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# Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.
# Kanga "Number Mosaics" rotated to
#
#                ...-16---15
#                           \
#                  6----5   14
#                 /      \    \
#                7   1    4   13
#               /   /    /    /
#              8   2----3   12
#               \           /
#                9---10---11
#
#
# Could go pointy end with same loop/step, or point to the right
#
#                    13--12--11
#                   /         |
#                 14  4---3  10
#                /  /     |   |
#              15  5  1---2   9
#                \  \         |
#                 16  6---7---8
#                   \             |
#                    17--18--19--20
#
package Math::PlanePath::HexSpiral;
use 5.004;
use strict;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 129;
use Math::PlanePath;
*_sqrtint = \&Math::PlanePath::_sqrtint;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
  'round_nearest',
  'xy_is_even';
# uncomment this to run the ### lines
#use Devel::Comments '###';
use Math::PlanePath::SquareSpiral;
*parameter_info_array = \&Math::PlanePath::SquareSpiral::parameter_info_array;
#      2w+3 --- 3w/2+3 -- w+4
#      /                     \
#    2w+4         0 -------- w+3  *
#      \                         /
#      2w+5 ----------------- 3w+7    w=2; 1+3*w+7=14
#                       ^
#                      X=0
sub x_negative_at_n {
  my ($self) = @_;
  return $self->n_start + ($self->{'wider'} ? 0 : 3);
}
sub y_negative_at_n {
  my ($self) = @_;
  return $self->n_start + 2*$self->{'wider'} + 5;
}
sub _UNDOCUMENTED__dxdy_list_at_n {
  my ($self) = @_;
  return $self->n_start + 3*$self->{'wider'} + 7;
}
sub rsquared_minimum {
  my ($self) = @_;
  return ($self->{'wider'} % 2
          ? 1   # odd "wider" minimum X=1,Y=0
          : 0); # even "wider" includes X=0,Y=0
}
*sumabsxy_minimum = \&rsquared_minimum;
use constant dx_minimum => -2;
use constant dx_maximum => 2;
use constant dy_minimum => -1;
use constant dy_maximum => 1;
*_UNDOCUMENTED__dxdy_list = \&Math::PlanePath::_UNDOCUMENTED__dxdy_list_six;
use constant absdx_minimum => 1;
*absdiffxy_minimum = \&rsquared_minimum;
use constant dsumxy_minimum => -2; # SW diagonal
use constant dsumxy_maximum => 2;  # dX=+2 and diagonal
use constant ddiffxy_minimum => -2; # NW diagonal
use constant ddiffxy_maximum => 2;  # SE diagonal
use constant dir_maximum_dxdy => (1,-1); # South-East
use constant turn_any_right => 0; # only left or straight
sub _UNDOCUMENTED__turn_any_left_at_n {
  my ($self) = @_;
  return $self->n_start + $self->{'wider'} + 1;
}
#------------------------------------------------------------------------------
sub new {
  my $self = shift->SUPER::new (@_);
  # parameters
  $self->{'wider'} ||= 0;  # default
  if (! defined $self->{'n_start'}) {
    $self->{'n_start'} = $self->default_n_start;
  }
  return $self;
}
# wider==0
# diagonal down and to the left
#   d = [ 0,  1,  2,  3 ]
#   N = [ 1,  6, 17,  34 ]
#   N = (3*$d**2 + 2*$d + 1)
#   d = -1/3 + sqrt(1/3 * $n + -2/9)
#     = (-1 + sqrt(3*$n - 2)) / 3
#
# wider==1
# diagonal down and to the left
#   d = [ 0,  1,  2,  3 ]
#   N = [ 1,  8, 21,  40 ]
#   N = (3*$d**2 + 4*$d + 1)
#   d = -2/3 + sqrt(1/3 * $n + 1/9)
#     = (-2 + sqrt(3*$n + 1)) / 3
#
# wider==2
# diagonal down and to the left
#   d = [ 0, 1,  2,  3,  4 ]
#   N = [ 1, 10, 25, 46, 73 ]
#   N = (3*$d**2 + 6*$d + 1)
#   d = -1 + sqrt(1/3 * $n + 2/3)
#     = (-3 + sqrt(3*$n + 6)) / 3
#
# N = 3*$d*$d + (2+2*$w)*$d + 1
#   = (3*$d + 2 + 2*$w)*$d + 1
# d = (-1-w + sqrt(3*$n + ($w+2)*$w - 2)) / 3
#   = (sqrt(3*$n + ($w+2)*$w - 2) -1-w) / 3
sub n_to_xy {
  my ($self, $n) = @_;
