This file is indexed.

/usr/share/perl5/Math/PlanePath/SierpinskiCurve.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
# 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/>.


package Math::PlanePath::SierpinskiCurve;
use 5.004;
use strict;
use List::Util 'sum','first';
#use List::Util 'min','max';
*min = \&Math::PlanePath::_min;
*max = \&Math::PlanePath::_max;

use vars '$VERSION', '@ISA';
$VERSION = 117;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
*_divrem_mutate = \&Math::PlanePath::_divrem_mutate;

use Math::PlanePath::Base::Generic
  'is_infinite',
  'round_nearest';
use Math::PlanePath::Base::Digits
  'round_down_pow',
  'digit_split_lowtohigh';

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


use constant n_start => 0;

sub x_negative {
  my ($self) = @_;
  return ($self->{'arms'} >= 3);
}
sub y_negative {
  my ($self) = @_;
  return ($self->{'arms'} >= 5);
}

use constant parameter_info_array =>
  [
   { name        => 'arms',
     share_key   => 'arms_8',
     display     => 'Arms',
     type        => 'integer',
     minimum     => 1,
     maximum     => 8,
     default     => 1,
     width       => 1,
     description => 'Arms',
   },

   { name        => 'straight_spacing',
     display     => 'Straight Spacing',
     type        => 'integer',
     minimum     => 1,
     default     => 1,
     width       => 1,
     description => 'Spacing of the straight line points.',
   },
   { name        => 'diagonal_spacing',
     display     => 'Diagonal Spacing',
     type        => 'integer',
     minimum     => 1,
     default     => 1,
     width       => 1,
     description => 'Spacing of the diagonal points.',
   },
  ];

# Ntop = (4^level)/2 - 1
# Xtop = 3*2^(level-1) - 1
# fill = Ntop / (Xtop*(Xtop-1)/2)
#      -> 2 * ((4^level)/2 - 1) / (3*2^(level-1) - 1)^2
#      -> 2 * ((4^level)/2) / (3*2^(level-1))^2
#      =  4^level / (9*4^(level-1)
#      =  4/9 = 0.444

sub x_negative_at_n {
  my ($self) = @_;
  return $self->arms_count >= 3 ? 2 : undef;
}
sub y_negative_at_n {
  my ($self) = @_;
  return $self->arms_count >= 5 ? 4 : undef;
}

{
  # Note: shared by Math::PlanePath::SierpinskiCurveStair
  my @x_minimum = (undef,
                   1,  # 1 arm
                   0,  # 2 arms
                  );   # more than 2 arm, X goes negative
  sub x_minimum {
    my ($self) = @_;
    return $x_minimum[$self->arms_count];
  }
}
{
  # Note: shared by Math::PlanePath::SierpinskiCurveStair
  my @sumxy_minimum = (undef,
                       1,  # 1 arm, octant and X>=1 so X+Y>=1
                       1,  # 2 arms, X>=1 or Y>=1 so X+Y>=1
                       0,  # 3 arms, Y>=1 and X>=Y, so X+Y>=0
                      );   # more than 3 arm, Sum goes negative so undef
  sub sumxy_minimum {
    my ($self) = @_;
    return $sumxy_minimum[$self->arms_count];
  }
}
use constant sumabsxy_minimum => 1;

# Note: shared by Math::PlanePath::SierpinskiCurveStair
#                 Math::PlanePath::AlternatePaper
#                 Math::PlanePath::AlternatePaperMidpoint
sub diffxy_minimum {
  my ($self) = @_;
  return ($self->arms_count == 1
          ? 1       # octant Y<=X-1 so X-Y>=1
          : undef); # more than 1 arm, DiffXY goes negative
}
use constant absdiffxy_minimum => 1; # X=Y never occurs
use constant rsquared_minimum => 1; # minimum X=1,Y=0

sub dx_minimum {
  my ($self) = @_;
  return - max($self->{'straight_spacing'},
               $self->{'diagonal_spacing'});
}
*dy_minimum = \&dx_minimum;

sub dx_maximum {
  my ($self) = @_;
  return max($self->{'straight_spacing'},
             $self->{'diagonal_spacing'});
}
*dy_maximum = \&dx_maximum;

sub _UNDOCUMENTED__dxdy_list {
  my ($self) = @_;
  my $s = $self->{'straight_spacing'};
  my $d = $self->{'diagonal_spacing'};
  return ($s,0,                    # E     eight scaled
          ($d ? ( $d, $d) : ()),   # NE    except s=0
          ($s ? (  0, $s) : ()),   # N     or d=0 skips
          ($d ? (-$d, $d) : ()),   # NW
          ($s ? (-$s,  0) : ()),   # W
          ($d ? (-$d,-$d) : ()),   # SW
          ($s ? (  0,-$s) : ()),   # S
          ($d ? ( $d,-$d) : ()));  # SE

