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# Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 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/>.
# http://www.woollythoughts.com/afghans/peano.html
#     Knitting
#
# http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node36.html
#     Closed path, curved parts
#
# http://www.wolframalpha.com/entities/calculators/Peano_curve/jh/4o/im/
#     Curved corners tilted to a diamond, or is it an 8-step pattern?
#
# http://www.davidsalomon.name/DC2advertis/AppendC.pdf
#
# http://www.cut-the-knot.org/do_you_know/hilbert.shtml
#     Java applet
#
package Math::PlanePath::HilbertCurve;
use 5.004;
use strict;
#use List::Util 'max','min';
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 129;
use Math::PlanePath;
use Math::PlanePath::Base::NSEW;
@ISA = ('Math::PlanePath::Base::NSEW',
        'Math::PlanePath');
use Math::PlanePath::Base::Generic
  'is_infinite',
  'round_nearest';
use Math::PlanePath::Base::Digits
  'round_down_pow',
  'round_up_pow',
  'bit_split_lowtohigh',
  'digit_split_lowtohigh',
  'digit_join_lowtohigh';
# uncomment this to run the ### lines
#use Smart::Comments;
use constant n_start => 0;
use constant class_x_negative => 0;
use constant class_y_negative => 0;
*xy_is_visited = \&Math::PlanePath::Base::Generic::xy_is_visited_quad1;
#------------------------------------------------------------------------------
# state=0    3--2   plain
#               |
#            0--1
#
# state=4    1--2  transpose
#            |  |
#            0  3
#
# state=8
#
# state=12   3  0  rot180 + transpose
#            |  |
#            2--1
#
# generated by tools/hilbert-curve-table.pl
my @next_state = (4,0,0,12, 0,4,4,8, 12,8,8,4, 8,12,12,0);
my @digit_to_x = (0,1,1,0, 0,0,1,1, 1,0,0,1, 1,1,0,0);
my @digit_to_y = (0,0,1,1, 0,1,1,0, 1,1,0,0, 1,0,0,1);
my @yx_to_digit = (0,1,3,2, 0,3,1,2, 2,3,1,0, 2,1,3,0);
my @min_digit = (0,0,1,0, 0,1,3,2, 2,undef,undef,undef,
                 0,0,3,0, 0,2,1,1, 2,undef,undef,undef,
                 2,2,3,1, 0,0,1,0, 0,undef,undef,undef,
                 2,1,1,2, 0,0,3,0, 0);
my @max_digit = (0,1,1,3, 3,2,3,3, 2,undef,undef,undef,
                 0,3,3,1, 3,3,1,2, 2,undef,undef,undef,
                 2,3,3,2, 3,3,1,1, 0,undef,undef,undef,
                 2,2,1,3, 3,1,3,3, 0);
sub n_to_xy {
  my ($self, $n) = @_;
  ### HilbertCurve n_to_xy(): $n
  ### hex: sprintf "%#X", $n
  if ($n < 0) { return; }
  if (is_infinite($n)) { return ($n,$n); }
  my $int = int($n);
  $n -= $int;   # fraction part
  my @ndigits = digit_split_lowtohigh($int,4);
  my $state = ($#ndigits & 1 ? 4 : 0);
  my $dirstate   = ($#ndigits & 1 ? 0 : 4); # default if all $ndigit==3
  my (@xbits, @ybits);
  foreach my $i (reverse 0 .. $#ndigits) {    # digits high to low
    my $ndigit = $ndigits[$i];
    $state += $ndigit;
    if ($ndigit != 3) {
      $dirstate = $state;  # lowest non-3 digit
    }
    $xbits[$i] = $digit_to_x[$state];
    $ybits[$i] = $digit_to_y[$state];
    $state = $next_state[$state];
  }
  my $zero = ($int * 0); # inherit bigint 0
  return ($n * ($digit_to_x[$dirstate+1] - $digit_to_x[$dirstate]) # frac
          + digit_join_lowtohigh (\@xbits, 2, $zero),
          $n * ($digit_to_y[$dirstate+1] - $digit_to_y[$dirstate]) # frac
          + digit_join_lowtohigh (\@ybits, 2, $zero));
}
sub xy_to_n {
  my ($self, $x, $y) = @_;
