,$ny) = $self->n_to_xy($n);
#        my $nh = &$rect_dist ($nx,$ny);
#
#        ### hi consider: "arm=$arm  i=$i  digits=".join(',',reverse @digits)."  is n=$n xy=$nx,$ny nh=$nh"
#
#        if ($i == 0 && $nh == 0) {
#          ### hi found inside: $n
#          if ($n > $n_hi) {
#            $n_hi = $n;
#            next ARM_HI;
#          }
#        }
#
#        if ($i == 0 || $nh > $sidemax[$i+2]) {
#          ### too far away: "$nx,$ny   nh=$nh vs ".$sidemax[$i+2]." at i=$i"
#
#          while (--$digits[$i] < 0) {
#            $digits[$i] = 1;
#            if (++$i < $level_limit) {
#              ### hi backtrack up ...
#            } else {
#              ### hi nothing within level limit for this arm ...
#              next ARM_HI;
#            }
#          }
#
#        } else {
#          ### hi descend
#          ### assert: $i > 0
#          $i--;
#          $digits[$i] = 1;
#        }
#      }
#    }
#
#    if ($n_hi == 0) {
#      ### oops, lo found but hi not found
#      $n_hi = $n_lo;
#    }
#
#    return ($n_lo, $n_hi);
# }


#------------------------------------------------------------------------------
# level ranges

# eg. arms=3 0 .. 3*2^k    by 3s
#            1 .. 3*2^k+1  by 3s
#            2 .. 3*2^k+2  by 3s     k=3 n_hi=26
#
# eg. arms=4 3 .. 4*2^k+3  by 4s
#   k=1 n_hi=11
#   k=2 n_hi=19
#   k=3 n_hi=35
#
sub level_to_n_range {
  my ($self, $level) = @_;
  return (0, 2**$level * $self->{'arms'} + ($self->{'arms'}-1));
}
# 0 .. 2^level
# -1 .. 2^level-1
# level = round_up_pow(N)
# eg N=13 -> 2^4=16 level=4
# level = round_down_pow(N-1) + 1
#
sub n_to_level {
  my ($self, $n) = @_;
  if ($n < 0) { return undef; }
  if (is_infinite($n)) { return $n; }
  $n = round_nearest($n);
  _divrem_mutate ($n, $self->{'arms'});
  my ($pow, $exp) = round_down_pow ($n-1, 2);
  return $exp + 1;
}

#------------------------------------------------------------------------------

{
  my @_UNDOCUMENTED_level_to_left_line_boundary = (1,2,4);
  sub _UNDOCUMENTED_level_to_left_line_boundary {
    my ($self, $level) = @_;
    if ($level < 0) { return undef; }
    if ($level <= 2) { return $_UNDOCUMENTED_level_to_left_line_boundary[$level]; }
    if (is_infinite($level)) { return $level; }

    my $l0 = 2;
    my $l1 = 4;
    my $l2 = 8;
    foreach (4 .. $level) {
      ($l2,$l1,$l0) = ($l2 + 2*$l0, $l2, $l1);
    }
    return $l2;
  }
}

{
  my @level_to_right_line_boundary = (1,2,4,8,undef);
  sub _UNDOCUMENTED_level_to_right_line_boundary {
    my ($self, $level) = @_;
    if ($level < 0) { return undef; }
    if ($level <= 3) { return $level_to_right_line_boundary[$level]; }
    if (is_infinite($level)) { return $level; }

    my $r0 =  2;
    my $r1 =  4;
    my $r2 =  8;
    my $r3 = 16;
    foreach (5 .. $level) {
      ($r3,$r2,$r1,$r0) = (2*$r3 - $r2 + 2*$r1 - 2*$r0,  $r3, $r2, $r1);
    }
    return $r3;
  }
}
sub _UNDOCUMENTED_level_to_line_boundary {
  my ($self, $level) = @_;
  if ($level < 0) { return undef; }
  return $self->_UNDOCUMENTED_level_to_right_line_boundary($level+1);
}

sub _UNDOCUMENTED_level_to_u_left_line_boundary {
  my ($self, $level) = @_;
  if ($level < 0) { return undef; }
  return ($level == 0 ? 3
          : $self->_UNDOCUMENTED_level_to_right_line_boundary($level) + 4);
}
sub _UNDOCUMENTED_level_to_u_right_line_boundary {
  my ($self, $level) = @_;
  if ($level < 0) { return undef; }
  return ($self->_UNDOCUMENTED_level_to_right_line_boundary($level)
          + $self->_UNDOCUMENTED_level_to_right_line_boundary($level+1));
}
sub _UNDOCUMENTED_level_to_u_line_boundary {
  my ($self, $level) = @_;
  if ($level < 0) { return undef; }
  return ($self->_UNDOCUMENTED_level_to_u_left_line_boundary($level)
          + $self->_UNDOCUMENTED_level_to_u_right_line_boundary($level));
}