}
{
  my @_UNDOCUMENTED__dxdy_list_at_n = (undef,
                                       21, 20, 27, 36,
                                       29, 12, 12, 13);
  sub _UNDOCUMENTED__dxdy_list_at_n {
    my ($self) = @_;
    return $_UNDOCUMENTED__dxdy_list_at_n[$self->{'arms'}];
  }
}
sub dsumxy_minimum {
  my ($self) = @_;
  return - max($self->{'straight_spacing'},
               2*$self->{'diagonal_spacing'});
}
sub dsumxy_maximum {
  my ($self) = @_;
  return max($self->{'straight_spacing'},
             2*$self->{'diagonal_spacing'});
}
*ddiffxy_minimum = \&dsumxy_minimum;
*ddiffxy_maximum = \&dsumxy_maximum;

use constant dir_maximum_dxdy => (1,-1); # South-East


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

sub new {
  my $self = shift->SUPER::new(@_);

  $self->{'arms'} = max(1, min(8, $self->{'arms'} || 1));
  $self->{'straight_spacing'} ||= 1;
  $self->{'diagonal_spacing'} ||= 1;
  return $self;
}

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

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

  my $int = int($n); # BigFloat int() gives BigInt, use that
  $n -= $int;   # preserve possible BigFloat
  ### $int
  ### $n

  my $arm = _divrem_mutate ($int, $self->{'arms'});

  my $s = $self->{'straight_spacing'};
  my $d = $self->{'diagonal_spacing'};
  my $base = 2*$d+$s;
  my $x = my $y = ($int * 0);  # inherit big 0
  my $len = $x + $base;      # inherit big

  foreach my $digit (digit_split_lowtohigh($int,4)) {
    ### at: "$x,$y  digit=$digit"

    if ($digit == 0) {
      $x = $n*$d + $x;
      $y = $n*$d + $y;
      $n = 0;

    } elsif ($digit == 1) {
      ($x,$y) = ($n*$s - $y + $len-$d-$s,   # rotate +90
                 $x + $d);
      $n = 0;

    } elsif ($digit == 2) {
      # rotate -90
      ($x,$y) = ($n*$d + $y  + $len-$d,
                 -$n*$d - $x + $len-$d-$s);
      $n = 0;

    } else { # digit==3
      $x += $len;
    }
    $len *= 2;
  }

  # n=0 or n=33..33
  $x = $n*$d + $x;
  $y = $n*$d + $y;

  $x += 1;
  if ($arm & 1) {
    ($x,$y) = ($y,$x);   # mirror 45
  }
  if ($arm & 2) {
    ($x,$y) = (-1-$y,$x);   # rotate +90
  }
  if ($arm & 4) {
    $x = -1-$x;   # rotate 180
    $y = -1-$y;
  }

  # use POSIX 'floor';
  # $x += floor($x/3);
  # $y += floor($y/3);

  # $x += floor(($x-1)/3) + floor(($x-2)/3);
  # $y += floor(($y-1)/3) + floor(($y-2)/3);


  ### final: "$x,$y"
  return ($x,$y);
}

my @digit_to_dir = (0, -2, 2, 0);
my @dir8_to_dx = (1, 1, 0,-1, -1, -1,  0, 1);
my @dir8_to_dy = (0, 1, 1, 1,  0, -1, -1,-1);
my @digit_to_nextturn = (-1,   # after digit=0
                         2,    #       digit=1
                         -1);  #       digit=2
sub n_to_dxdy {
  my ($self, $n) = @_;
  ### n_to_dxdy(): $n

  if ($n < 0) {
    return;  # first direction at N=0
  }

  my $int = int($n);
  $n -= $int;

  my $arm = _divrem_mutate($int,$self->{'arms'});
  my $lowbit = _divrem_mutate($int,2);
  ### $lowbit
  ### $int

  if (is_infinite($int)) {
    return ($int,$int);
  }
  my @ndigits = digit_split_lowtohigh($int,4);
  ### @ndigits

  my $dir8 = sum(0, map {$digit_to_dir[$_]} @ndigits);
  if ($arm & 1) {
    $dir8 = - $dir8;  # mirrored on second,fourth,etc arm
  }
  $dir8 += ($arm|1);  # NE,NW,SW, or SE

  my $turn;
  if ($n || $lowbit) {
    # next turn

    # lowest non-3 digit, or zero if all 3s (implicit 0 above high digit)
    $turn = $digit_to_nextturn[ first {$_!=3} @ndigits, 0 ];
    if ($arm & 1) {
      $turn = - $turn;  # mirrored on second,fourth,etc arm
    }
  }

  if ($lowbit) {
    $dir8 += $turn;
  }

  my $s = $self->{'straight_spacing'};
  my $d = $self->{'diagonal_spacing'};

