This file is indexed.

/usr/share/perl5/Math/PlanePath/KnightSpiral.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
# Copyright 2010, 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::KnightSpiral;
use 5.004;
use strict;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;

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

use Math::PlanePath::Base::Generic
  'round_nearest';

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


use constant xy_is_visited => 1;
sub x_negative_at_n {
  my ($self) = @_;
  return $self->n_start + 3;
}
sub y_negative_at_n {
  my ($self) = @_;
  return $self->n_start + 1;
}
sub _UNDOCUMENTED__dxdy_list_at_n {
  my ($self) = @_;
  return $self->n_start + 8;
}
use constant dx_minimum => -2;
use constant dx_maximum => 2;
use constant dy_minimum => -2;
use constant dy_maximum => 2;
use constant _UNDOCUMENTED__dxdy_list => (2,1,   # ENE
                           1,2,   # NNE
                           -1,2,  # NNW
                           -2,1,  # WNW
                           -2,-1, # WSW
                           -1,-2, # SSW
                           1,-2,  # SSE
                           2,-1,  # ESE
                          );
use constant absdx_minimum => 1;
use constant absdy_minimum => 1;
use constant dsumxy_minimum => -3; # -2,-1
use constant dsumxy_maximum => 3;  # +2,+1
use constant ddiffxy_minimum => -3;
use constant ddiffxy_maximum => 3;
use constant dir_minimum_dxdy => (2,1);  # X=2,Y=1 angle
use constant dir_maximum_dxdy => (2,-1);

# Maybe ...
# use constant parameter_info_array =>
#   [
#    Math::PlanePath::Base::Generic::parameter_info_nstart1(),
#   ];

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

sub new {
  my $self = shift->SUPER::new(@_);
  if (! defined $self->{'n_start'}) {
    $self->{'n_start'} = $self->default_n_start;
  }
  return $self;
}

sub _odd {
  my ($n) = @_;
  ### _odd(): $n
  $n -= 2*int($n/2);
  ### rem: "$n"
  if ($n > 1) {
    return 2-$n;
  } else {
    return $n;
  }
  # return (int($n) % 2);
}

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

  # adjust to N=1 at origin X=0,Y=0
  $n = $n - $self->{'n_start'} + 1;

  if ($n < 2) {
    if ($n < 1) { return; }
    $n--;
    return (2*$n, -$n);
  }

  my $d = int ((7 + sqrt(int($n) - 1)) / 4);
  my $d1 = $d-1;
  my $outer = 2*$d1;
  my $inner = $outer - 1;
  my $p = 2*$d1;
  my $p1 = $p - 1;

  # use Smart::Comments;

  #### s frac: .25 * (7 + sqrt($n - 1))
  #### $d
  #### $d1
  #### $inner
  #### $outer
  #### $p
  #### $p1

  $n -= $d*(16*$d - 56) + 50;
  #### remainder: $n

  # one
  #
  if ($n < $p1) {
    #### right upwards, eg 2 ...
    return (- _odd($n) + $outer,
            2*$n - $inner);
  }
  $n -= $p1;

  if ($n < $p1) {
    #### top leftwards, eg 3 ...
    return (-2*$n + $inner,
            _odd($n) + $inner);
  }
  $n -= $p1;

  if ($n < $p) {
    #### left downwards ...
    return ( - _odd($n) - $inner,
             -2*$n + $outer);
  }
  $n -= $p;

  if ($n < $p1) {
    #### bottom rightwards: $n
    return (2*$n - $inner,
            _odd($n) - $outer);
  }
  $n -= $p1;



  ### two ...
  #
  if ($n < $p1) {
    ### right upwards ...
    return (_odd($n) + $inner,
            2*$n - $inner);
  }
  $n -= $p1;

  if ($n < $p) {
    #### top leftwards
    return (-2*$n + $outer,
            _odd($n) + $inner);
  }
  $n -= $p;

  if ($n < $p1) {
    #### left downwards
    return (_odd($n) - $outer,
            -2*$n + $inner);
  }
  $n -= $p1;

  if ($n < $p1) {
    #### bottom rightwards: $n
    return (2*$n - $inner,
            - _odd($n) - $inner);
  }
  $n -= $p1;



  ### three ...
  #
  if ($n < $p) {
    ### right upwards, eg 12 ...
    return (_odd($n) + $inner,
            2*$n - $outer);
  }
  $n -= $p;

  if ($n < $p1) {
    ### top leftwards, eg 14 ...
    return (-2*$n + $inner,
            - _odd($n) + $outer);
  }
  $n -= $p1;

  if ($n < $p1) {
    ### left downwards, eg 15 ...
    return (- _odd($n) - $inner,
            -2*$n + $inner);
  }
  $n -= $p1;

  if ($n < $p1) {
    ### bottom rightwards, eg 16 ...
    return (2*$n - $outer,
            - _odd($n) - $inner);
  }
  $n -= $p1;