sub _UNDOCUMENTED_level_to_enclosed_area {
  my ($self, $level) = @_;
  # A[k] = 2^(k-1) - B[k]/4
  if ($level < 0) { return undef; }
  if ($level == 0) { return 0; } # avoid 2**(-1)
  return 2**($level-1) - $self->_UNDOCUMENTED_level_to_line_boundary($level) / 4;
}
*_UNDOCUMENTED_level_to_doubled_points = \&_UNDOCUMENTED_level_to_enclosed_area;

{
  my @_UNDOCUMENTED_level_to_single_points = (2,3,5);
  sub _UNDOCUMENTED_level_to_single_points {
    my ($self, $level) = @_;
    if ($level < 0) { return undef; }
    if ($level <= 2) { return $_UNDOCUMENTED_level_to_single_points[$level]; }
    if (is_infinite($level)) { return $level; }

    my $l0 = 3;
    my $l1 = 5;
    my $l2 = 9;
    foreach (4 .. $level) {
      ($l2,$l1,$l0) = ($l2 + 2*$l0, $l2, $l1);
    }
    return $l2;
  }
}

{
  my @_UNDOCUMENTED_level_to_enclosed_area_join = (0,0,0,1);
  sub _UNDOCUMENTED_level_to_enclosed_area_join {
    my ($self, $level) = @_;
    if ($level < 0) { return undef; }
    if ($level <= 3) { return $_UNDOCUMENTED_level_to_enclosed_area_join[$level]; }
    if (is_infinite($level)) { return $level; }

    my ($j0,$j1,$j2,$j3) = @_UNDOCUMENTED_level_to_enclosed_area_join;
    $j3 += $level*0;
    foreach (4 .. $level) {
      ($j3,$j2,$j1,$j0) = (2*$j3 - $j2 + 2*$j1 - 2*$j0,  $j3, $j2, $j1);
    }
    return $j3;
  }
}

#------------------------------------------------------------------------------
# points visited

{
  my @_UNDOCUMENTED_level_to_visited = (2, 3, 5, 9, 16);
  sub _UNDOCUMENTED_level_to_visited {
    my ($self, $level) = @_;

    if ($level < 0) { return undef; }
    if ($level <= $#_UNDOCUMENTED_level_to_visited) { return $_UNDOCUMENTED_level_to_visited[$level]; }
    if (is_infinite($level)) { return $level; }

    my ($p0,$p1,$p2,$p3,$p4) = @_UNDOCUMENTED_level_to_visited;
    foreach (5 .. $level) {
      ($p4,$p3,$p2,$p1,$p0) = (4*$p4 - 5*$p3 + 4*$p2 - 6*$p1 + 4*$p0,  $p4, $p3, $p2, $p1);
    }
    return $p4;
  }
}

#------------------------------------------------------------------------------
{
  my @_UNDOCUMENTED__n_segment_is_right_boundary
    #         R M A B C D F G H
    #         1 2 3 4 5 6 7 8 9
    = ([undef,1,3,1,6,7,9,3     ],
       [undef,2,4,5,4,8,5,0,0,4 ]);

  sub _UNDOCUMENTED__n_segment_is_right_boundary {
    my ($self, $n) = @_;
    if (is_infinite($n)) { return 0; }
    unless ($n >= 0) { return 0; }
    $n = int($n);

    my $state = 1;
    foreach my $bit (reverse bit_split_lowtohigh($n)) { # high to low
      $state = $_UNDOCUMENTED__n_segment_is_right_boundary[$bit][$state]
        || return 0;
    }
    return 1;
  }
}