  ### HilbertCurve xy_to_n(): "$x, $y"
  $x = round_nearest ($x);
  if (is_infinite($x)) { return $x; }
  $y = round_nearest ($y);
  if (is_infinite($y)) { return $y; }
  if ($x < 0 || $y < 0) {
    return undef;
  }
  my @xbits = bit_split_lowtohigh($x);
  my @ybits = bit_split_lowtohigh($y);
  my $numbits = max($#xbits,$#ybits);
  my @ndigits;
  my $state = ($numbits & 1 ? 4 : 0);
  foreach my $i (reverse 0 .. $numbits) {   # high to low
    ### at: "state=$state  xbit=".($xbits[$i]||0)." ybit=".($ybits[$i]||0)
    my $ndigit = $yx_to_digit[$state + 2*($ybits[$i]||0) + ($xbits[$i]||0)];
    $ndigits[$i] = $ndigit;
    $state = $next_state[$state+$ndigit];
  }
  ### @ndigits
  return digit_join_lowtohigh(\@ndigits, 4,
                              $x * 0 * $y); # inherit bignum 0
}
# rect_to_n_range() finds the exact minimum/maximum N in the given rectangle.
#
# The strategy is similar to xy_to_n(), except that at each bit position
# instead of taking a bit of x,y from the input instead those bits are
# chosen from among the 4 sub-parts according to which has the maximum N and
# is within the given target rectangle.  The final result is both an $n_max
# and a $x_max,$y_max which is its position, but only the $n_max is
# returned.
#
# At a given sub-part the comparisons ask whether x1 is above or below the
# midpoint, and likewise x2,y1,y2.  Since x2>=x1 and y2>=y1 there's only 3
# combinations of x1>=cmp,x2>=cmp, not 4.
# exact
sub rect_to_n_range {
  my ($self, $x1,$y1, $x2,$y2) = @_;
  ### HilbertCurve rect_to_n_range(): "$x1,$y1, $x2,$y2"
  $x1 = round_nearest ($x1);
  $y1 = round_nearest ($y1);
  $x2 = round_nearest ($x2);
  $y2 = round_nearest ($y2);
  ($x1,$x2) = ($x2,$x1) if $x1 > $x2;
  ($y1,$y2) = ($y2,$y1) if $y1 > $y2;
  if ($x2 < 0 || $y2 < 0) {
    return (1, 0); # rectangle outside first quadrant
  }
  my $n_min = my $n_max
    = my $x_min = my $y_min
      = my $x_max = my $y_max
        = ($x1 * 0 * $x2 * $y1 * $y2); # inherit bignum 0
  my ($len, $level) = round_down_pow (($x2 > $y2 ? $x2 : $y2),
                                      2);
  ### $len
  ### $level
  if (is_infinite($level)) {
    return (0, $level);
  }
  my $min_state = my $max_state = ($level & 1 ? 4 : 0);
  while ($level >= 0) {
    {
      my $x_cmp = $x_min + $len;
      my $y_cmp = $y_min + $len;
      my $digit = $min_digit[3*$min_state
                             + ($x1 >= $x_cmp ? 2 : $x2 >= $x_cmp ? 1 : 0)
                             + ($y1 >= $y_cmp ? 6 : $y2 >= $y_cmp ? 3 : 0)];
      $n_min = 4*$n_min + $digit;
      $min_state += $digit;
      if ($digit_to_x[$min_state]) { $x_min += $len; }
      if ($digit_to_y[$min_state]) { $y_min += $len; }
      $min_state = $next_state[$min_state];
    }
    {
      my $x_cmp = $x_max + $len;
      my $y_cmp = $y_max + $len;
      my $digit = $max_digit[3*$max_state
                             + ($x1 >= $x_cmp ? 2 : $x2 >= $x_cmp ? 1 : 0)
                             + ($y1 >= $y_cmp ? 6 : $y2 >= $y_cmp ? 3 : 0)];
      $n_max = 4*$n_max + $digit;
      $max_state += $digit;
      if ($digit_to_x[$max_state]) { $x_max += $len; }
      if ($digit_to_y[$max_state]) { $y_max += $len; }
      $max_state = $next_state[$max_state];
    }
    $len = int($len/2);
    $level--;
  }
  return ($n_min, $n_max);
}
#------------------------------------------------------------------------------
