This file is indexed.

/usr/share/perl5/Math/PlanePath/SierpinskiArrowheadCentres.pm is in libmath-planepath-perl 113-1.

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
# Copyright 2011, 2012, 2013 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=SierpinskiArrowheadCentres --lines --scale=10
#
# math-image --path=SierpinskiArrowheadCentres --all --output=numbers_dash
# math-image --path=SierpinskiArrowheadCentres --all --text --size=80


package Math::PlanePath::SierpinskiArrowheadCentres;
use 5.004;
use strict;

#use List::Util 'max';
*max = \&Math::PlanePath::_max;

use vars '$VERSION', '@ISA';
$VERSION = 113;
use Math::PlanePath;
@ISA = ('Math::PlanePath');

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

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


use Math::PlanePath::SierpinskiArrowhead;
*parameter_info_array  # align parameter
  = \&Math::PlanePath::SierpinskiArrowhead::parameter_info_array;
*new = \&Math::PlanePath::SierpinskiArrowhead::new;

use constant n_start => 0;
use constant class_y_negative => 0;
*x_negative = \&Math::PlanePath::SierpinskiArrowhead::x_negative;
*x_maximum  = \&Math::PlanePath::SierpinskiArrowhead::x_maximum;
use constant sumxy_minimum => 0;  # triangular X>=-Y
use Math::PlanePath::SierpinskiTriangle;
*diffxy_maximum = \&Math::PlanePath::SierpinskiTriangle::diffxy_maximum;

use constant dy_minimum => -1;
use constant dy_maximum => 1;
*dx_minimum = \&Math::PlanePath::SierpinskiArrowhead::dx_minimum;
*dx_maximum = \&Math::PlanePath::SierpinskiArrowhead::dx_maximum;
*absdx_minimum = \&Math::PlanePath::SierpinskiArrowhead::absdx_minimum;
*absdx_maximum = \&Math::PlanePath::SierpinskiArrowhead::absdx_maximum;
*dsumxy_minimum = \&Math::PlanePath::SierpinskiArrowhead::dsumxy_minimum;
*dsumxy_maximum = \&Math::PlanePath::SierpinskiArrowhead::dsumxy_maximum;
sub ddiffxy_minimum {
  my ($self) = @_;
  return ($self->{'align'} eq 'right' ? -1 : -2);
}
sub ddiffxy_maximum {
  my ($self) = @_;
  return ($self->{'align'} eq 'right' ? 1 : 2);
}
*dir_maximum_dxdy = \&Math::PlanePath::SierpinskiArrowhead::dir_maximum_dxdy;

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

# States as multiples of 3 so that state+digit is the lookup for next state
# and x,y bit.
#
# 0        3           6        9           12       15
#
# 8        0           4        4           0        8
# |        |           |\       |\           \        \
# 7-6      1-2         3 5      5 3         2-1      6-7
#    \        \        |  \     |  \        |        |
# 1   5    7   3       2   6    6   2       3   7    5   1
# |\   \   |\   \       \  |     \  |       |   |\   |   |\
# 0 2-3-4  8 6-5-4     0-1 7-8  8-7 1-0     4-5-6 8  4-3-2 0

#  15                   6                    3
#  6  0                 0 12                 12 6

#  0,1                  0,2                   1,2

my @next_state = (6,0,15,  12,3,9,   # 3,6
                  0,6,12,  15,9,3,   # 6,9
                  3,12,6,  9,15,0);  # 12,15
my @state_to_xbit = (0,1,0, 0,1,0,
                     0,0,1, 1,0,0,
                     0,0,1, 1,0,0);  # 12,15
my @state_to_ybit = (0,0,1, 1,0,0,
                     0,1,0, 0,1,0,
                     1,0,0, 0,0,1);  # 12,15

# dx,dy for digit==0 and digit==1 in each stage
my @state_to_dx;
my @state_to_dy;
foreach my $state (0,1, 3,4, 6,7, 9,10, 12,13, 15,16) {
  $state_to_dx[$state] = $state_to_xbit[$state+1] - $state_to_xbit[$state];
  $state_to_dy[$state] = $state_to_ybit[$state+1] - $state_to_ybit[$state];
}

sub n_to_xy {
  my ($self, $n) = @_;
  ### SierpinskiArrowheadCentres n_to_xy(): $n
  if ($n < 0) {
    return;
  }
  if (is_infinite($n)) {
    return ($n,$n);
  }