  $dir8 &= 7;
  my $spacing = ($dir8 & 1 ? $d : $s);
  my $dx = $spacing * $dir8_to_dx[$dir8];
  my $dy = $spacing * $dir8_to_dy[$dir8];

  if ($n) {
    $dir8 += $turn;
    $dir8 &= 7;
    $spacing = ($dir8 & 1 ? $d : $s);
    $dx += $n*($spacing * $dir8_to_dx[$dir8]
               - $dx);
    $dy += $n*($spacing * $dir8_to_dy[$dir8]
               - $dy);
  }

  return ($dx, $dy);
}

# 2| . 3 .
# 1| 1 . 2
# 0| . 0 .
#  +------
#    0 1 2
#
# 4| . . . 3 .          # diagonal_spacing == 3
# 3| . . . . 2 4        # mod=2*3+1=7
# 2| . . . . . . .
# 1| 1 . . . . . . .
# 0| . 0 . . . . . . 6
#  +------------------
#    0 1 2 3 4 5 6 7 8
#
sub _NOTWORKING__xy_is_visited {
  my ($self, $x, $y) = @_;
  $x = round_nearest($x);
  $y = round_nearest($y);
  my $mod = 2*$self->{'diagonal_spacing'} + $self->{'straight_spacing'};
  return (_rect_within_arms($x,$y, $x,$y, $self->{'arms'})
          && ((($x%$mod)+($y%$mod)) & 1));
}

#   x1    *  x2 *
#    +-----*-+y2*
#    |      *|  *
#    |       *  *
#    |       |* *
#    |       | **
#    +-------+y1*
#   ----------------
#
# arms=5 x1,y2 after X=Y-1 line, so x1 > y2-1, x1 >= y2
# ************
#      x1   *   x2
#      +---*----+y2
#      |  *     |
#      | *      |
#      |*       |
#      *        |
#     *+--------+y1
#    *
#
# arms=7 x1,y1 after X=-2-Y line, so x1 > -2-y1
# ************
# ** +------+
# * *|      |
# *  *      |
# *  |*     |
# *  | *    |
# *y1+--*---+
# * x1   *
#
# _rect_within_arms() returns true if rectangle x1,y1,x2,y2 has some part
# within the extent of the $arms set of octants.
#
sub _rect_within_arms {
  my ($x1,$y1, $x2,$y2, $arms) = @_;
  return ($arms <= 4
          ? ($y2 >= 0  # y2 top edge must be positive
             && ($arms <= 2
                 ? ($arms == 1 ? $x2 > $y1   # arms==1  bottom right
                    :            $x2 >= 0)   # arms==2  right edge
                 : ($arms == 4               # arms==4  anything
                    || $x2 >= -$y2)))        # arms==3  top right

          # arms >= 5
          : ($y2 >= 0  # y2 top edge positive is good, otherwise check
             || ($arms <= 6
                 ? ($arms == 5 ? $x1 < $y2   # arms==5  top left
                    :            $x1 < 0)    # arms==6  left edge
                 : ($arms == 8               # arms==8  anything
                    || $x1 <= -2-$y1))));    # arms==7  bottom left
}

sub xy_to_n {
  my ($self, $x, $y) = @_;
  ### SierpinskiCurve xy_to_n(): "$x, $y"

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

  my $arm = 0;
  if ($y < 0) {
    $arm = 4;
    $x = -1-$x;  # rotate -180
    $y = -1-$y;
  }
  if ($x < 0) {
    $arm += 2;
    ($x,$y) = ($y, -1-$x);  # rotate -90
  }
  if ($y > $x) {       # second octant
    $arm++;
    ($x,$y) = ($y,$x); # mirror 45
  }

  my $arms = $self->{'arms'};
  if ($arm >= $arms) {
    return undef;
  }

  $x -= 1;
  if ($x < 0 || $x < $y) {
    return undef;
  }
  ### x adjust to zero: "$x,$y"
  ### assert: $x >= 0
  ### assert: $y >= 0

  my $s = $self->{'straight_spacing'};
  my $d = $self->{'diagonal_spacing'};
  my $base = (2*$d+$s);
  my ($len,$level) = round_down_pow (($x+$y)/$base || 1,  2);
  ### $level
  ### $len
  if (is_infinite($level)) {
    return $level;
  }

  # Xtop = 3*2^(level-1)-1
  #
  $len *= 2*$base;
  ### initial len: $len

  my $n = 0;
  foreach (0 .. $level) {
    $n *= 4;
    ### at: "loop=$_ len=$len   x=$x,y=$y  n=$n"
    ### assert: $x >= 0
    ### assert: $y >= 0

    my $len_sub_d = $len - $d;
    if ($x < $len_sub_d) {
      ### digit 0 or 1...
      if ($x+$y+$s < $len) {
        ### digit 0 ...
      } else {
        ### digit 1 ...
        ($x,$y) = ($y-$d, $len-$s-$d-$x);   # shift then rotate -90
        $n += 1;
      }
    } else {
      $x -= $len_sub_d;
      ### digit 2 or 3 to: "x=$x y=$y"
      if ($x < $y) {   # before diagonal
        ### digit 2...
        ($x,$y) = ($len-$d-$s-$y, $x);     # shift y-len then rotate +90
        $n += 2;
      } else {
        #### digit 3...
        $x -= $d;
        $n += 3;
      }
      if ($x < 0) {
        return undef;
      }
    }
    $len /= 2;
  }