1;
__END__


#------------------------------------------------------------------------------

# n_to_xy() old code based on i+1 multiply up.
#
# my @dir4_to_dx = (1,0,-1,0);
# my @dir4_to_dy = (0,1,0,-1);
#
# sub n_to_xy {
#   my ($self, $n) = @_;
#   ### DragonCurve n_to_xy(): $n
#
#   if ($n < 0) { return; }
#   if (is_infinite($n)) { return ($n, $n); }
#
#   my $int = int($n);   # integer part
#   $n -= $int;          # fraction part
#   my $zero = ($int * 0);  # inherit bignum 0
#
#   # arm as initial rotation
#   my $rot = _divrem_mutate ($int, $self->{'arms'});
#
#   my @digits = bit_split_lowtohigh($int);
#   ### @digits
#
#   my @sx;
#   my @sy;
#   {
#     my $sy = $zero;     # inherit BigInt 0
#     my $sx = $zero + 1; # inherit BigInt 1
#     ### $sx
#     ### $sy
#
#     foreach (@digits) {
#       push @sx, $sx;
#       push @sy, $sy;
#       # (sx,sy) + rot+90(sx,sy), is multiply (i+1)
#       ($sx,$sy) = ($sx - $sy,
#                    $sy + $sx);
#     }
#   }
#
#   my $rev = 0;
#   my $x = $zero;
#   my $y = $zero;
#   while (defined (my $digit = pop @digits)) {  # high to low
#     my $sx = pop @sx;
#     my $sy = pop @sy;
#     ### at: "$x,$y  $digit   side $sx,$sy"
#     ### $rot
#
#     if ($rot & 2) {
#       ($sx,$sy) = (-$sx,-$sy);
#     }
#     if ($rot & 1) {
#       ($sx,$sy) = (-$sy,$sx);
#     }
#
#     if ($rev) {
#       if ($digit) {
#         $x -= $sy;
#         $y += $sx;
#         ### rev add to: "$x,$y next is still rev"
#       } else {
#         $rot ++;
#         $rev = 0;
#       }
#     } else {
#       if ($digit) {
#         $rot ++;
#         $x += $sx;
#         $y += $sy;
#         $rev = 1;
#         ### add to: "$x,$y next is rev"
#       }
#     }
#   }
#
#   $rot &= 3;
#   $x = $n * $dir4_to_dx[$rot] + $x;
#   $y = $n * $dir4_to_dy[$rot] + $y;
#
#   ### final: "$x,$y"
#   return ($x,$y);
# }

# n_to_dxdy() by separate direction and frac next turn.
#
# my @dir4_to_dx = (1,0,-1,0);
# my @dir4_to_dy = (0,1,0,-1);
#
# sub n_to_dxdy {
#   my ($self, $n) = @_;
#   ### n_to_dxdy(): $n
#
#   my $int = int($n);
#   $n -= $int;  # $n fraction part
#   ### $int
#   ### $n
#
#   my $dir = _divrem_mutate ($int, $self->{'arms'});
#   ### arm as initial dir: $dir
#
#   my @digits = bit_split_lowtohigh($int);
#   ### @digits
#
#   my $prev = 0;
#   foreach my $digit (reverse @digits) {
#     $dir += ($digit != $prev);
#     $prev = $digit;
#   }
#   $dir &= 3;
#   my $dx = $dir4_to_dx[$dir];
#   my $dy = $dir4_to_dy[$dir];
#   ### $dx
#   ### $dy
#
#   if ($n) {
#     ### apply fraction part: $n
#
#     # maybe:
#     # +/- $n as dx or dy
#     # +/- (1-$n) as other dy or dx
#
#     # strip any low 1-bits, and the 0-bit above them
#     while (shift @digits) { }
#
#     $dir += ($digits[0] ? -1 : 1); # bit above lowest 0-bit, 1=right,0=left
#     $dir &= 3;
#     $dx += $n*($dir4_to_dx[$dir] - $dx);
#     $dy += $n*($dir4_to_dy[$dir] - $dy);
#
#     # my $sign = ($digits[0] ? 1 : -1); # bit above lowest 0-bit
#     # ($dx,$dy) = ($dx - $n*($dx - $sign*$dy),
#     #              $dy - $n*($dy + $sign*$dx));
#
#     # my ($next_dx, $next_dy);
#     # if ($digits[0]) {   # bit above lowest 0-bit
#     #   # right
#     #   $next_dx = $dy;
#     #   $next_dy = -$dx;
#     # } else {
#     #   # left
#     #   $next_dx = -$dy;
#     #   $next_dy = $dx;
#     # }
#     # ### $next_dx
#     # ### $next_dy
#     #
#     # $dx += $n*($next_dx - $dx);
#     # $dy += $n*($next_dy - $dy);
#   }
#
#   ### result: "$dx, $dy"
#   return ($dx,$dy);
# }

#------------------------------------------------------------------------------

=for stopwords eg Ryde Dragon Math-PlanePath Heighway Harter et al vertices doublings OEIS Online Jorg Arndt fxtbook versa Nlevel Nlevel-1 Xlevel,Ylevel lengthways Lmax Lmin Wmin Wmax Ns Shallit Kmosek Seminumerical dX,dY bitwise lookup dx dy ie Xmid,Ymid Xlevel Xmid

=head1 NAME

Math::PlanePath::DragonCurve -- dragon curve

=head1 SYNOPSIS

 use Math::PlanePath::DragonCurve;
 my $path = Math::PlanePath::DragonCurve->new;
 my ($x, $y) = $path->n_to_xy (123);

=head1 DESCRIPTION

This is the dragon or paper folding curve by Heighway, Harter, et al.

