This file is indexed.

/usr/share/perl5/Math/PlanePath/ComplexRevolving.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
# Copyright 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::ComplexRevolving;
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
  'is_infinite',
  'round_nearest';
use Math::PlanePath::Base::Digits
  'bit_split_lowtohigh',
  'digit_join_lowtohigh';

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


use constant n_start => 0;
use constant xy_is_visited => 1;
use constant x_negative_at_n => 5;
use constant y_negative_at_n => 7;
# use constant dir_maximum_dxdy => (0,0);  # supremum, almost full way


#------------------------------------------------------------------------------
# b=i+1
# X+iY = b^e0 + i*b^e1 + ... + i^t * b^et
#
sub n_to_xy {
  my ($self, $n) = @_;
  ### ComplexRevolving n_to_xy(): $n

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

  {
    my $int = int($n);
    ### $int
    ### $n
    if ($n != $int) {
      my ($x1,$y1) = $self->n_to_xy($int);
      my ($x2,$y2) = $self->n_to_xy($int+1);
      my $frac = $n - $int;  # inherit possible BigFloat
      my $dx = $x2-$x1;
      my $dy = $y2-$y1;
      return ($frac*$dx + $x1, $frac*$dy + $y1);
    }
    $n = $int;       # BigFloat int() gives BigInt, use that
  }

  my $x = my $y = ($n * 0);  # inherit bignum 0

  if (my @digits = bit_split_lowtohigh($n)) {
    my $bx = $x + 1;    # inherit bignum 1
    my $by = $x;       # 0
    for (;;) {
      if (shift @digits) { # low to high
        $x += $bx;
        $y += $by;
        ($bx,$by) = (-$by,$bx);  # (bx+by*i)*i = bx*i - by,  is rotate +90
      }
      @digits || last;

      # (bx+by*i) * (i+1)
      #   = bx*i+bx + -by + by*i
      #   = (bx-by) + i*(bx+by)
      ($bx,$by) = ($bx - $by,
                   $bx + $by);
    }
  }

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

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

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

  foreach my $overflow ($x+$y, $x-$y) {
    if (is_infinite($overflow)) { return $overflow; }
  }

  my $zero = $x * 0 * $y;  # inherit bignum 0

  my @n;
  while ($x || $y) {
    ### at: "$x,$y  power=$power  n=$n"

    # (a+bi)*(i+1) = (a-b)+(a+b)i
    #
    if (($x % 2) == ($y % 2)) {  # x+y even
      push @n, 0;
    } else {
      ### not multiple of 1+i, take e0=0 for b^e0=1
      # [(x+iy)-1]/i
      #   = [(x-1)+yi]/i
      #   = y + (x-1)/i
      #   = y + (1-x)*i    # rotate -90

      push @n, 1;
      ($x,$y) = ($y, 1-$x);

      ### sub and div to: "$x,$y"
    }

    # divide i+1 = mul (i-1)/(i^2 - 1^2)
    #            = mul (i-1)/-2
    # is (i*y + x) * (i-1)/-2
    #  x = (-x - y)/-2  = (x + y)/2
    #  y = (-y + x)/-2  = (y - x)/2
    #
    ### assert: (($x+$y)%2)==0
    ($x,$y) = (($x+$y)/2, ($y-$x)/2);
  }

  return digit_join_lowtohigh(\@n,2,$zero);
}

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

  my $xm = max(abs($x1),abs($x2));
  my $ym = max(abs($y1),abs($y2));

  return (0, int (32*($xm*$xm + $ym*$ym)));
}

#------------------------------------------------------------------------------
# levels

sub level_to_n_range {
  my ($self, $level) = @_;
  return (0, 2**$level - 1);
}
sub n_to_level {
  my ($self, $n) = @_;
  if ($n < 0) { return undef; }
  if (is_infinite($n)) { return $n; }
  $n = round_nearest($n);
  my ($pow, $exp) = round_down_pow ($n, 2);
  return $exp + 1;
}

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

=for stopwords eg Ryde Math-PlanePath ie Nstart Nlevel Seminumerical et

=head1 NAME

Math::PlanePath::ComplexRevolving -- points in revolving complex base i+1

=head1 SYNOPSIS

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

=head1 DESCRIPTION

X<Knuth, Donald>This path traverses points by a complex number base i+1 with
turn factor i (+90 degrees) at each 1 bit.  This is the "revolving binary
representation" of Knuth's Seminumerical Algorithms section 4.1 exercise 28.

=cut

# math-image --path=ComplexRevolving --expression='i<64?i:0' --output=numbers --size=79x30

=pod

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

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

The 1 bits in N are exponents e0 to et, in increasing order,

    N = 2^e0 + 2^e1 + ... + 2^et        e0 < e1 < ... < et

and are applied to a base b=i+1 as

    X+iY = b^e0 + i * b^e1 + i^2 * b^e2 + ... + i^t * b^et

Each 2^ek has become b^ek base b=i+1.  The i^k is an extra factor i at each
1 bit of N, causing a rotation by +90 degrees for the bits above it.  Notice
the factor is i^k not i^ek, ie. it increments only with the 1-bits of N, not
the whole exponent.

A single bit N=2^k is the simplest and is X+iY=(i+1)^k.  These
N=1,2,4,8,16,etc are at successive angles 45, 90, 135, etc degrees (the same
as in C<ComplexPlus>).  But points N=2^k+1 with two bits means X+iY=(i+1) +
i*(i+1)^k and that factor "i*" is a rotation by 90 degrees so points
N=3,5,9,17,33,etc are in the next quadrant around from their preceding
2,4,8,16,32.

As per the exercise in Knuth it's reasonably easy to show that this
calculation is a one-to-one mapping between integer N and complex integer
X+iY, so the path covers the plane and visits all points once each.

=head1 FUNCTIONS

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

=over 4

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

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.

=back

=head2 Level Methods

=over

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

Return C<(0, 2**$level - 1)>.

=back

=head1 SEE ALSO

L<Math::PlanePath>,
L<Math::PlanePath::ComplexMinus>,
L<Math::PlanePath::ComplexPlus>,
L<Math::PlanePath::DragonCurve>

Donald Knuth, "The Art of Computer Programming", volume 2 "Seminumerical
Algorithms", section 4.1 exercise 28.

=head1 HOME PAGE

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

=head1 LICENSE

Copyright 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