  ### end at: "x=$x,y=$y   n=$n"
  ### assert: $x >= 0
  ### assert: $y >= 0

  $n *= 4;
  if ($y == 0 && $x == 0) {
    ### final digit 0 ...
  } elsif ($x == $d && $y == $d) {
    ### final digit 1 ...
    $n += 1;
  } elsif ($x == $d+$s && $y == $d) {
    ### final digit 2 ...
    $n += 2;
  } elsif ($x == $base && $y == 0) {
    ### final digit 3 ...
    $n += 3;
  } else {
    return undef;
  }

  return $n*$arms + $arm;
}

# not exact
sub rect_to_n_range {
  my ($self, $x1,$y1, $x2,$y2) = @_;
  ### SierpinskiCurve rect_to_n_range(): "$x1,$y1  $x2,$y2"

  $x1 = round_nearest ($x1);
  $x2 = round_nearest ($x2);
  $y1 = round_nearest ($y1);
  $y2 = round_nearest ($y2);
  ($x1,$x2) = ($x2,$x1) if $x1 > $x2;
  ($y1,$y2) = ($y2,$y1) if $y1 > $y2;

  my $arms = $self->{'arms'};
  unless (_rect_within_arms($x1,$y1, $x2,$y2, $arms)) {
    ### rect outside octants, for arms: $arms
    return (1,0);
  }

  my $max = ($x2 + $y2);
  if ($arms >= 3) {
    _apply_max ($max, -1-$x1 + $y2);

    if ($arms >= 5) {
      _apply_max ($max, -1-$x1 - $y1-1);

      if ($arms >= 7) {
        _apply_max ($max, $x2 - $y1-1);
      }
    }
  }

  # base=2d+s
  # level begins at
  #   base*(2^level-1)-s = X+Y     ... maybe
  #   base*2^level = X+base
  #   2^level = (X+base)/base
  #   level = log2((X+base)/base)
  # then
  #   Nlevel = 4^level-1

  my $base = 2 * $self->{'diagonal_spacing'} + $self->{'straight_spacing'};
  my ($power) = round_down_pow (int(($max+$base-2)/$base),
                                2);
  return (0, 4*$power*$power * $arms - 1);
}

sub _apply_max {
  ### _apply_max(): "$_[0] cf $_[1]"
  unless ($_[0] > $_[1]) {
    $_[0] = $_[1];
  }
}

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

sub level_to_n_range {
  my ($self, $level) = @_;
  return (0, 4**$level * $self->{'arms'} - 1);
}
sub n_to_level {
  my ($self, $n) = @_;
  if ($n < 0) { return undef; }
  if (is_infinite($n)) { return $n; }
  $n = round_nearest($n) + ($self->{'arms'} - 1);
  _divrem_mutate ($n, $self->{'arms'});
  my ($pow, $exp) = round_down_pow ($n, 4);
  return $exp + 1;
}

#------------------------------------------------------------------------------
1;
__END__



# #      ...0    ...1
# #      ...1    ...2
# #      ...2    ...3
# #    ..0333  ..1000    any low 3s
# #      ..02    ..03
# #      ..12    ..13
# #      ..22    ..23
# #   ..03332 ..03333
# #   ..13332 ..13333
# #   ..23332 ..23333
#
# my @lowdigit_to_dir = (1,-2, 1, 0);
# my @digit_to_dir    = (0, 2,-2, 0);
# my @dir8_to_dx = (1, 1, 0,-1, -1, -1,  0, 1);
# my @dir8_to_dy = (0, 1, 1, 1,  0, -1, -1,-1);
# my @digit_to_nextturn  = (-1,-1,2);
# my @digit_to_nextturn2 = (2,-1,2);
#
# sub _WORKING_BUT_HAIRY__n_to_dxdy {
#   my ($self, $n) = @_;
#   ### n_to_dxdy(): $n
#
#   if ($n < 0) {
#     return;  # first direction at N=0
#   }
#   if (is_infinite($n)) {
#     return ($n,$n);
#   }
#
#   my $int = int($n);
#   $n -= $int;
#   my @digits = digit_split_lowtohigh($int,4);
#   ### @digits
#
#   # strip low 3s
#   my $any_low3s;
#   while (($digits[0]||0) == 3) {
#     shift @digits;
#     $any_low3s = 1;
#   }
#
#   my $dir8 = $lowdigit_to_dir[$digits[0] || 0];
#   $dir8 += sum(0, map {$digit_to_dir[$_]} @digits);
#   $dir8 &= 7;
#   my $dx = $dir8_to_dx[$dir8];
#   my $dy = $dir8_to_dy[$dir8];
#
#   if ($n) {
#     # fraction part
#
#     if ($any_low3s) {
#       $dir8 += $digit_to_nextturn2[$digits[0]||0];
#     } else {
#       my $digit = $digits[0] || 0;
#       if ($digit == 2) {
#         shift @digits;
#         # lowest non-3 digit
#         do {
#           $digit = shift @digits || 0;  # zero if all 3s or no digits at all
#         } until ($digit != 3);
#         $dir8 += $digit_to_nextturn2[$digit];
#       } else {
#         $dir8 += $digit_to_nextturn[$digit];
#       }
#     }
#     $dir8 &= 7;
#     $dx += $n*($dir8_to_dx[$dir8] - $dx);
#     $dy += $n*($dir8_to_dy[$dir8] - $dy);
#   }
#   return ($dx, $dy);
# }