=cut

# math-image --path=DragonCurve --all --output=numbers_dash --size=70x30

=pod

                 9----8    5---4               2
                 |    |    |   |
                10--11,7---6   3---2           1
                      |            |
      17---16   13---12        0---1       <- Y=0
       |    |    |
      18-19,15-14,22-23                       -1
            |    |    |
           20--21,25-24                       -2
                 |
                26---27                       -3
                      |
         --32   29---28                       -4
            |    |
           31---30                            -5

       ^    ^    ^    ^    ^   ^   ^
      -5   -4   -3   -2   -1  X=0  1 ...

The curve visits "inside" X,Y points twice.  The first of these is X=-2,Y=1
which is N=7 and also N=11.  The segments N=6,7,8 and N=10,11,12 have
touched, but the path doesn't cross itself.  The doubled vertices are all
like this, touching but not crossing and no edges repeating.

=head2 Arms

The curve fills a quarter of the plane and four copies mesh together
perfectly when rotated by 90, 180 and 270 degrees.  The C<arms> parameter
can choose 1 to 4 curve arms successively advancing.

For example arms=4 begins as follows, with N=0,4,8,12,etc being the first
arm, N=1,5,9,13 the second, N=2,6,10,14 the third and N=3,7,11,15 the
fourth.

    arms => 4

            20 ------ 16
                       |
             9 ------5/12 -----  8       23
             |         |         |        |
    17 --- 13/6 --- 0/1/2/3 --- 4/15 --- 19
     |       |         |         |
    21      10 ----- 14/7 ----- 11
                       |
                      18 ------ 22

With four arms every X,Y point is visited twice (except the origin 0,0 where
all four begin) and every edge between the points is traversed once.

=head2 Level Angle

The first step N=1 is to the right along the X axis and the path then slowly
spirals anti-clockwise and progressively fatter.  The end of each
replication is N=2^level which is at level*45 degrees around,

    N       X,Y     angle   radial dist
   ----    -----    -----   -----------
     1      1,0        0         1
     2      1,1       45       sqrt(2)
     4      0,2       90       sqrt(4)=2
     8     -2,2      135       sqrt(8)
    16     -4,0      180       sqrt(16)=4
    32     -4,-4     225       sqrt(32)
   ...

Here's points N=0 to N=2^9=512.  "0" is the origin and "+" is N=512.  Notice
it's spiralled around full-circle to angle 45 degrees up again, like the
initial N=2.

                                    * *     * *
                                  * * *   * * *
                                  * * * * * * * * *
                                  * * * * * * * * *
                            * *   * * * *       * *
                          * * *   * * * *     + * *
                          * * * * * *         * *
                          * * * * * * *
                          * * * * * * * *
                              * * * * * *
                              * * * *
                                  * * * * * * *
                            * *   * * * * * * * *
                          * * *   * * * * * * * *
                          * * * * * * * * * *
                          * * * * * * * * * * * * * * *
                          * * * * * * * * * * * * * * * *
                              * * * * * * * * * * * * * *
                              * * * * * * * * * * * *
        * * * *                   * * * * * * * * * * *
        * * * * *           * *   * * * *       * * * * *
    * * * *   0 *         * * *   * * * *   * * * * * * *
    * * * *               * * * * * *       * * * * *
      * * *               * * * * * * *       * * * *
        * * * *     * *   * * * * * * * *
    * * * * * *   * * *   * * * * * * * *
    * * * * * * * * * * * * * * * * *
      * * * * * * * * * * * * * * * * *
                * * * * *       * * * * *
            * * * * * * *   * * * * * * *
            * * * * *       * * * * *
              * * * *         * * * *

At a power of two Nlevel=2^level for N=2 or higher, the curve always goes
upward from Nlevel-1 to Nlevel, and then goes to the left for Nlevel+1.  For
example at N=16 the curve goes up N=15 to N=16, then left for N=16 to N=17.
Likewise at N=32, etc.  The spiral curls around ever further but the
self-similar twist back means the Nlevel endpoint is always at this same
up/left orientation.  See L</Total Turn> below for the net direction in
general.