# shared by Math::PlanePath::AR2W2Curve and others
sub level_to_n_range {
  my ($self, $level) = @_;
  return (0, 4**$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, $exp) = round_up_pow ($n+1, 4);
  return $exp;
}
#------------------------------------------------------------------------------
1;
__END__
HideShow 28 lines of Pod
=for :stopwords Ryde Math-PlanePath PlanePaths OEIS ZOrderCurve ZOrder Peano Gosper's HAKMEM Jorg Arndt's bitwise bignums fxtbook Ueber stetige Abbildung einer Linie auf ein FlE<228>chenstE<252>ck Mathematische Annalen DOI ascii lookup Arndt PlanePath ie Sergey Kitaev Toufik Mansour Automata Combinatorics Preprint localized
=head1 NAME
Math::PlanePath::HilbertCurve -- 2x2 self-similar quadrant traversal
=head1 SYNOPSIS
 use Math::PlanePath::HilbertCurve;
 my $path = Math::PlanePath::HilbertCurve->new;
 my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path is an integer version of the curve described by Hilbert in 1891
for filling a unit square.  It traverses a quadrant of the plane one step at
a time in a self-similar 2x2 pattern,
=over
David Hilbert, "Ueber die stetige Abbildung einer Linie auf ein
FlE<228>chenstE<252>ck", Mathematische Annalen, volume 38, number 3, 1891,
pages 459-460, DOI 10.1007/BF01199431.
=back
=cut
# Someone's copy:
# https://www.maths.ed.ac.uk/~v1ranick/papers/hilbert.pdf
#
# Someone's copy existed in the past:
# http://notendur.hi.is/oddur/hilbert/gcs-wrapper-1.pdf
HideShow 283 lines of Pod
=pod
           ...
        |   |
      7 |  63--62  49--48--47  44--43--42
        |       |   |       |   |       |
      6 |  60--61  50--51  46--45  40--41
        |   |           |           |
      5 |  59  56--55  52  33--34  39--38
        |   |   |   |   |   |   |       |
      4 |  58--57  54--53  32  35--36--37
        |                   |
      3 |   5---6   9--10  31  28--27--26
        |   |   |   |   |   |   |       |
      2 |   4   7---8  11  30--29  24--25
        |   |           |           |
      1 |   3---2  13--12  17--18  23--22
        |       |   |       |   |       |
    Y=0 |   0---1  14--15--16  19--20--21
        +----------------------------------
          X=0   1   2   3   4   5   6   7
The start is a sideways U shape N=0 to N=3, then four of those are put
together in an upside-down U as
    5,6    9,10
    4,7--- 8,11
      |      |
    3,2   13,12
    0,1   14,15--
The orientation of the sub parts ensure the starts and ends are adjacent, so
3 next to 4, 7 next to 8, and 11 next to 12.
The process repeats, doubling in size each time and alternately sideways or
upside-down U with invert and/or transpose as necessary in the sub-parts.
The pattern is sometimes drawn with the first step 0->1 upwards instead of
to the right.  First step right is used here for consistency with other
PlanePaths.  Swap X and Y for upwards first instead.
See F<examples/hilbert-path.pl> for a sample program printing the path
pattern in ascii.
=head2 Level Ranges
Within a power-of-2 square 2x2, 4x4, 8x8, 16x16 etc (2^k)x(2^k) at the
origin, all the N values 0 to 2^(2*k)-1 are within the square.  The maximum
3, 15, 63, 255 etc 2^(2*k)-1 is alternately at the top left or bottom right
corner.
Because each step is by 1, the distance along the curve between two X,Y
points is the difference in their N values (as from C<xy_to_n()>).