   #                                              63-64            14
   #                                               |  |
   #                                              62 65            13
   #                                             /     \
   #                                        60-61       66-67      12
   #                                         |              |
   #                                        59-58       69-68      11
   #                                             \     /
   #                                  51-52       57 70            10
   #                                   |  |        |  |
   #                                  50 53       56 71       ...   9
   #                                 /     \     /     \     /
   #                            48-49       54-55       72-73       8
   #                             |
   #                            47-46       41-40                   7
   #                                 \     /     \
   #                      15-16       45 42       39                6
   #                       |  |        |  |        |
   #                      14 17       44-43       38                5
   #                     /     \                 /
   #                12-13       18-19       36-37                   4
   #                 |              |        |
   #                11-10       21-20       35-34                   3
   #                     \     /                 \
   #           3--4        9 22       27-28       33                2
   #           |  |        |  |        |  |        |
   #           2  5        8 23       26 29       32                1
   #         /     \     /     \     /     \     /
   #     0--1        6--7       24-25       30-31                 Y=0
   #
   #  ^
   # X=0 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 ...


# The factor of 3 arises because there's a gap between each level, increasing
# it by a fixed extra each time,
#
#     length(level) = 2*length(level-1) + 2
#                   = 2^level + (2^level + 2^(level-1) + ... + 2)
#                   = 2^level + (2^(level+1)-1 - 1)
#                   = 3*2^level - 2




=for stopwords eg Ryde Waclaw Sierpinski Sierpinski's Math-PlanePath Nlevel Nend Ntop Xlevel OEIS dX dY dX,dY nextturn

=head1 NAME

Math::PlanePath::SierpinskiCurve -- Sierpinski curve

=head1 SYNOPSIS

 use Math::PlanePath::SierpinskiCurve;
 my $path = Math::PlanePath::SierpinskiCurve->new (arms => 2);
 my ($x, $y) = $path->n_to_xy (123);

=head1 DESCRIPTION

X<Sierpinski, Waclaw>This is an integer version of the self-similar curve by
Waclaw Sierpinski traversing the plane by right triangles.  The default is a
single arm of the curve in an eighth of the plane.

=cut

# math-image --path=SierpinskiCurve --all --output=numbers_dash --size=79x26

=pod

    10  |                                  31-32
        |                                 /     \
     9  |                               30       33
        |                                |        |
     8  |                               29       34
        |                                 \     /
     7  |                         25-26    28 35    37-38
        |                        /     \  /     \  /     \
     6  |                      24       27       36       39
        |                       |                          |
     5  |                      23       20       43       40
        |                        \     /  \     /  \     /
     4  |                 7--8    22-21    19 44    42-41    55-...
        |               /     \           /     \           /
     3  |              6        9       18       45       54
        |              |        |        |        |        |
     2  |              5       10       17       46       53
        |               \     /           \     /           \
     1  |        1--2     4 11    13-14    16 47    49-50    52
        |      /     \  /     \  /     \  /     \  /     \  /
    Y=0 |  .  0        3       12       15       48       51
        |
        +-----------------------------------------------------------
           ^
          X=0 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16

The tiling it represents is

                    /
                   /|\
                  / | \
                 /  |  \
                /  7| 8 \
               / \  |  / \
              /   \ | /   \
             /  6  \|/  9  \
            /-------|-------\
           /|\  5  /|\ 10  /|\
          / | \   / | \   / | \
         /  |  \ /  |  \ /  |  \
        /  1| 2 X 4 |11 X 13|14 \
       / \  |  / \  |  / \  |  / \ ...
      /   \ | /   \ | /   \ | /   \
     /  0  \|/  3  \|/  12 \|/  15 \
    ----------------------------------

The points are on a square grid with integer X,Y.  4 points are used in each
3x3 block.  In general a point is used if

    X%3==1 or Y%3==1 but not both

    which means
    ((X%3)+(Y%3)) % 2 == 1

The X axis N=0,3,12,15,48,etc are all the integers which use only digits 0
and 3 in base 4.  For example N=51 is 303 base4.  Or equivalently the values
all have doubled bits in binary, for example N=48 is 110000 binary.
(Compare the C<CornerReplicate> which also has these values along the X
axis.)