=head2 Level Ranges

The X,Y extents of the path through to Nlevel=2^level can be expressed as a
"length" in the direction of the Xlevel,Ylevel endpoint and a "width"
across.

    level even, so endpoint is a straight line
    k = level/2

       +--+      <- Lmax
       |  |
       |  E      <- Lend = 2^k at Nlevel=2^level
       |
       +-----+
             |
          O  |   <- Lstart=0
          |  |
          +--+   <- Lmin

       ^     ^
    Wmin     Wmax

    Lmax = (7*2^k - 4)/6 if k even
           (7*2^k - 2)/6 if k odd

    Lmin = - (2^k - 1)/3 if k even
           - (2^k - 2)/3 if k odd

    Wmax = (2*2^k - 2) / 3 if k even
           (2*2^k - 1) / 3 if k odd

    Wmin = Lmin

For example level=2 is to Nlevel=2^2=4 and k=level/2=1 is odd so it measures
as follows,

    4      <- Lmax = (7*2^1 - 2)/6 = 2
    |
    3--2
       |
    0--1   <- Lmin = -(2^1 - 2)/3 = 0

    ^  ^Wmax = (2*2^1 - 1)/3 = 1
    |
    Wmin = Lmin = 0

Or level=4 is to Nlevel=2^4=16 and k=4/2=2 is even.  It measures as follows.
The lengthways "L" measures are in the direction of the N=16 endpoint and
the "W" measures are across.

          9----8    5---4        <- Wmax = (2*2^2 - 2)/3 = 2
          |    |    |   |
         10--11,7---6   3---2
               |            |
    16   13---12        0---1
     |    |
    15---14                      <- Wmin = -(2^2 - 1)/3 = -1

     ^                      ^Lmin = Wmin = -1
     |
     Lmax = (7*2^2 - 4)/6 = 4

The formulas are all integer values, but the fractions 7/6, 1/3 and 2/3 show
the limits as the level increases.  If scaled so that length Lend=2^k is
reckoned as 1 unit then Lmax extends 1/6 past the end, Lmin and Wmin extend
1/3, and Wmax extends across 2/3.

    +--------+ --
    | -      | 1/6   total length
    || |     |          = 1/6+1+1/3 = 3/2
    || E     | --
    ||       |
    ||       |
    | \      |  1
    |  \     |
    |   --\  |
    |      \ |
    |       ||
    |  O    || --
    |  |    ||
    |  |    || 1/3
    |   ---- |
    +--------+ --
    1/3|  2/3

    total width = 1/3+2/3 = 1

=head2 Paper Folding

The path is called a paper folding curve because it can be generated by
thinking of a long strip of paper folded in half repeatedly and then
unfolded so each crease is a 90 degree angle.  The effect is that the curve
repeats in successive doublings turned by 90 degrees and reversed.

The first segment unfolds, pivoting at the "1",

                                          2
                                     ->   |
                     unfold         /     |
                      ===>         |      |
                                          |
    0-------1                     0-------1

Then the same again with that L shape, pivoting at the "2", then after that
pivoting at the "4", and so on.

                                 4
                                 |
                                 |
                                 |
                                 3--------2
           2                              |
           |        unfold          ^     |
           |         ===>            \_   |
           |                              |
    0------1                     0--------1

It can be shown that this unfolding doesn't overlap itself but the corners
may touch, such as at the X=-2,Y=1 etc noted above.

=head1 FUNCTIONS

See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.

=over 4

=item C<$path = Math::PlanePath::DragonCurve-E<gt>new ()>

=item C<$path = Math::PlanePath::DragonCurve-E<gt>new (arms =E<gt> $int)>

Create and return a new path object.

The optional C<arms> parameter can make 1 to 4 copies of the curve, each arm
successively advancing.

=item C<($x,$y) = $path-E<gt>n_to_xy ($n)>

Return the X,Y coordinates of point number C<$n> on the path.  Points begin
at 0 and if C<$n E<lt> 0> then the return is an empty list.

Fractional positions give an X,Y position along a straight line between the
integer positions.

=item C<$n = $path-E<gt>xy_to_n ($x,$y)>

Return the point number for coordinates C<$x,$y>.  If there's nothing at
C<$x,$y> then return C<undef>.

The curve visits an C<$x,$y> twice for various points (all the "inside"
points).  The smaller of the two N values is returned.

=item C<@n_list = $path-E<gt>xy_to_n_list ($x,$y)>

Return a list of N point numbers for coordinates C<$x,$y>.

The origin 0,0 has C<arms_count()> many N since it's the starting point for
each arm.  Other points have up to two Ns for a given C<$x,$y>.  If arms=4
then every C<$x,$y> has exactly two Ns.

=item C<$n = $path-E<gt>n_start()>

Return 0, the first N in the path.

=back

=head2 Level Methods

=over

=item C<($n_lo, $n_hi) = $path-E<gt>level_to_n_range($level)>

Return C<(0, 2**$level)>, or for multiple arms return C<(0, $arms *
2**$level + ($arms-1))>.

