# Copyright 2010, 2011, 2012, 2013, 2014, 2015 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::PentSpiralSkewed;
use 5.004;
use strict;
#use List::Util 'min','max';
*min = \&Math::PlanePath::_min;
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 122;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'round_nearest';
# uncomment this to run the ### lines
#use Smart::Comments;
use constant parameter_info_array =>
[
Math::PlanePath::Base::Generic::parameter_info_nstart1(),
];
use constant xy_is_visited => 1;
sub x_negative_at_n {
my ($self) = @_;
return $self->n_start + 3;
}
sub y_negative_at_n {
my ($self) = @_;
return $self->n_start + 4;
}
sub _UNDOCUMENTED__dxdy_list_at_n {
my ($self) = @_;
return $self->n_start + 6;
}
use constant dx_minimum => -1;
use constant dx_maximum => 1;
use constant dy_minimum => -1;
use constant dy_maximum => 1;
use constant _UNDOCUMENTED__dxdy_list => (1,0, # E
0,1, # N
-1,1, # NW
-1,-1, # SW
1,-1, # SE
);
use constant dsumxy_minimum => -2; # SW diagonal
use constant dsumxy_maximum => 1;
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 new {
my $self = shift->SUPER::new(@_);
if (! defined $self->{'n_start'}) {
$self->{'n_start'} = $self->default_n_start;
}
return $self;
}
sub n_to_xy {
my ($self, $n) = @_;
#### n_to_xy: $n
# adjust to N=0 at origin X=0,Y=0
$n = $n - $self->{'n_start'};
if ($n < 0) { return; }
my $d = int( (sqrt(40*$n+9)+7) / 10);
$n -= (5*$d-1)*$d/2;
if ($n < -$d) {
$n += 2*$d;
if ($n < 1) {
# bottom horizontal
return ($n+$d-1, -$d+1);
} else {
# lower right vertical ...
return ($d, $n-$d);
}
} else {
if ($n <= $d) {
### top diagonals left and right ...
return (-$n,
-abs($n) + $d);
} else {
### lower left diagonal ...
return ($n - 2*$d,
-$n + $d);
}
}
}
sub xy_to_n {
my ($self, $x, $y) = @_;
$x = round_nearest ($x);
$y = round_nearest ($y);
if ($x > 0 && $y < 0) {
# vertical downwards at x=0
# d = [ 1, 2, 3 ]
# n = [ 5, 14, 28 ]
# n = (5/2*$d**2 + 3/2*$d + 1)
# so
my $d = max($x-1, -$y);
### lower right square part
### $d
return ((5*$d + 3)*$d/2
+ $x
+ ($x > $d ? $y+$d : 0)
+ $self->{'n_start'});
}
# vertical at x=0
# d = [ 1, 2, 3 ]
# n = [ 3, 10, 22 ]
# n = (5/2*$d**2 + -1/2*$d + 1)
#
my $d = abs($x)+abs($y);
return ((5*$d - 1)*$d/2
- $x
+ ($y < 0 ? 2*($d+$x) : 0)
+ $self->{'n_start'});
}
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### PentSpiralSkewed rect_to_n_range(): $x1,$y1, $x2,$y2
my $d = 0;
foreach my $x ($x1, $x2) {
$x = round_nearest ($x);
foreach my $y ($y1, $y2) {
$y = round_nearest ($y);
my $this_d = 1 + ($x > 0 && $y < 0
? max($x,-$y)
: abs($x)+abs($y));
### $x
### $y
### $this_d
$d = max($d, $this_d);
}
}
### $d
return ($self->{'n_start'},
$self->{'n_start'} + 5*$d*($d-1)/2 + 2);
}
1;
__END__
=for stopwords Ryde Math-PlanePath OEIS
=head1 NAME
Math::PlanePath::PentSpiralSkewed -- integer points in a pentagonal shape
=head1 SYNOPSIS
use Math::PlanePath::PentSpiralSkewed;
my $path = Math::PlanePath::PentSpiralSkewed->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path makes a pentagonal (five-sided) spiral with points skewed so as to
fit a square grid and fully cover the plane.
10 ... 2
/ \ \
11 3 9 20 1
/ / \ \ \
12 4 1--2 8 19 <- Y=0
\ \ | |
13 5--6--7 18 -1
\ |
14-15-16-17 -2
^ ^ ^ ^ ^ ^
-2 -1 X=0 1 2 3 ...
The pattern is similar to the C<SquareSpiral> but cuts three corners which
makes each cycle is faster. Each cycle is just 5 steps longer than the
previous (where it's 8 for a C<SquareSpiral>).
=head2 N Start
The default is to number points starting N=1 as shown above. An optional
C<n_start> can give a different start, in the same pattern. For example to
start at 0,
=cut
# math-image --path=PentSpiralSkewed,n_start=0 --expression='i<=57?i:0' --output=numbers --size=60x11
=pod
38 n_start => 0
39 21 37 ...
40 22 9 20 36 57
41 23 10 2 8 19 35 56
42 24 11 3 0 1 7 18 34 55
43 25 12 4 5 6 17 33 54
44 26 13 14 15 16 32 53
45 27 28 29 30 31 52
46 47 48 49 50 51
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::PentSpiral-E<gt>new ()>
=item C<$path = Math::PlanePath::PentSpiral-E<gt>new (n_start =E<gt> $n)>
Create and return a new path object.
=item C<$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
point in the path as a square of side 1.
=back
=head1 OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this
path include
=over
L<http://oeis.org/A192136> (etc)
=back
n_start=1 (the default)
A192136 N on X axis, (5*n^2 - 3*n + 2)/2
A140066 N on Y axis
A116668 N on X negative axis, (5n^2 + n + 2)/2
A134238 N on Y negative axis
A158187 N on North-West diagonal, 10*n^2 + 1
A005891 N on South-East diagonal, centred pentagonals
n_start=0
A000566 N on X axis, heptagonal numbers
A005476 N on Y axis
A005475 N on X negative axis
A147875 N on Y negative axis, second heptagonals
A033583 N on North-West diagonal, 10*n^2
A028895 N on South-East diagonal, 5*triangular
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::SquareSpiral>,
L<Math::PlanePath::DiamondSpiral>,
L<Math::PlanePath::HexSpiralSkewed>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015 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