  #### n_to_xy: "$n   wider=$self->{'wider'}"
  $n = $n - $self->{'n_start'};  # N=0 basis
  if ($n < 0) { return; }
  my $w = $self->{'wider'};
  my $d = int((_sqrtint(3*$n + ($w+2)*$w + 1) - 1 - $w) / 3);
  #### d frac: ((_sqrtint(3*$n + ($w+2)*$w + 1) - 1 - $w) / 3)
  #### $d
  $n += 1; # N=1 basis
  $n -= (3*$d + 2 + 2*$w)*$d + 1;
  #### remainder: $n
  $d = $d + 1; # no warnings if $d==inf
  if ($n <= $d+$w) {
    #### bottom horizontal
    $d = -$d + 1;
    return (2*$n + $d - $w,
            $d);
  }
  $n -= $d+$w;
  if ($n <= $d-1) {
    #### right lower diagonal, being 1 shorter: $n
    return ($n + $d + 1 + $w,
            $n - $d + 1);
  }
  $n -= $d-1;
  if ($n <= $d) {
    #### right upper diagonal: $n
    return (-$n + 2*$d + $w,
            $n);
  }
  $n -= $d;
  if ($n <= $d+$w) {
    #### top horizontal
    return (-2*$n + $d + $w,
            $d);
  }
  $n -= $d+$w;
  if ($n <= $d) {
    #### left upper diagonal
    return (-$n - $d - $w,
            -$n + $d );
  }
  #### left lower diagonal
  $n -= $d;
  return ($n - 2*$d - $w,
          -$n);
}
sub xy_is_visited {
  my ($self, $x, $y) = @_;
  return xy_is_even($self,$x+$self->{'wider'},$y);
}
sub xy_to_n {
  my ($self, $x, $y) = @_;
  ### xy_to_n(): "$x, $y"
  $x = round_nearest ($x);
  $y = round_nearest ($y);
  my $w = $self->{'wider'};
  if (($x ^ $y ^ $w) & 1) {
    return undef;  # nothing on odd squares
  }
  my $ay = abs($y);
  my $ax = abs($x) - $w;
  if ($ax > $ay) {
    my $d = ($ax + $ay)/2;  # x+y is even
    if ($x > 0) {
      ### right ends
      ### $d
      return ((3*$d - 2 + 2*$w)*$d - $w  # horizontal to the right
              + $y                       # offset up or down
              + $self->{'n_start'});
    } else {
      ### left ends
      return ((3*$d + 1 + 2*$w)*$d    # horizontal to the left
              - $y                    # offset up or down
              + $self->{'n_start'});
    }
  } else {
    my $d = $ay;
    if ($y > 0) {
      ### top horizontal
      ### $d
      return ((3*$d + 2*$w)*$d      # diagonal up to the left
              + (-$d - $x-$w) / 2   # negative offset rightwards
              + $self->{'n_start'});
    } else {
      ### bottom horizontal, and centre horizontal
      ### $d
      ### offset: $d
      return ((3*$d + 2 + 2*$w)*$d   # diagonal down to the left
              + ($x + $w + $d)/2     # offset rightwards
              + $self->{'n_start'});
    }
  }
}
# not exact
sub rect_to_n_range {
  my ($self, $x1,$y1, $x2,$y2) = @_;
  ### HexSpiral rect_to_n_range(): $x1,$y1, $x2,$y2
  my $w = $self->{'wider'};
  # symmetric in +/-y, and biggest y is biggest n
  my $y = max (abs($y1), abs($y2));
  # symmetric in +/-x, and biggest x
  my $x = max (abs($x1), abs($x2));
  if ($x >= $w) {
    $x -= $w;
  }
  # in the middle horizontal path parts y determines the loop number
  # in the end parts diagonal distance, 2 apart
  my $d = ($y >= $x
           ? $y                 # middle
           : ($x + $y + 1)/2);  # ends
  $d = int($d) + 1;
  # diagonal downwards bottom left being the end of a revolution
  # s=0
  # s=1  n=7
  # s=2  n=19
  # s=3  n=37
  # s=4  n=61
  # n = 3*$d*$d + 3*$d + 1
  #
  # ### gives: "sum $d is " . (3*$d*$d + 3*$d + 1)
  # ENHANCE-ME: find actual minimum if rect doesn't cover 0,0
  return ($self->{'n_start'},
          (3*$d + 3 + 2*$w)*$d + $self->{'n_start'});
}
1;
__END__
HideShow 86 lines of Pod
=for stopwords PlanePath Ryde Math-PlanePath ie OEIS
=head1 NAME
Math::PlanePath::HexSpiral -- integer points around a hexagonal spiral
=head1 SYNOPSIS
 use Math::PlanePath::HexSpiral;
 my $path = Math::PlanePath::HexSpiral->new;
 my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path makes a hexagonal spiral, with points spread out horizontally to
fit on a square grid.