=head2 Level Ranges

Counting the N=0 point as level=0, and with each level being 4 copies of the
previous, the levels end at

    Nlevel = 4^level - 1     = 0, 3, 15, ...
    Xlevel = 3*2^level - 2   = 1, 4, 10, ...
    Ylevel = 0

For example level=2 is Nlevel = 2^(2*2)-1 = 15 at X=3*2^2-2 = 10.

=for Test-Pari-DEFINE  Nlevel(level) = 4^level - 1

=for Test-Pari-DEFINE  Xlevel(level) = 3*2^level - 2

=for Test-Pari  Nlevel(0) == 0

=for Test-Pari  Nlevel(1) == 3

=for Test-Pari  Nlevel(2) == 15

=for Test-Pari  Xlevel(0) == 1

=for Test-Pari  Xlevel(1) == 4

=for Test-Pari  Xlevel(2) == 10

Doubling a level is the middle of the next level and is the top of the
triangle in that next level.

    Ntop = 2*4^level - 1               = 1, 7, 31, ...
    Xtop = 3*2^level - 1               = 2, 5, 11, ...
    Ytop = 3*2^level - 2  = Xlevel     = 1, 4, 10, ...

For example doubling level=2 is Ntop = 2*4^2-1 = 31 at X=3*2^2-1 = 11 and
Y=3*2^2-2 = 10.

=for Test-Pari-DEFINE  Ntop(level) = 2*4^level - 1

=for Test-Pari-DEFINE  Xtop(level) = 3*2^level - 1

=for Test-Pari-DEFINE  Ytop(level) = 3*2^level - 2

=for Test-Pari  2*4^2-1 == 31

=for Test-Pari  Ntop(2) == 31

=for Test-Pari  X=3*2^2-1 == 11

=for Test-Pari  Xtop(2) == 11

=for Test-Pari  3*2^2-2 == 10

=for Test-Pari  Ytop(2) == 10

The factor of 3 arises from the three steps which make up the N=0,1,2,3
section.  The Xlevel width grows as

    Xlevel(1) = 3
    Xlevel(level) = 2*Xwidth(level-1) + 3

which dividing out the factor of 3 is 2*w+1, giving 2^k-1 (in binary a left
shift and bring in a new 1 bit).

Notice too the Nlevel points as a fraction of the triangular area
Xlevel*(Xlevel-1)/2 gives the 4 out of 9 points filled,

    FillFrac = Nlevel / (Xlevel*(Xlevel-1)/2)
            -> 4/9

=head2 Arms

The optional C<arms> parameter can draw multiple curves, each advancing
successively.  For example 2 arms,


    arms => 2                            ...
                                          |
    11  |     33       39       57       63
        |    /  \     /  \     /  \     /
    10  |  31    35-37    41 55    59-61    62-...
        |    \           /     \           /
     9  |     29       43       53       60
        |      |        |        |        |
     8  |     27       45       51       58
        |    /           \     /           \
     7  |  25    21-19    47-49    50-52    56
        |    \  /     \           /     \  /
     6  |     23       17       48       54
        |               |        |
     5  |      9       15       46       40
        |    /  \     /           \     /  \
     4  |   7    11-13    14-16    44-42    38
        |    \           /     \           /
     3  |      5       12       18       36
        |      |        |        |        |
     2  |      3       10       20       34
        |    /           \     /           \
     1  |   1     2--4     8 22    26-28    32
        |       /     \  /     \  /     \  /
    Y=0 |      0        6       24       30
        |
        +-----------------------------------------
            ^
           X=0 1  2  3  4  5  6  7  8  9 10 11

The N=0 point is at X=1,Y=0 (in all arms forms) so that the second arm is
within the first quadrant.

1 to 8 arms can be done this way.  For example 8 arms are

    arms => 8

           ...                       ...           6
            |                          |
           58       34       33       57           5
             \     /  \     /  \     /
    ...-59    50-42    26 25    41-49    56-...    4
          \           /     \           /
           51       18       17       48           3
            |        |        |        |
           43       10        9       40           2
          /           \     /           \
        35    19-11     2  1     8-16    32        1
          \  /     \           /     \  /
           27        3     .  0       24       <- Y=0

           28        4        7       31          -1
          /  \     /           \     /  \
        36    20-12     5  6    15-23    39       -2
          \           /     \           /
           44       13       14       47          -3
            |        |        |        |
           52       21       22       55          -4
          /           \     /           \
    ...-60    53-45    29 30    46-54    63-...   -5
             /     \  /     \  /     \
           61       37       38       62          -6
            |                          |
           ...                       ...          -7

                           ^
     -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6

The middle "." is the origin X=0,Y=0.  It would be more symmetrical to make
the origin the middle of the eight arms, at X=-0.5,Y=-0.5 in the above, but
that would give fractional X,Y values.  Apply an offset X+0.5,Y+0.5 to
centre it if desired.