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

  my @digits = digit_split_lowtohigh($int,3);
  my $state = ($#digits & 1 ? 6 : 0);
  ### @digits
  ### $state

  my (@x,@y); # bits low to high
  my $dirstate = $state ^ 6;  # if all digits==2

  foreach my $i (reverse 0 .. $#digits) {
    ### at: "x=".join(',',@x[($i+1)..$#digits])." y=".join(',',@y[($i+1)..$#digits])."  apply  i=$i state=$state digit=$digits[$i]"

    my $digit = $digits[$i];  # high to low
    $state += $digit;
    $x[$i] = $state_to_xbit[$state];
    $y[$i] = $state_to_ybit[$state];
    if ($digit != 2) {
      $dirstate = $state; # lowest non-2 digit
    }
    $state = $next_state[$state];
  }

  my $zero = $int * 0;
  my $x = $n*$state_to_dx[$dirstate] + digit_join_lowtohigh(\@x,2,$zero);
  my $y = $n*$state_to_dy[$dirstate] + digit_join_lowtohigh(\@y,2,$zero);

  if ($self->{'align'} eq 'right') {
    $y += $x;
  } elsif ($self->{'align'} eq 'left') {
    ($x,$y) = (-$y,$x+$y);
  } elsif ($self->{'align'} eq 'triangular') {
    ($x,$y) = ($x-$y,$x+$y);
  }
  return ($x,$y);
}

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

  if ($y < 0) {
    return undef;
  }

  if ($self->{'align'} eq 'left') {
    if ($x > 0) {
      return undef;
    }
    $x = 2*$x + $y; # adjust to triangular style

  } elsif ($self->{'align'} eq 'triangular') {
    if (($x%2) != ($y%2)) {
      return undef;
    }

  } else {
    # right or diagonal
    if ($x < 0) {
      return undef;
    }
    if ($self->{'align'} eq 'right') {
      $x = 2*$x - $y;
    } else { # diagonal
      ($x,$y) = ($x-$y, $x+$y);
      }
  }
  ### adjusted xy: "$x,$y"


  my ($len, $level) = round_down_pow ($y, 2);
  ### pow2 round up: ($y + ($y==$x || $y==-$x))
  ### $len
  ### $level
  $level += 1;

  if (is_infinite($level)) {
    return $level;
  }

  my $n = 0;
  while ($level) {
    $n *= 3;
    ### at: "$x,$y  level=$level len=$len"

    if ($y < 0 || $x < -$y || $x > $y) {
      ### out of range ...
      return undef;
    }

    if ($y < $len) {
      ### digit 0, first triangle, no change ...

    } else {
      if ($level & 1) {
        ### odd level ...
        if ($x > 0) {
          ### digit 1, right triangle ...
          $n += 1;
          $y -= $len;
          $x = - ($x-$len);
          ### shift right and mirror to: "$x,$y"
        } else {
          ### digit 2, left triangle ...
          $n += 2;
          $x += 1;
          $y -= 2*$len-1;
          ### shift down to: "$x,$y"
          ($x,$y) = ((3*$y-$x)/2,   # rotate -120
                     ($x+$y)/-2);
          ### rotate to: "$x,$y"
        }
      } else {
        ### even level ...
        if ($x < 0) {
          ### digit 1, left triangle ...
          $n += 1;
          $y -= $len;
          $x = - ($x+$len);
          ### shift right and mirror to: "$x,$y"
        } else {
          ### digit 2, right triangle ...
          $n += 2;
          $x -= 1;
          $y -= 2*$len-1;
          ### shift down to: "$x,$y"
          ($x,$y) = (($x+3*$y)/-2,             # rotate +120
                     ($x-$y)/2);
          ### now: "$x,$y"
        }
      }
    }