             28 -- 27 -- 26 -- 25                  3
            /                    \
          29    13 -- 12 -- 11    24               2
         /     /              \     \
       30    14     4 --- 3    10    23            1
      /     /     /         \     \    \
    31    15     5     1 --- 2     9    22    <- Y=0
      \     \     \              /     /
       32    16     6 --- 7 --- 8    21           -1
         \     \                    /
          33    17 -- 18 -- 19 -- 20              -2
            \
             34 -- 35 ...                         -3
     ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
    -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6
Each horizontal gap is 2, so for instance n=1 is at X=0,Y=0 then n=2 is at
X=2,Y=0.  The diagonals are just 1 across, so n=3 is at X=1,Y=1.  Each
alternate row is offset from the one above or below.  The result is a
triangular lattice per L<Math::PlanePath/Triangular Lattice>.
The octagonal numbers 8,21,40,65, etc 3*k^2-2*k fall on a horizontal
straight line at Y=-1.  In general straight lines are 3*k^2 + b*k + c.
A plain 3*k^2 goes diagonally up to the left, then b is a 1/6 turn
anti-clockwise, or clockwise if negative.  So b=1 goes horizontally to the
left, b=2 diagonally down to the left, b=3 diagonally down to the right,
etc.
=head2 Wider
An optional C<wider> parameter makes the path wider, stretched along the top
and bottom horizontals.  For example
    $path = Math::PlanePath::HexSpiral->new (wider => 2);
gives
                                ... 36----35                   3
                                            \
                21----20----19----18----17    34               2
               /                          \     \
             22     8---- 7---- 6---- 5    16    33            1
            /     /                    \     \    \
          23     9     1---- 2---- 3---- 4    15    32    <- Y=0
            \     \                          /     /
             24    10----11----12----13----14    31           -1
               \                               /
                25----26----27----28---29----30               -2
           ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
          -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7
The centre horizontal from N=1 is extended by C<wider> many extra places,
then the path loops around that shape.  The starting point N=1 is shifted to
the left by wider many places to keep the spiral centred on the origin
X=0,Y=0.  Each horizontal gap is still 2.
Each loop is still 6 longer than the previous, since the widening is
basically a constant amount added into each loop.
=head2 N Start
The default is to number points starting N=1 as shown above.  An optional
C<n_start> can give a different start with the same shape etc.  For example
to start at 0,
=cut
# math-image --path=HexSpiral,n_start=0 --all --output=numbers --size=70x9
HideShow 119 lines of Pod
=pod
    n_start => 0
             27    26    25    24                    3
          28    12    11    10    23                 2
       29    13     3     2     9    22              1
    30    14     4     0     1     8    21      <- Y=0
       31    15     5     6     7    20   ...       -1
          32    16    17    18    19    38          -2
             33    34    35    36    37             -3
                       ^
    -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6
In this numbering the X axis N=0,1,8,21,etc is the octagonal numbers
3*X*(X+1).
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::HexSpiral-E<gt>new ()>
=item C<$path = Math::PlanePath::HexSpiral-E<gt>new (wider =E<gt> $w)>
Create and return a new hex spiral object.  An optional C<wider> parameter
widens the path, it defaults to 0 which is no widening.
=item C<($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<$y> are
each rounded to the nearest integer, which has the effect of treating each
C<$n> in the path as a square of side 1.
Only every second square in the plane has an N, being those where X,Y both
odd or both even.  If C<$x,$y> is a position without an N, ie. one of X,Y
odd the other even, then the return is C<undef>.
=back
=head1 OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to
this path include
=over
L<http://oeis.org/A056105> (etc)
=back
    A328818    X coordinate
    A307012    Y coordinate
    A307011    (X-Y)/2
    A307013    (X+Y)/2
    A056105    N on X axis
    A056106    N on X=Y diagonal
    A056107    N on North-West diagonal
    A056108    N on negative X axis
    A056109    N on South-West diagonal
    A003215    N on South-East diagonal
    A063178    total sum N previous row or diagonal
    A135711    boundary length of N hexagons 
    A135708    grid sticks of N hexagons 
    n_start=0
      A001399    N positions of turns (extra initial 1)
      A000567    N on X axis, octagonal numbers
      A049451    N on X negative axis
      A049450    N on X=Y diagonal north-east
      A033428    N on north-west diagonal, 3*k^2
      A045944    N on south-west diagonal, octagonal numbers second kind
      A063436    N on WSW slope dX=-3,dY=-1
      A028896    N on south-east diagonal
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::HexSpiralSkewed>,
L<Math::PlanePath::HexArms>,
L<Math::PlanePath::TriangleSpiral>,
L<Math::PlanePath::TriangularHypot>
=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 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