There are 2^level segments in a curve level, so 2^level+1 points numbered
from 0.  For multiple arms there are arms*(2^level+1) points, numbered from
0 so n_hi = arms*(2^level+1)-1.

=back

=head1 FORMULAS

Various formulas for boundary length and area can be found in the author's
mathematical write-up

=over

L<http://user42.tuxfamily.org/dragon/index.html>

=back

=head2 X,Y to N

The current code uses the C<DragonMidpoint> C<xy_to_n()> by rotating -45
degrees and offsetting to the midpoints of the four edges around the target
X,Y.  The C<DragonMidpoint> algorithm then gives four candidate N values and
those which convert back to the desired X,Y in the C<DragonCurve>
C<n_to_xy()> are the results for C<xy_to_n_list()>.

    Xmid,Ymid = X+Y, Y-X    # rotate -45 degrees
    for dx = 0 or -1
      for dy = 0 or 1
        N candidate = DragonMidpoint xy_to_n(Xmid+dx,Ymid+dy)

Since there's at most two C<DragonCurve> Ns at a given X,Y the loop can stop
when two Ns are found.

Only the "leaving" edges will convert back to the target N, so only two of
the four edges actually need to be considered.  Is there a way to identify
them?  For arm 1 and 3 the leaving edges are up,down on odd points (meaning
sum X+Y odd) and right,left for even points (meaning sum X+Y even).  But for
arm 2 and 4 it's the other way around.  Without an easy way to determine the
arm this doesn't seem to help.

=head2 X,Y is Visited

Whether a given X,Y is visited by the curve can be determined from one or
two segments (rather then up to four for X,Y to N).

            |             S midpoint Xmid = X+Y
            |                        Ymid = Y-X
    *---T--X,Y--S---*
            |             T midpoint Xmid-1
            |                        Ymid+1

Segment S is to the East of X,Y.  The arm it falls on can be determined as
per L<Math::PlanePath::DragonMidpoint/X,Y to N>.  Numbering arm(S) = 0,1,2,3
then

                                     X,Y Visited
                                     -----------
    if arms_count()==4                  yes     # whole plane
    if arm(S) < arms_count()            yes
    if arm(S)==2 and arms_count()==1    no
    if arm(T) < arms_count()            yes

This works because when two arms touch they approach and leave by a right
angle, without crossing.  So two opposite segments S and T identify the two
possible arms coming to the X,Y point.

           |
           |
            \
      ----   ----
          \
           |
           |

An arm only touches its immediate neighbour, ie. arm-1 or arm+1 mod 4.  This
means if arm(S)==2 then arm(T) can only be 1,2,3, not 0.  So if
C<arms_count()> is 1 then arm(T) cannot be on the curve and no need to run
its segment check.

The only exception to the right-angle touching rule is at the origin X,Y =
0,0.  In that case Xmid,Ymid = 0,0 is on the first arm and X,Y is correctly
determined to be on the curve.  If S was not to the East but some other
direction away from X,Y then this wouldn't be so.

=head2 Turn

At each point the curve always turns either left or right, it never goes
straight ahead.  The bit above the lowest 1-bit in N gives the turn
direction.

    N = 0b...z10000   (possibly no trailing 0s)

    z bit    Turn
    -----    ----
      0      left
      1      right

For example N=12 is binary 0b1100, the lowest 1 bit is 0b_1__ and the bit
above that is a 1, which means turn to the right.  Or N=18 is binary
0b10010, the lowest 1 is 0b___1_ and the bit above that is 0, so turn left
there.

This z bit can be picked out with some bit twiddling

    $mask = $n & -$n;          # lowest 1 bit, 000100..00
    $z = $n & ($mask << 1);    # the bit above it
    $turn = ($z == 0 ? 'left' : 'right');

This sequence is in Knuth volume 2 "Seminumerical Algorithms" answer to
section 4.5.3 question 41 and is called the "dragon sequence".  It's
expressed there recursively as

    d(0) = 1       # unused, since first turn at N=1
    d(2N) = d(N)   # shift down looking for low 1-bit
    d(4N+1) = 0    # bit above lowest 1-bit is 0
    d(4N+3) = 1    # bit above lowest 1-bit is 1

=head2 Next Turn

The bits also give the turn after next by looking at the bit above the
lowest 0-bit.  This works because 011..11 + 1 = 100..00 so the bit above the
lowest 0 becomes the bit above the lowest 1.