    $level--;
    $len /= 2;
  }

  ### final: "$x,$y with n=$n"
  if ($x == 0 && $y == 0) {
    return $n;
  } else {
    return undef;
  }
}

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

  $y1 = round_nearest ($y1);
  $y2 = round_nearest ($y2);
  if ($y1 > $y2) { ($y1,$y2) = ($y2,$y1); } # swap to y1<=y2

  if ($self->{'align'} eq 'diagonal') {
    $y2 += max (round_nearest ($x1),
                round_nearest ($x2));
  }

  unless ($y2 >= 0) {
    ### rect all negative, no N ...
    return (1, 0);
  }

  my ($len,$level) = round_down_pow ($y2, 2);
  ### $y2
  ### $level
  return (0, 3**($level+1) - 1);
}

1;
__END__

#------------------------------------------------------------------------------
# Old n_to_xy() triangular with explicit add/sub.

  # my $x = my $y = ($int * 0); # inherit bigint 0
  # my $len = $x + 1;           # inherit bigint 1
  # 
  # my @digits = digit_split_lowtohigh($int,3);
  # for (;;) {
  #   unless (@digits) {
  #     $x = $n + $x;
  #     $y = $n + $y;
  #     last;
  #   }
  #   my $digit = shift @digits; # low to high
  # 
  #   ### odd right: "$x, $y  len=$len  frac=$n"
  #   ### $digit
  #   if ($digit == 0) {
  #     $x = $n + $x;
  #     $y = $n + $y;
  #     $n = 0;
  # 
  #   } elsif ($digit == 1) {
  #     $x = -2*$n -$x + $len;  # mirror and offset
  #     $y += $len;
  #     $n = 0;
  # 
  #   } else {
  #     ($x,$y) = (($x+3*$y)/-2 - 1,             # rotate +120
  #                ($x-$y)/2    + 2*$len-1);
  #   }
  # 
  #   unless (@digits) {
  #     $x = -$n + $x;
  #     $y = $n + $y;
  #     last;
  #   }
  #   $digit = shift @digits; # low to high
  #   $len *= 2;
  # 
  #   ### odd left: "$x, $y  len=$len  frac=$n"
  #   ### $digit
  #   if ($digit == 0) {
  #     $x = -$n + $x;
  #     $y = $n + $y;
  #     $n = 0;
  # 
  #   } elsif ($digit == 1) {
  #     $x = 2*$n + -$x - $len;  # mirror and offset
  #     $y += $len;
  #     $n = 0;
  # 
  #   } else {
  #     ($x,$y) = ((3*$y-$x)/2 + 1,              # rotate -120
  #                ($x+$y)/-2  + 2*$len-1);
  #   }
  #   $len *= 2;
  # }
  # 
  # ### final: "$x,$y"
  # if ($self->{'align'} eq 'right') {
  #   return (($x+$y)/2, $y);
  # } elsif ($self->{'align'} eq 'left') {
  #   return (($x-$y)/2, $y);
  # } elsif ($self->{'align'} eq 'diagonal') {
  #   return (($x+$y)/2, ($y-$x)/2);
  # } else { # triangular
  #   return ($x,$y);
  # }

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

=for stopwords eg Ryde Sierpinski Nlevel ie Math-PlanePath bitand dX,dY

=head1 NAME

Math::PlanePath::SierpinskiArrowheadCentres -- self-similar triangular path traversal

=head1 SYNOPSIS

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

=head1 DESCRIPTION

X<Sierpinski, Waclaw>This is a version of the Sierpinski arrowhead path
taking the centres of each triangle represented by the arrowhead segments.
The effect is to traverse the C<SierpinskiTriangle> points in a connected
sequence.

              ...                                 ...
               /                                   /
        .    30     .     .     .     .     .    65     .   ...
            /                                      \        /
    28----29     .     .     .     .     .     .    66    68      9
      \                                               \  /
       27     .     .     .     .     .     .     .    67         8
         \
          26----25    19----18----17    15----14----13            7
               /        \           \  /           /
             24     .    20     .    16     .    12               6
               \        /                       /
                23    21     .     .    10----11                  5
                  \  /                    \
                   22     .     .     .     9                     4
                                          /
                       4---- 5---- 6     8                        3
                        \           \  /
                          3     .     7                           2
                           \
                             2---- 1                              1
                                 /
                                0                             <- Y=0

    -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7

The base figure is the N=0 to N=2 shape.  It's repeated up in mirror image
as N=3 to N=6 then rotated across as N=6 to N=9.  At the next level the same
is done with N=0 to N=8 as the base, then N=9 to N=17 up mirrored, and N=18
to N=26 across, etc.