On the X=Y diagonal N=0,2,8,10,32,etc is the integers using only digits 0
and 2 in base 4, or equivalently have even-numbered bits 0, like x0y0...z0.
=head2 Locality
The Hilbert curve is fairly well localized in the sense that a small
rectangle (or other shape) is usually a small range of N.  This property is
used in some database systems to store X,Y coordinates using the resulting
Hilbert curve N as an index.  A search through a 2-D region is then usually
a fairly modest linear search through N values.  C<rect_to_n_range()> gives
exact N range for a rectangle, or see notes L<Rectangle to N Range> below
for calculating on any shape.
The N range can be large when crossing sub-parts.  In the sample above it
can be seen for instance adjacent points X=0,Y=3 and X=0,Y=4 have rather
widely spaced N values 5 and 58.
Fractional X,Y values can be indexed by extending the N calculation down
into X,Y binary fractions.  The code here doesn't do that, but could be
pressed into service by moving the binary point in X and Y an even number of
places, the same in each.  (An odd number of bits would require swapping X,Y
to compensate for the alternating transpose in part 0.)  The resulting
integer N is then divided down by a corresponding multiple-of-4 binary
places.
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::HilbertCurve-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.
Fractional positions give an X,Y position along a straight line between the
integer positions.  Integer positions are always just 1 apart either
horizontally or vertically, so the effect is that the fraction part is an
offset along either C<$x> or C<$y>.
=item C<$n = $path-E<gt>xy_to_n ($x,$y)>
Return an integer point number for coordinates C<$x,$y>.  Each integer N is
considered the centre of a unit square and an C<$x,$y> within that square
returns N.
=item C<($n_lo, $n_hi) = $path-E<gt>rect_to_n_range ($x1,$y1, $x2,$y2)>
The returned range is exact, meaning C<$n_lo> and C<$n_hi> are the smallest
and biggest in the rectangle.
=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)>.
=back
=head1 FORMULAS
=head2 N to X,Y
Converting N to X,Y coordinates is reasonably straightforward.  The top two
bits of N is a configuration
    3--2                    1--2
       |    or transpose    |  |
    0--1                    0  3
according to whether it's an odd or even bit-pair position.  Then within
each of the "3" sub-parts there's also inverted forms
    1--0        3  0
    |           |  |
    2--3        2--1
Working N from high to low with a state variable can record whether there's
a transpose, an invert, or both, being four states altogether.  A bit pair
0,1,2,3 from N then gives a bit each of X,Y according to the configuration
and a new state which is the orientation of that sub-part.  Bill Gosper's
HAKMEM item 115 has this with either bit operations or a table for the state
and X,Y bits,
=over
L<https://dspace.mit.edu/handle/1721.1/6086>,
L<http://www.inwap.com/pdp10/hbaker/hakmem/topology.html#item115>
=back
X<fxtbook>And C++ code based on that in Jorg Arndt's book,
=over
L<http://www.jjj.de/fxt/#fxtbook> (section 1.31.1)
=back
It also works to process N from low to high, at each stage applying any
transpose (swap X,Y) and/or invert (bitwise NOT) to the low X,Y bits
generated so far.  This works because there's no "reverse" sections, or
equivalently the curve is the same forward and reverse.  Low to high saves
locating the top bits of N, but if using bignums then the bitwise inverts of
the full X,Y values will be much more work.
=head2 X,Y to N
X,Y to N can follow the table approach from high to low taking one bit from
X and Y each time.  The state table of N-pair -> X-bit,Y-bit is reversible,
and a new state is based on the N-pair thus obtained (or could be based on
the X,Y bits if that mapping is combined into the state transition table).
=head2 Rectangle to N Range
An easy over-estimate of the maximum N in a region can be had by finding the
next bigger (2^k)x(2^k) square enclosing the region.  This means the biggest
X or Y rounded up to the next power of 2, so
    find lowest k with 2^k > max(X,Y)
    N_max = 2^(2k) - 1
Or equivalently rounding down to the next lower power of 2,
    find highest k with 2^k <= max(X,Y)
    N_max = 2^(2*(k+1)) - 1
An exact N range can be found by following the high to low N to X,Y
procedure above.  Start at the 2^(2k) bit pair position in an N bigger than
the desired region and choose 2 bits for N to give a bit each of X and Y.