    N = 0b...w01111    (possibly no trailing 1s)

    w bit    Next Turn
    ----     ---------
      0       left
      1       right

For example at N=12=0b1100 the lowest 0 is the least significant bit 0b___0,
and above that is a 0 too, so at N=13 the turn is to the left.  Or for
N=18=0b10010 the lowest 0 is again the least significant bit, but above it
is a 1, so at N=19 the turn is to the right.

This too can be found with some bit twiddling, as for example

    $mask = $n ^ ($n+1);      # low one and below 000111..11
    $w = $n & ($mask + 1);    # the bit above there
    $turn = ($w == 0 ? 'left' : 'right');

=head2 Total Turn

The total turn is the count of 0E<lt>-E<gt>1 transitions in the runs of bits
of N, which is the same as how many bit pairs of N (including overlaps) are
different so "01" or "10".

This can be seen from the segment replacements resulting from bits of N,

    N bits from high to low, start in "plain" state

    plain state
     0 bit -> no change
     1 bit -> count left, and go to reversed state

    reversed state
     0 bit -> count left, and go to plain state
     1 bit -> no change

The 0 or 1 counts are from the different side a segment expands on in plain
or reversed state.  Segment A to B expands to an "L" shape bend which is on
the right in plain state, but on the left in reversed state.

      plain state             reverse state

      A = = = = B                    +
       \       ^              0bit  / \
        \     /               turn /   \ 1bit
    0bit \   / 1bit           left/     \
          \ /  turn              /       v
           +   left             A = = = = B

In both cases a rotate of +45 degrees keeps the very first segment of the
whole curve in a fixed direction (along the X axis), which means the
south-east slope shown is no-change.  This is the 0 of plain or the 1 of
reversed.  And the north-east slope which is the other new edge is a turn
towards the left.

It can be seen the "state" above is simply the previous bit, so the effect
for the bits of N is to count a left turn at each transition from 0-E<gt>1
or 1-E<gt>0.  Initial "plain" state means the infinite zero bits at the high
end of N are included.  For example N=9 is 0b1001 so three left turns for
curve direction south to N=10 (as can be seen in the diagram above).

     1 00 1   N=9
    ^ ^  ^
    +-+--+---three transitions,
             so three left turns for direction south

The transitions can also be viewed as a count of how many runs of contiguous
0s or 1s, up to the highest 1-bit.

    1 00 1   three blocks of 0s and 1s

X<Arndt, Jorg>X<fxtbook>This can be calculated by some bit twiddling with a
shift and xor to turn transitions into 1-bits which can then be counted, as
per Jorg Arndt (fxtbook section 1.31.3.1 "The Dragon Curve").

    total turn = count_1_bits ($n ^ ($n >> 1))

The reversing structure of the curve shows up in the total turn at each
point.  The total turns for a block of 2^N is followed by its own reversal
plus 1.  For example,

                    ------->
    N=0 to N=7    0, 1, 2, 1, 2, 3, 2, 1

    N=15 to N=8   1, 2, 3, 2, 3, 4, 3, 2    each is +1
                               <-------

=head2 N to dX,dY

C<n_to_dxdy()> is the "total turn" per above, or for fractional N then an
offset according to the "next turn" above.  If using the bit twiddling
operators described then the two can be calculated separately.

The current C<n_to_dxdy()> code tries to support floating point or other
number types without bitwise XOR etc by processing bits high to low with a
state table which combines the calculations for total turn and next turn.
The state encodes

    total turn       0 to 3
    next turn        0 or 1
    previous bit     0 or 1  (the bit above the current bit)

The "next turn" remembers the bit above lowest 0 seen so far (or 0
initially).  The "total turn" counts 0-E<gt>1 or 1-E<gt>0 transitions.  The
"previous bit" shows when there's a transition, or what bit is above when a
0 is seen.  It also works not to have this previous bit in the state but
instead pick out two bits each time.

At the end of bit processing any "previous bit" in state is no longer needed
and can be masked out to lookup the final four dx, dy, next dx, next dy.

=head1 OEIS

The Dragon curve is in Sloane's Online Encyclopedia of Integer Sequences in
many forms (and see C<DragonMidpoint> for its forms too),

=over

L<http://oeis.org/A014577> (etc)

=back

    A038189   turn, 0=left,1=right, bit above lowest 1, extra 0
    A089013    same as A038189, but initial extra 1
    A082410   turn, 1=left,0=right, reversing complement, extra 0
    A099545   turn, 1=left,3=right, as [odd part n] mod 4
               so turn by 90 degrees * 1 or 3
    A034947   turn, 1=left,-1=right, Jacobi (-1/n)
    A112347   turn, 1=left,-1=right, Kronecker (-1/n), extra 0
    A121238   turn, 1=left,-1=right, -1^(n + some partitions) extra 1
    A014577   next turn, 1=left,0=right
    A014707   next turn, 0=left,1=right
    A014709   next turn, 1=left,2=right
    A014710   next turn, 2=left,1=right

These numerous turn sequences differ only in having left or right
represented as 0, 1, -1, etc, and possibly "extra" initial 0 or 1 at n=0
arising from the definitions and the first turn being at n=N=1.  The "next
turn" forms begin at n=0 for turn at N=1 and so are the turn at N=n+1.