The X,Y coordinates are on a triangular lattice using every second integer
X, per L<Math::PlanePath/Triangular Lattice>.

The base pattern is a triangle like

      .-------.-------.
       \     / \     /
        \ 2 /   \ 1 /
         \ /     \ /
          .- - - -.
           \     /
            \ 0 /
             \ /
              .

Higher levels replicate this within the triangles 0,1,2 but the middle is
not traversed.  The result is the familiar Sierpinski triangle by connected
steps of either 2 across or 1 diagonal.

    * * * * * * * * * * * * * * * *
     *   *   *   *   *   *   *   *
      * *     * *     * *     * *
       *       *       *       *
        * * * *         * * * *
         *   *           *   *
          * *             * *
           *               *
            * * * * * * * *
             *   *   *   *
              * *     * *
               *       *
                * * * *
                 *   *
                  * *
                   *

See the C<SierpinskiTriangle> path to traverse by rows instead.

=head2 Level Ranges

Counting the N=0,1,2 part as level 1, each replication level goes from

    Nstart = 0
    Nlevel = 3^level - 1     inclusive

For example level 2 from N=0 to N=3^2-1=9.  Each level doubles in size,

                 0  <= Y <= 2^level - 1
    - (2^level - 1) <= X <= 2^level - 1

The Nlevel position is alternately on the right or left,

    Xlevel = /  2^level - 1      if level even
             \  - 2^level + 1    if level odd

The Y axis ie. X=0, is crossed just after N=1,5,17,etc which is is 2/3
through the level, which is after two replications of the previous level,

    Ncross = 2/3 * 3^level - 1
           = 2 * 3^(level-1) - 1

=head2 Align Parameter

An optional C<align> parameter controls how the points are arranged relative
to the Y axis.  The default shown above is "triangular".  The choices are
the same as for the C<SierpinskiTriangle> path.

"right" means points to the right of the axis, packed next to each other and
so using an eighth of the plane.

=cut

# math-image --path=SierpinskiArrowheadCentres,align=right --all --output=numbers_dash

=pod

    align => "right"

        |   |
     7  |  26-25 19-18-17 15-14-13     
        |    /    |     |/     /       
     6  |  24    20    16    12        
        |   |   /           /          
     5  |  23 21       10-11           
        |   |/          |              
     4  |  22           9              
        |             /                
     3  |   4--5--6  8                 
        |   |     |/                   
     2  |   3     7                    
        |   |                          
     1  |   2--1                       
        |    /                         
    Y=0 |   0                          
        +--------------------------
           X=0 1  2  3  4  5  6  7

"left" is similar but skewed to the left of the Y axis, ie. into negative X.

=cut

# math-image --path=SierpinskiArrowheadCentres,align=left --all --output=numbers_dash

=pod

    align => "left"

    \                         |
     26-25 19-18-17 15-14-13  |  7 
         |   \     \ |     |  |    
        24    20    16    12  |  6 
          \    |           |  |    
           23 21       10-11  |  5 
             \ |         \    |    
              22           9  |  4 
                           |  |    
                  4--5--6  8  |  3 
                   \     \ |  |    
                     3     7  |  2 
                      \       |    
                        2--1  |  1 
                           |  |    
                           0  | Y=0
    --------------------------+

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

"diagonal" puts rows on diagonals down from the Y axis to the X axis.  This
uses the whole of the first quadrant, with gaps.

=cut

# math-image --expression='i<=26?i:0' --path=SierpinskiArrowheadCentres,align=diagonal --output=numbers_dash

=pod

    align => "diagonal"

        |   |                     
     7  |  26                     
        |    \                    
     6  |  24-25                  
        |   |                     
     5  |  23    19               
        |   |     |\              
     4  |  22-21-20 18            
        |             \           
     3  |   4          17         
        |   |\          |         
     2  |   3  5       16-15      
        |   |   \           \     
     1  |   2     6    10    14   
        |    \    |     |\     \  
    Y=0 |   0--1  7--8--9 11-12-13
        +--------------------------
           X=0 1  2  3  4  5  6  7

These diagonals visit all points X,Y where X and Y written in binary have no
1-bits in the same places, ie. where S<X bitand Y> = 0.  This is the same as
the C<SierpinskiTriangle> with align=diagonal.