The X,Y bits are based on the state table as above and the bits chosen for N
are those for which the resulting X,Y sub-square overlaps some of the target
region.  The smallest N similarly, choosing the smallest bit pair for N
which overlaps.
The biggest and smallest N digit for a sub-part can be found with a lookup
table.  The X range might cover one or both sub-parts, and the Y range
similarly, for a total 4 possible configurations.  Then a table of
state+coverage -E<gt> digit gives the minimum and maximum N bit-pair, and
state+digit gives a new state the same as X,Y to N.
Biggest and smallest N must be calculated with separate state and X,Y values
since they track down different N bits and thus different states.  But they
take the same number of steps from an enclosing level down to level 0 and
can thus be done in a single loop.
The N range for any shape can be found this way, not just a rectangle like
C<rect_to_n_range()>.  At each level the procedure only depends on asking
which of the four sub-parts overlaps some of the target area.
=head2 Direction
Each step between successive N values is always 1 up, down, left or right.
The next direction can be calculated from N in the high-to-low procedure
above by watching for the lowest non-3 digit and noting the direction from
that digit towards digit+1.  That can be had from the state+digit -E<gt> X,Y
table looking up digit and digit+1, or alternatively a further table
encoding state+digit -E<gt> direction.
The reason for taking only the lowest non-3 digit is that in a 3 sub-part
the direction it goes is determined by the next higher level.  For example
at N=11 the direction is down for the inverted-U of the next higher level
N=0,4,8,12.
This non-3 (or non whatever highest digit) is a general procedure and can be
used on any state-based high-to-low procedure of self-similar curves.  In
the current code, it's used to apply a fractional part of N in the correct
direction but is not otherwise made directly available.
Because the Hilbert curve has no "reverse" sections it also works to build a
direction from low to high N digits.  1 and 2 digits make no change to the
orientation, 0 digit is a transpose, and a 3 digit is a rotate and
transpose, except that low 3s are transpose-only (no rotate) for the same
reason as taking the lowest non-3 above.
Jorg Arndt in the fxtbook above notes the direction can be obtained just by
counting 3s in n and -n (the twos-complement).  This numbers segments
starting n=1, unlike PlanePath here starting N=0, so it becomes
    N+1 count 3s  / 0 mod 2   S or E
                  \ 1 mod 2   N or W
    -(N+1) count 3s  / 0 mod 2   N or E
                     \ 1 mod 2   S or W
For the twos-complement negation, an even number of base-4 digits of N must
be taken.  Because -(N+1) = ~N, ie. a ones-complement, the second part is
also
    N count 0s          / 0 mod 2   N or E
    in even num digits  \ 1 mod 2   S or W
Putting the two together then
    N count 0s   N+1 count 3s    direction (0=E,1=N,etc)
    in base 4    in base 4
      0 mod 2      0 mod 2          0
      1 mod 2      0 mod 2          3
      0 mod 2      1 mod 2          1
      1 mod 2      1 mod 2          2
=head2 Segments in Direction
The number of segments in each direction is calculated in
=over
Sergey Kitaev, Toufik Mansour and Patrice SE<233>E<233>bold, "Generating the
Peano Curve and Counting Occurrences of Some Patterns", Journal of Automata,
Languages and Combinatorics, volume 9, number 4, 2004, pages 439-455.
L<http://personal.strath.ac.uk/sergey.kitaev/publications.html>,
L<http://www.jalc.de/issues/2004/issue_9_4/jalc-2004-439-455.php>
(Preprint as Sergey Kitaev and Toufik Mansour, "The Peano Curve and Counting
Occurrences of Some Patterns", October 2002.
L<http://arxiv.org/abs/math/0210268/>.)