    A005811   total turn
    A088748   total turn + 1
    A164910   cumulative [total turn + 1]
    A166242   2^(total turn), by double/halving

    A088431   turn sequence run lengths
    A007400     2*runlength

    A091072   N positions of the left turns, being odd part form 4K+1
    A003460   turns N=1 to N=2^n-1 packed as bits 1=left,0=right
                low to high, then written in octal
    A126937   points numbered like SquareSpiral (start N=0 and flip Y)

    A146559   X at N=2^k, for k>=1, being Re((i+1)^k)
    A009545   Y at N=2^k, for k>=1, being Im((i+1)^k)

    A227036   boundary length N=0 to N=2^k
                also right boundary length to N=2^(k+1)
    A203175   left boundary length N=0 to N=2^k
                also differences of total boundary

    A003230   area enclosed N=0 to N=2^k, for k=4 up
               same as double points
    A003478   area enclosed by left side,
               also area increment
    A003477   area of each connected block

    A003479   join area between N=2^k replications
    A003229   join area increment,
               also area left side extra over doubling
    A077949    same

    A003476   squares on right boundary
               also single points N=0 to N=2^(k-1) inclusive
    A203175   squares on left boundary
    A164395   single points N=0 to N=2^k-1 inclusive, for k=4 up

For reference, "dragon-like" A059125 is similar to the turn sequence
A014707, but differs in having the "middle" values for each replication come
from successive values of the sequence itself, or some such.

=head2 A088431 and A007400

The run lengths A088431 and A007400 are from a continued fraction expansion
of an infinite sum

        1   1   1     1      1              1
    1 + - + - + -- + --- + ----- + ... + ------- + ...
        2   4   16   256   65536         2^(2^k)

X<Shallit, Jeffrey>X<Kmosek>Jeffrey Shallit and independently M. Kmosek show
how continued fraction terms repeated in reverse give rise to this sort of
power sum,

=over

Jeffrey Shallit, "Simple Continued Fractions for Some Irrational Numbers",
Journal of Number Theory, volume 11, 1979, pages 209-217.
L<http://www.cs.uwaterloo.ca/~shallit/papers.html>
L<http://www.cs.uwaterloo.ca/~shallit/Papers/scf.ps>

(And which appears in Knuth "Art of Computer Programming", volume 2, section
4.5.3 exercise 41.)

=cut

# M. Kmosek, "Rozwiniecie niektorych liczb niewymiernych na ulamki lancuchowe", Master's
# thesis, Uniwersytet Warszawski, 1979.  -- is that right?

=pod

=back

=head2 A126937

The A126937 C<SquareSpiral> numbering has the dragon curve and square
spiralling with their Y axes in opposite directions, as shown in its
F<a126937.pdf>.  So the dragon curve turns up towards positive Y but the
square spiral is numbered down towards negative Y (or vice versa).
C<PlanePath> code for this starting at C<$i=0> would be

      my $dragon = Math::PlanePath::DragonCurve->new;
      my $square = Math::PlanePath::SquareSpiral->new (n_start => 0);
      my ($x, $y) = $dragon->n_to_xy ($i);
      my $A126937_of_i = $square->xy_to_n ($x, -$y);

=head1 SEE ALSO

L<Math::PlanePath>,
L<Math::PlanePath::DragonRounded>,
L<Math::PlanePath::DragonMidpoint>,
L<Math::PlanePath::R5DragonCurve>,
L<Math::PlanePath::TerdragonCurve>

L<Math::PlanePath::ComplexMinus>,
L<Math::PlanePath::ComplexPlus>,
L<Math::PlanePath::CCurve>,
L<Math::PlanePath::AlternatePaper>

L<http://rosettacode.org/wiki/Dragon_curve>

=for comment http://wiki.tcl.tk/10745 recursive curves

=head1 HOME PAGE

L<http://user42.tuxfamily.org/math-planepath/index.html>

=head1 LICENSE

Copyright 2011, 2012, 2013, 2014 Kevin Ryde

Math-PlanePath is free software; 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, or (at your option) any later
version.

Math-PlanePath is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License for
more details.

You should have received a copy of the GNU General Public License along with
Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.

=cut