=head1 FUNCTIONS

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

=over 4

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

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

Create and return a new arrowhead path object.  C<align> is a string, one of
the following as described above.

    "triangular"       the default
    "right"
    "left"
    "diagonal"

=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.

If C<$n> is not an integer then the return is on a straight line between the
integer points.

=back

=head1 FORMULAS

=head2 N to X,Y

The align="diagonal" style is the most convenient to calculate.  Each
ternary digit of N becomes a bit of X and Y.

    ternary digits of N, high to low
        xbit = state_to_xbit[state+digit]
        ybit = state_to_ybit[state+digit]
        state = next_state[state+digit]

There's a total of 6 states which are the permutations of 0,1,2 in the three
triangular parts.  The states are in pairs as forward and reverse, but that
has no particular significance.  Numbering the states by "3"s allows the
digit 0,1,2 to be added to make an index into tables for X,Y bit and next
state.

    state=0     state=3      
    +---------+ +---------+  
    |^ 2 |    | |\ 0 |    |  
    | \  |    | | \  |    |  
    |  \ |    | |  v |    |  
    |----+----| |----+----|  
    |    |^   | |    ||   |  
    | 0  || 1 | | 0  || 1 |  
    |--->||   | |<---|v   |  
    +---------+ +---------+  

    state=6      state=9     
    +---------+  +---------+ 
    |    |    |  |    |    | 
    | 1  |    |  | 1  |    | 
    |--->|    |  |<---|    | 
    |----+----|  |----+----| 
    |^   |\ 2 |  ||   |^   | 
    ||0  | \  |  || 2 | \0 | 
    ||   |  v |  |v   |  \ | 
    +---------+  +---------+ 

    state=12     state=15    
    +---------+  +---------+ 
    || 0 |    |  |^   |    | 
    ||   |    |  || 2 |    | 
    |v   |    |  ||   |    | 
    |----+----|  |----+----| 
    |\ 1 |    |  |^ 1 |    | 
    | \  | 2  |  | \  |  0 | 
    |  v |--->|  |  \ |<---| 
    +---------+  +---------+ 

The initial state is 0 if an even number of ternary digits, or 6 if odd.  In
the samples above it can be seen for example that N=0 to N=8 goes upwards as
per state 0, whereas N=0 to N=2 goes across as per state 6.

Having calculated an X,Y in align="diagonal" style it can be mapped to the
other alignments by

    align        coordinates from diagonal X,Y
    -----        -----------------------------
    triangular      X-Y, X+Y
    right           X, X+Y
    left            -Y, X+Y    

=head2 N to dX,dY

For fractional N the direction of the curve towards the N+1 point can be
found from the least significant digit 0 or 1 (ie. a non-2 digit) and the
state at that point.

This works because if the least significant ternary digit of N is a 2 then
the direction of the curve is determined by the next level up, and so on for
all trailing 2s until reaching a non-2 digit.

If N is all 2s then the direction should be reckoned from an initial 0 digit
above them, which means the opposite 6 or 0 of the initial state.

=head2 Rectangle to N Range

An easy over-estimate of the range can be had from inverting the Nlevel
formulas in L</Level Ranges> above.

    level = floor(log2(Ymax)) + 1
    Nmax = 3^level - 1

For example Y=5, level=floor(log2(11))+1=3, so Nmax=3^3-1=26, which is the
left end of the Y=7 row, ie. rounded up to the end of the Y=4 to Y=7
replication.

=head1 SEE ALSO

L<Math::PlanePath>,
L<Math::PlanePath::SierpinskiArrowhead>,
L<Math::PlanePath::SierpinskiTriangle>

=head1 HOME PAGE

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

=head1 LICENSE

Copyright 2011, 2012, 2013 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