=head2 Spacing

The optional C<diagonal_spacing> and C<straight_spacing> can increase the
space between points diagonally or vertically+horizontally.  The default for
each is 1.

=cut

# math-image --path=SierpinskiCurve,straight_spacing=2,diagonal_spacing=1 --all --output=numbers_dash --size=79x26
# math-image --path=SierpinskiCurve,straight_spacing=3,diagonal_spacing=3 --all --output=numbers_dash --size=79x26

=pod

    straight_spacing => 2
    diagonal_spacing => 1

                        7 ----- 8
                     /           \
                    6               9
                    |               |
                    |               |
                    |               |
                    5              10              ...
                     \           /                   \
        1 ----- 2       4      11      13 ---- 14      16
     /           \   /           \   /           \   /
    0               3              12              15

   X=0  1   2   3   4   5   6   7   8   9  10  11  12  13 ...


The effect is only to spread the points.  The straight lines are both
horizontal and vertical so when they're stretched the curve remains on a 45
degree angle in an eighth of the plane.

In the level formulas above the "3" factor becomes 2*d+s, effectively being
the N=0 to N=3 section sized as d+s+d.

    d = diagonal_spacing
    s = straight_spacing

    Xlevel = (2d+s)*(2^level - 1)  + 1

    Xtop = (2d+s)*2^(level-1) - d - s + 1
    Ytop = (2d+s)*2^(level-1) - d - s

=head2 Closed Curve

Sierpinski's original conception was a closed curve filling a unit square by
ever greater self-similar detail,

    /\_/\ /\_/\ /\_/\ /\_/\
    \   / \   / \   / \   /
     | |   | |   | |   | |
    / _ \_/ _ \ / _ \_/ _ \
    \/ \   / \/ \/ \   / \/
       |  |         | |
    /\_/ _ \_/\ /\_/ _ \_/\
    \   / \   / \   / \   /
     | |   | |   | |   | |
    / _ \ / _ \_/ _ \ / _ \
    \/ \/ \/ \   / \/ \/ \/
              | |
    /\_/\ /\_/ _ \_/\ /\_/\
    \   / \   / \   / \   /
     | |   | |   | |   | |
    / _ \_/ _ \ / _ \_/ _ \
    \/ \   / \/ \/ \   / \/
       |  |         | |
    /\_/ _ \_/\ /\_/ _ \_/\
    \   / \   / \   / \   /
     | |   | |   | |   | |
    / _ \ / _ \ / _ \ / _ \
    \/ \/ \/ \/ \/ \/ \/ \/

The code here might be pressed into use for this by drawing a mirror image
of the curve N=0 through Nlevel.  Or using the C<arms=E<gt>2> form N=0 to
N=4^level - 1, inclusive, and joining up the ends.

The curve is also usually conceived as scaling down by quarters.  This can
be had with C<straight_spacing =E<gt> 2> and then an offset to X+1,Y+1 to
centre in a 4*2^level square

=head2 Koch Curve Midpoints

The replicating structure is the same as the Koch curve
(L<Math::PlanePath::KochCurve>) in that the curve repeats four times to make
the next level.

The Sierpinski curve points are midpoints of a Koch curve of 90 degree
angles with a unit gap between verticals.

     Koch Curve                  Koch Curve
                          90 degree angles, unit gap

           /\                       |  |
          /  \                      |  |
         /    \                     |  |
    -----      -----          ------    ------

=cut

=pod

   Sierpinski curve points "*" as midpoints

                      |  |
                      7  8
                      |  |
               ---6---    ---9---

               ---5---    --10---
           |  |       |  |       |  |
           1  2       4  11     13  14
           |  |       |  |       |  |
    ---0---    ---3---    --12---    --15---


=head2 Koch Curve Rounded

The Sierpinski curve in mirror image across the X=Y diagonal and rotated -45
degrees is pairs of points on the lines of the Koch curve 90 degree angles
unit gap from above.

    Sierpinski curve mirror image and turn -45 degrees
    two points on each Koch line segment

                          15   16
                           |    |
                          14   17

                  12--13   .    .   18--19

                  11--10   .    .   21--20

           3   4           9   22            27   28
           |   |           |    |             |    |
           2   5           8   23            26   29

    0---1  .   .   6---7   .    .   24--25    .    .   30--31

This is a kind of "rounded" form of the 90-degree Koch, similar what
C<DragonRounded> does for the C<DragonCurve>.  Each 90 turn of the Koch
curve is done by two turns of 45 degrees in the Sierpinski curve here, and
each 180 degree turn in the Koch is two 90 degree turns here.  So the
Sierpinski turn sequence is pairs of the Koch turn sequence, as follows.
The mirroring means a swap leftE<lt>-E<gt>right between the two.