  ### four ...
  #
  if ($n <= 1) {
    ### special 17 upwards ...
    return ($n + $outer - 2,
            2*$n - $outer);
  }
  if ($n < $p) {
    ### right upwards ...
    return (- _odd($n) + $outer,
            2*$n - $outer);
  }
  $n -= $p;

  if ($n < $p) {
    ### top leftwards, eg 19 ...
    return (-2*$n + $outer,
            - _odd($n) + $outer);
  }
  $n -= $p;

  if ($n < $p) {
    ### left downwards, eg 21 ...
    return (_odd($n) - $outer,
            -2*$n + $outer);
  }
  $n -= $p;

  if ($n < $p) {
    ### bottom rightwards, eg 23 ...
    return (2*$n - $outer,
            _odd($n) - $outer);
  }
  $n -= $p;

  ### step outwards, eg 25 ...
  return (2*$n + $outer,
          - _odd($n) - $outer);
}


#   157   92  113  134  155   90  111  132  153   88  109  130  151
#   114  135  156   91  112  133  154   89  110  131  152   87  108
#    93  158   73   32   45   58   71   30   43   56   69  150  129
#   136  115   46   59   72   31   44   57   70   29   42  107   86
#   159   94   33   74   21    4    9   14   19   68   55  128  149
#   116  137   60   47   10   15   20    3    8   41   28   85  106
#    95  160|  75   34 |  5   22    1   18   13 | 54   67| 148  127
#   138  117   48   61   16   11   24    7    2   27   40  105   84
#   161   96   35   76   23    6   17   12   25   66   53  126  147
#   118  139   62   49   78   37   64   51   80   39   26   83  104
#    97  162   77   36   63   50   79   38   65   52   81  146  125
#   140  119  164   99  142  121  166  101  144  123  168  103   82
#   163   98  141  120  165  100  143  122  167  102  145  124  169

sub xy_to_n {
  my ($self, $x, $y) = @_;
  $x = round_nearest ($x);
  $y = round_nearest ($y);
  if ($x == 0 && $y == 0) {
    return $self->{'n_start'};
  }

  my $r = max(abs($x),abs($y));
  my $d = int (($r+1)/2);  # ring number, counting $x=1,2 as $d==1
  $r -= (~$r & 1);  # next lower odd number
  ### $d
  ### $r

  if ($y >= $r) {
    ### top horizontal
    my $xodd = ($x & 1);
    $x = ($x - $xodd) / 2;
    ### $xodd
    ### $x

    # x odd
    # [3,30,89,180,303]         (16*$d**2 + -21*$d + 8)
    # [14,57,132,239,378,549]   (16*$d**2 + -5*$d + 3)
    # 
    # [9,44,111,210,341,504]    (16*$d**2 + -13*$d + 6)
    # [20,71,154,269,416]       (16*$d**2 + 3*$d + 1)

    my $n = 16*$d*$d - $x;
    if (($x ^ $y ^ $d) & 1) {
      if ($xodd) {
        return $n -5*$d + 2 + $self->{'n_start'};
      } else {
        return $n -13*$d + 5 + $self->{'n_start'};
      }
    } else {
      if ($xodd) {
        return $n -21*$d + 7 + $self->{'n_start'};
      } else {
        return $n + 3*$d + $self->{'n_start'};
      }
    }
  }

  # the lower left outer corner 25,81,169,etc belongs on the bottom
  # horizontal, it's not an extension downwards from the right vertical
  # (positions N=18,66,146,etc), hence $x!=-$y
  #
  if ($x >= $r && $x != -$y) {
    ### right vertical
    my $yodd = ($y & 1);
    $y = ($y - $yodd) / 2;
    ### $yodd
    ### $y

    # y odd
    # [3, 28,85, 174,295, 448,633]  (16*$d**2 + -23*$d + 10)
    # [8,41, 106,203, 332,493]      (16*$d**2 + -15*$d + 7)
    #
    # y even
    # [13,54,127,232,369,538]      (16*$d**2 + -7*$d + 4)
    # [18,67,148,261,406,583,792]  (16*$d**2 + $d + 1)
    #
    my $n = 16*$d*$d + $y;
    if (($x ^ $y ^ $d) & 1) {
      if ($yodd) {
        return $n -15*$d + 6 + $self->{'n_start'};
      } else {
        return $n -7*$d + 3 + $self->{'n_start'};
      }
    } else {
      if ($yodd) {
        return $n -23*$d + 9 + $self->{'n_start'};
      } else {
        return $n + $d + $self->{'n_start'};
      }
    }
  }

  if ($y <= -$r) {
    ### bottom horizontal
    my $xodd = ($x & 1);
    $x = ($x - $xodd) / 2;
    ### $xodd
    ### $x