=cut
# http://personal.strath.ac.uk/sergey.kitaev/
# http://personal.strath.ac.uk/sergey.kitaev/publications.html
# https://www.researchgate.net/publication/230802620_Generating_the_Peano_curve_and_counting_occurrences_of_some_patterns
#     full text html
# https://www.researchgate.net/profile/Toufik_Mansour/publication/230802620_Generating_the_Peano_curve_and_counting_occurrences_of_some_patterns/links/00b7d5299d5a89bb02000000/Generating-the-Peano-curve-and-counting-occurrences-of-some-patterns.pdf
#
# https://personal.cis.strath.ac.uk/sergey.kitaev/publications.html
#     old home
# https://personal.cis.strath.ac.uk/sergey.kitaev/index_files/Papers/peano.ps
#     old postscript
# http://web.archive.org/web/1id_/http://personal.cis.strath.ac.uk/sergey.kitaev/index_files/Papers/peano.ps
#     not in wayback
#
# http://www.math.chalmers.se/Math/Research/Combinatorics/preprints/kitaev/kitmansel.ps
#    216253
# https://hal.archives-ouvertes.fr/hal-01645132
#    journal ref only (?)
#
# http://www.jalc.de/issues/2004/issue_9_4/jalc-2004-439-455.php
# http://www.jalc.de/issues/2004/issue_9_4/fulltext/pp-439-455.pdf
# https://doi.org/10.25596/jalc-2004-439
#     account required
#
# http://www.ms.uky.edu/~kitaev/       [no such ?]
# http://www.ms.uky.edu/~readdy/DM/DM-RESEARCH/kitaev_research.html
#
# Toufik Mansour, list only no links:
# http://math.haifa.ac.il/toufik/manpubff.html
HideShow 57 lines of Pod
=pod
=back
Their form is based on keeping the top-most U shape fixed and expanding
sub-parts.  This means the end segments alternate vertical and horizontal in
successive expansion levels.
    direction            k=1              2       2
      1 to 4                            *---*   *---*
                           2           1|  3|   |1  |3
        1                *---*          *   *---*   *
        |               1|   |3        1| 4   2   4 |3
    4--- ---2            *   *          *---*   *---*
        |                                  1|   |3       k=2
        3                               *---*   *---*
                                          2       2
    count segments in direction, for k >= 1
    d(1,k) = 4^(k-1)                = 1,4,16,64,256,1024,4096,...
    d(2,k) = 4^(k-1) + 2^(k-1) - 1  = 1,5,19,71,271,1055,4159,...
    d(3,k) = 4^(k-1)                = 1,4,16,64,256,1024,4096,...
    d(4,k) = 4^(k-1) - 2^(k-1)      = 0,2,12,56,240, 992,4032,...
                             (A000302, A099393, A000302, A020522)
    total segments d(1,k)+d(2,k)+d(3,k)+d(4,k) = 4^k - 1
The path form here keeps the first segment direction fixed.  This means a
transpose 1E<lt>-E<gt>2 and 3E<lt>-E<gt>4 in odd levels.  The result is to
take the alternate d values as follows.  For k=0 there is a single point N=0
so no line segments at all and so c(dir,0)=0.
    first 4^k-1 segments
    c(1,k) = / 0                        if k=0
     North   | 4^(k-1) + 2^(k-1) - 1    if k odd >= 1
             \ 4^(k-1)                  if k even >= 2
      = 0, 1, 4, 19, 64, 271, 1024, 4159, 16384, ...
    c(2,k) = / 0                        if k=0
     East    | 4^(k-1)                  if k odd >= 1
             \ 4^(k-1) + 2^(k-1) - 1    if k even >= 2
      = 0, 1, 5, 16, 71, 256, 1055, 4096, 16511, ...
    c(3,k) = / 0                        if k=0
     South   | 4^(k-1) - 2^(k-1)        if k odd >= 1
             \ 4^(k-1)                  if k even >= 2
      = 0, 0, 4, 12, 64, 240, 1024, 4032, 16384, ...
    c(4,k) = / 0                        if k=0
     West    | 4^(k-1)                  if k odd >= 1
             \ 4^(k-1) - 2^(k-1)        if k even >= 2
      = 0, 1, 2, 16, 56, 256, 992, 4096, 16256, ...