           N=1    2    3    4    5     6      7      8
    Koch     L    R    L    L    L     R      L      R     ...

           N=1,2  3,4  5,6  7,8  9,10  11,12  13,14  15,16
    Sierp    R R  L L  R R  R R  R R   L  L   R  R   L  L  ...

=head1 FUNCTIONS

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

=over 4

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

=item C<$path = Math::PlanePath::SierpinskiCurve-E<gt>new (arms =E<gt> $integer, diagonal_spacing =E<gt> $integer, straight_spacing =E<gt> $integer)>

Create and return a new path object.

=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>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, 4**$level - 1)>, or for multiple arms return C<(0, $arms *
4**$level - 1)>.

There are 4^level points in a level, or arms*4^level when multiple arms,
numbered starting from 0.

=back

=head1 FORMULAS

=head2 N to dX,dY

The curve direction at N even can be calculated from the base-4 digits of
N/2 in a fashion similar to the Koch curve (L<Math::PlanePath::KochCurve/N
to Direction>).  Counting direction in eighths so 0=East, 1=North-East,
2=North, etc,

    digit     direction
    -----     ---------
      0           0
      1          -2
      2           2
      3           0

    direction = 1 + sum direction[base-4 digits of N/2]
      for N even

For example the direction at N=10 has N/2=5 which is "11" in base-4, so
direction = 1+(-2)+(-2) = -3 = south-west.

The 1 in 1+sum is direction north-east for N=0, then -2 or +2 for the digits
follow the curve.  For an odd arm the curve is mirrored and the sign of each
digit direction is flipped, so a subtract instead of add,

    direction
    mirrored  = 1 - sum direction[base-4 digits of N/2]
       for N even

For odd N=2k+1 the direction at N=2k is calculated and then also the turn
which is made from N=2k to N=2(k+1).  This is similar to the Koch curve next
turn (L<Math::PlanePath::KochCurve/N to Next Turn>).

   lowest non-3      next turn
   digit of N/2   (at N=2k+1,N=2k+2)
   ------------   ----------------
        0           -1 (right)
        1           +2 (left)
        2           -1 (right)

Again the turn is in eighths, so -1 means -45 degrees (to the right).  For
example at N=14 has N/2=7 which is "13" in base-4 so lowest non-3 is "1"
which is turn +2, so at N=15 and N=16 turn by 90 degrees left.


   direction = 1 + sum direction[base-4 digits of k]
                 + if N odd then nextturn[low-non-3 of k]
     for N=2k or 2k+1

   dX,dY = direction to 1,0 1,1 0,1 etc

For fractional N the same nextturn is applied to calculate the direction of
the next segment, and combined with the integer dX,dY as per
L<Math::PlanePath/N to dX,dY -- Fractional>.

   N=2k or 2k+1 + frac

   direction = 1 + sum direction[base-4 digits of k]

   if (frac != 0 or N odd)
     turn = nextturn[low-non-3 of k]

   if N odd then direction += turn
   dX,dY = direction to 1,0 1,1 0,1 etc

   if frac!=0 then
     direction += turn
     next_dX,next_dY = direction to 1,0 1,1 0,1 etc

     dX += frac*(next_dX - dX)
     dY += frac*(next_dY - dY)

For the C<straight_spacing> and C<diagonal_spacing> options the dX,dY values
are not units like dX=1,dY=0 but instead are the spacing amount, either
straight or diagonal so

    direction      delta with spacing
    ---------    -------------------------
        0        dX=straight_spacing, dY=0
        1        dX=diagonal_spacing, dY=diagonal_spacing
        2        dX=0, dY=straight_spacing
        3        dX=-diagonal_spacing, dY=diagonal_spacing
       etc

As an alternative, it's possible to take just base-4 digits of N, without
separate handling for the low-bit of N, but it requires an adjustment on the
low base-4 digit, and the next turn calculation for fractional N becomes
hairier.  A little state table could encode the cumulative and lowest
whatever if desired, to take N by base-4 digits high to low, or equivalently
by bits high to low with an initial state based on high bit at an odd or
even bit position.

=head1 OEIS

The Sierpinski curve is in Sloane's Online Encyclopedia of Integer Sequences
as,

=over

L<http://oeis.org/A039963> (etc)

=back

    A039963   turn 1=right,0=left, doubling the KochCurve turns
    A081706   N-1 of left turn positions
               (first values 2,3 whereas N=3,4 here)
    A127254   abs(dY), so 0=horizontal, 1=vertical or diagonal,
                except extra initial 1
    A081026   X at N=2^k, being successively 3*2^j-1, 3*2^j

A039963 is numbered starting n=0 for the first turn, which is at the point
N=1 in the path here.

=head1 SEE ALSO

L<Math::PlanePath>,
L<Math::PlanePath::SierpinskiCurveStair>,
L<Math::PlanePath::SierpinskiArrowhead>,
L<Math::PlanePath::KochCurve>

=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