    # x odd
    # [7,38,101,196,323]         (16*$d**2 + -17*$d + 8)
    # [12,51,122,225,360,527]    (16*$d**2 + -9*$d + 5)
    #
    # x even
    # [17,64,143,254,397,572]    (16*$d**2 + -1*$d + 2)
    # [24,79,166,285,436]        (16*$d**2 + 7*$d + 1)

    my $n = 16*$d*$d + $x;
    if (($x ^ $y ^ $d) & 1) {
      if ($xodd) {
        return $n -9*$d + 4 + $self->{'n_start'};
      } else {
        return $n -1*$d + 1 + $self->{'n_start'};
      }
    } else {
      if ($xodd) {
        return $n -17*$d + 7 + $self->{'n_start'};
      } else {
        return $n + 7*$d + $self->{'n_start'};
      }
    }
  }

  if ($x <= -$r) {
    ### left vertical
    my $yodd = ($y & 1);
    $y = ($y - $yodd) / 2;
    ### $yodd
    ### $y

    # y odd
    # [10,47,116,217,350,515]  (16*$d**2 + -11*$d + 5)
    # [15,60,137,246,387]      (16*$d**2 + -3*$d + 2)
    #
    # y even
    # [5,34,95,188,313]    (16*$d**2 + -19*$d + 8)
    # [22,75,160,277,426]  (16*$d**2 + 5*$d + 1)
    #
    my $n = 16*$d*$d - $y;
    if (($x ^ $y ^ $d) & 1) {
      if ($yodd) {
        return $n -11*$d + 4 + $self->{'n_start'};
      } else {
        return $n -19*$d + 7 + $self->{'n_start'};
      }
    } else {
      if ($yodd) {
        return $n -3*$d + 1 + $self->{'n_start'};
      } else {
        return $n + 5*$d + $self->{'n_start'};
      }
    }
  }
}

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

  $x1 = round_nearest ($x1);
  $y1 = round_nearest ($y1);
  $x2 = round_nearest ($x2);
  $y2 = round_nearest ($y2);

  my $x = max(abs($x1),abs($x2));
  my $y = max(abs($y1),abs($y2));

  my $d = max(abs($x),abs($y));
  $d += ($d & 1);  # next even number if not already even
  ### $x
  ### $y
  ### $d
  ### is: $d*$d

  $d = 2*$d+1;  # width of whole square
  # ENHANCE-ME: find actual minimum if rect doesn't cover 0,0
  return ($self->{'n_start'},
          $self->{'n_start'} + $d*$d);
}

1;
__END__

=for stopwords versa Ryde Math-PlanePath OEIS

=head1 NAME

Math::PlanePath::KnightSpiral -- integer points around a square, by chess knight moves

=head1 SYNOPSIS

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

=head1 DESCRIPTION

This path traverses the plane by an infinite "knight's tour" in the form
of a square spiral.

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

                 ^
        -2  -1  X=0  1   2   3

Each step is a chess knight's move 1 across and 2 along, or vice versa.  The
pattern makes 4 cycles on a 2-wide path around a square before stepping
outwards to do the same again to a now bigger square.  The above sample
shows the first 4-cycle around the central 1, then stepping out at 26 and
beginning to go around the outside of the 5x5 square.

An attractive traced out picture of the path can be seen at the following
page (quarter way down under "Open Knight's Tour"),

=over

L<http://www.borderschess.org/KTart.htm>
L<http://www.borderschess.org/KTinfinity.gif>
L<http://www.borderschess.org/Infinite.gif>

=back

See L<math-image> to draw the path lines too.

=head1 FUNCTIONS

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

=over 4

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

Create and return a new knight spiral 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.

For C<$n < 1> the return is an empty list, it being considered the path
starts at 1.

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

Return the point number for coordinates C<$x,$y>.  C<$x> and C<$y> are
each rounded to the nearest integer, which has the effect of treating each N
in the path as centred in a square of side 1, so the entire plane is
covered.

=back

=head1 OEIS

This Knight's tour is in Sloane's OEIS following the Knight spiral and
giving the resulting X,Y location by the C<SquareSpiral> numbering.  There's
eight forms for 4 rotations and spiralling the same or opposite directions.

=over

L<http://oeis.org/A068608> (etc)

=back

    permutations
      A068608   same knight and square spiral directions
      A068609   rotate 90 degrees
      A068610   rotate 180 degrees
      A068611   rotate 270 degrees
      A068612   rotate 180 degrees, spiral opp dir (X negate)
      A068613   rotate 270 degrees, spiral opp dir
      A068614   spiral opposite direction (Y negate)
      A068615   rotate 90 degrees, spiral opp dir (X,Y transpose)

See F<examples/knights-oeis.pl> for a sample program printing the values of
A068608.

=head1 SEE ALSO

L<Math::PlanePath>,
L<Math::PlanePath::SquareSpiral>

=head1 HOME PAGE

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

=head1 LICENSE

Copyright 2010, 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/>.

=cut