=cut
# not in OEIS: 4, 19, 64, 271, 1024, 4159, 16384
# not in OEIS: 5, 16, 71, 256, 1055, 4096, 16511
# not in OEIS: 4, 12, 64, 240, 1024, 4032, 16384
# not in OEIS: 2, 16, 56, 256, 992, 4096, 16256
HideShow 8 lines of Pod
=pod
The segment N=4^k-1 to N=4^k is North (direction 1) when k odd, or East
(direction 2) when k even.  That could be added to the respective cases in
c(1,k) and c(2,k) if desired.
=cut
# (d1(k) = 4^(k-1));               for(k=0,8,print1(d1(k),","))
# (d2(k) = 4^(k-1) + 2^(k-1) - 1); for(k=0,8,print1(d2(k),","))
# (d3(k) = 4^(k-1));               for(k=0,8,print1(d3(k),","))
# (d4(k) = 4^(k-1) - 2^(k-1));     for(k=0,8,print1(d4(k),","))
# (c1(k) = if(k==0,0,if(k%2,d2(k),d1(k)))); for(k=0,8,print1(c1(k),", "))
# (c2(k) = if(k==0,1,if(k%2,d1(k),d2(k)))); for(k=0,8,print1(c2(k),", "))
# (c3(k) = if(k==0,0,if(k%2,d3(k),d4(k)))); for(k=0,8,print1(c3(k),", "))
# (c4(k) = if(k==0,0,if(k%2,d4(k),d3(k)))); for(k=0,8,print1(c4(k),", "))
#
# N=0 to N=4^k so first 4^k segments
# (east4k(k) = c2(k) + if(k>=2&&k%2==0,1,0)); for(k=0,8,print1(east4k(k),", "))
# 1,1,6,16,72,256,1056,4096,16512,
# 1,1,6,16,72,256,1056,4096
HideShow 34 lines of Pod
=pod
=head2 Hamming Distance
The Hamming distance between integers X and Y is the number of bit positions
where the two values differ when written in binary.  On the Hilbert curve
each bit-pair of N becomes a bit of X and a bit of Y,
       N      X   Y
    ------   --- ---
    0 = 00    0   0
    1 = 01    1   0     <- difference 1 bit
    2 = 10    1   1
    3 = 11    0   1     <- difference 1 bit
So the Hamming distance for N=0to3 is 1 at N=1 and N=3.  At higher levels,
these X,Y bits may be transposed (swapped) or rotated by 180 or both.
A transpose swapping XE<lt>-E<gt>Y doesn't change the bit difference.
A rotate by 180 is a flip 0E<lt>-E<gt>1 of the bit in both X and Y, so that
doesn't change the bit difference either.
On that basis, the Hamming distance X,Y is the number of base4 digits of N
which are 01 or 11.  If bit positions are counted from 0 for the least
significant bit then
    X,Y coordinates of N
    HammingDist(X,Y) = count 1-bits at even bit positions in N
           = 0,1,0,1, 1,2,1,2, 0,1,0,1, 1,2,1,2, ... (A139351)
See also L<Math::PlanePath::CornerReplicate/Hamming Distance> which is the
same formula, but arising directly from 01 or 11, no transpose or rotate.
