# Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016 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';
$VERSION = 123;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
*_sqrtint = \&Math::PlanePath::_sqrtint;
use Math::PlanePath::Base::Generic
'round_nearest';
# uncomment this to run the ### lines
#use Smart::Comments;
use constant xy_is_visited => 1;
sub x_negative_at_n {
my ($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 $d = int ((7 + _sqrtint($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, 2015, 2016 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