=cut
# (d1(k) = 4^(k-1));               for(k=0,8,print1(d1(k),","))
# (d2(k) = 4^(k-1) + 2^(k-1) - 1); for(k=0,8,print1(d2(k),","))
# (d3(k) = 4^(k-1));               for(k=0,8,print1(d3(k),","))
# (d4(k) = 4^(k-1) - 2^(k-1));     for(k=0,8,print1(d4(k),","))
# (c1(k) = if(k==0,0,if(k%2,d1(k),d2(k)))); for(k=0,8,print1(c1(k),", "))
# (c2(k) = if(k==0,1,if(k%2,d2(k),d1(k)))); for(k=0,8,print1(c2(k),", "))
# (c3(k) = if(k==0,0,if(k%2,d3(k),d4(k)))); for(k=0,8,print1(c3(k),", "))
# (c4(k) = if(k==0,0,if(k%2,d4(k),d3(k)))); for(k=0,8,print1(c4(k),", "))
HideShow 130 lines of Pod
=pod
=head1 OEIS
This path is in Sloane's OEIS in several forms,
=over
L<http://oeis.org/A059252> (etc)
=back
    A059253    X coord
    A059252    Y coord
    A059261    X+Y
    A059285    X-Y
    A163547    X^2+Y^2 = radius squared
    A139351    HammingDist(X,Y)
    A059905    X xor Y, being ZOrderCurve X
    A163365    sum N on diagonal
    A163477    sum N on diagonal, divided by 4
    A163482    N values on X axis
    A163483    N values on Y axis
    A062880    N values on diagonal X=Y (digits 0,2 in base 4)
    A163538    dX -1,0,1 change in X
    A163539    dY -1,0,1 change in Y
    A163540    absolute direction of each step (0=E,1=N,2=W,3=S)
    A163541    absolute direction, swapped X,Y
    A163542    relative direction (ahead=0,right=1,left=2)
    A163543    relative direction, swapped X,Y
    A083885    count East segments N=0 to N=4^k (first 4^k segs)
    A163900    distance dX^2+dY^2 between Hilbert and ZOrder
    A165464    distance dX^2+dY^2 between Hilbert and Peano
    A165466    distance dX^2+dY^2 between Hilbert and transposed Peano
    A165465    N where Hilbert and Peano have same X,Y
    A165467    N where Hilbert and Peano have transposed same X,Y
The following take points of the plane in various orders, each value in the
sequence being the N of the Hilbert curve at those positions.
    A163355    N by the ZOrderCurve points sequence
    A163356      inverse, ZOrderCurve by Hilbert points order
    A166041    N by the PeanoCurve points sequence
    A166042      inverse, PeanoCurve N by Hilbert points order
    A163357    N by diagonals like Math::PlanePath::Diagonals with
                 first Hilbert step along same axis the diagonals start
    A163358      inverse
    A163359    N by diagonals, transposed start along the opposite axis
    A163360      inverse
    A163361    A163357 + 1, numbering the Hilbert N's from N=1
    A163362      inverse
    A163363    A163355 + 1, numbering the Hilbert N's from N=1
    A163364      inverse
The above sequences are permutations of the integers since all X,Y positions
of the first quadrant are covered by each path (Hilbert, ZOrder, Peano).
The inverse permutations can be thought of taking X,Y positions in the
Hilbert order and asking what N the ZOrder, Peano or Diagonals path would
put there.
The A163355 permutation by ZOrderCurve can be considered for repeats or
cycles,
    A163905    ZOrderCurve permutation A163355 applied twice
    A163915    ZOrderCurve permutation A163355 applied three times
    A163901    fixed points (N where X,Y same in both curves)
    A163902    2-cycle points
    A163903    3-cycle points
    A163890    cycle lengths, points by N
    A163904    cycle lengths, points by diagonals
    A163910    count of cycles in 4^k blocks
    A163911    max cycle length in 4^k blocks
    A163912    LCM of cycle lengths in 4^k blocks
    A163914    count of 3-cycles in 4^k blocks
    A163909      those counts for even k only
    A163891    N of previously unseen cycle length
    A163893      first differences of those A163891
    A163894    smallest value not an n-cycle
    A163895      position of new high in A163894
    A163896      value of new high in A163894
    A163907    ZOrderCurve permutation twice, on points by diagonals
    A163908      inverse of this
See F<examples/hilbert-oeis.pl> for a sample program printing the A163359
permutation values.
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::HilbertSides>,
L<Math::PlanePath::HilbertSpiral>
L<Math::PlanePath::PeanoCurve>,
L<Math::PlanePath::ZOrderCurve>,
L<Math::PlanePath::BetaOmega>,
L<Math::PlanePath::KochCurve>
L<Math::Curve::Hilbert>,
L<Algorithm::SpatialIndex::Strategy::QuadTree>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 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
# Local variables:
# compile-command: "math-image --path=HilbertCurve --lines --scale=20"
# End:
# math-image --path=HilbertCurve --all --output=numbers_dash --size=70x30