# 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::HeptSpiralSkewed;
use 5.004;
use strict;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 120;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'round_nearest';
# uncomment this to run the ### lines
# use Smart::Comments;
use constant xy_is_visited => 1;
sub x_negative_at_n {
my ($self) = @_;
return $self->n_start + 3;
}
sub y_negative_at_n {
my ($self) = @_;
return $self->n_start + 5;
}
sub _UNDOCUMENTED__dxdy_list_at_n {
my ($self) = @_;
return $self->n_start + 8;
}
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 four plus NW
0,1, # N
-1,1, # NW
-1,0, # W
0,-1); # S
use constant dsumxy_minimum => -1; # W,S straight
use constant dsumxy_maximum => 1; # N,E straight
use constant ddiffxy_minimum => -2; # NW diagonal
use constant ddiffxy_maximum => 1;
use constant dir_maximum_dxdy => (0,-1); # South
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 $self;
}
# base South-West diagonal
# d = [ 1, 2, 3, 4 ]
# n = [ 0, 5, 17, 36 ]
# N = (7/2 d^2 - 11/2 d + 2)
# = (7/2*$d**2 - 11/2*$d + 2)
# = ((7/2*$d - 11/2)*$d + 2)
# d = 11/14 + sqrt(2/7 * $n + 9/196)
# = (11 + 14*sqrt(2/7 * $n + 9/196))/14
# = (sqrt(56*$n + 9) + 11)/14
#
# split North Y axis
# d = [ 1, 2, 3 ]
# n = [ 2, 11, 27 ]
# N = (7*$d-3)*$d/2
sub n_to_xy {
my ($self, $n) = @_;
#### HeptSpiralSkewed n_to_xy: $n
$n = $n - $self->{'n_start'}; # adjust to N=0 at origin X=0,Y=0
if ($n < 0) { return; }
my $d = int((sqrt(56*$n+9) + 11) / 14);
### $d
### d frac: (sqrt(56*$n+9) + 11) / 14
$n -= (7*$d-3)*$d/2;
### remainder: $n
if ($n < 0) { # split at Y axis
if ($n >= -$d) {
#### right diagonal ...
return (-$n,
$n + $d);
}
$n += $d;
if ($n < 1-$d) {
### bottom horizontal ...
return ($n + 2*$d-1, 1-$d);
}
### right vertical ...
return ($d, $n);
}
if ($n <= $d) {
### top horizontal ...
return (-$n, $d);
}
#### left vertical ...
return (-$d,
-$n + 2*$d);
}
sub xy_to_n {
my ($self, $x, $y) = @_;
$x = round_nearest ($x);
$y = round_nearest ($y);
if ($x >= 0 && $y >= 0) {
### slope
# relative to the y=0 base same as above
# d = [ 1, 2, 3, 4 ]
# n = [ 2, 10, 25, 47 ]
# n = (7/2*$d**2 + -5/2*$d + 1)
# = (3.5*$d - 2.5)*$d + 1
#
my $d = $x + $y;
return (7*$d - 5)*$d/2 + $y + $self->{'n_start'};
}
my $d = max(abs($x),abs($y));
my $n = (7*$d - 5)*$d/2;
if ($y == $d) {
### top horizontal
return $n+$d - $x + $self->{'n_start'};
}
if ($y == -$d) {
### bottom horizontal
return $n + 5*$d + $x + $self->{'n_start'};
}
if ($x == $d) {
### right vertical
return $n + $y + $self->{'n_start'};
}
# ($x == - $d)
### left vertical
return $n + 3*$d - $y + $self->{'n_start'};
}
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
$x1 = round_nearest ($x1);
$y1 = round_nearest ($y1);
$x2 = round_nearest ($x2);
$y2 = round_nearest ($y2);
my $d = 0;
foreach my $x ($x1, $x2) {
foreach my $y ($y1, $y2) {
$d = max ($d,
1 + ($x > 0 && $y > 0
? $x+$y # slope
: max(abs($x),abs($y)))); # square corners
}
}
# ENHANCE-ME: find actual minimum if rect doesn't cover 0,0
return ($self->{'n_start'},
$self->{'n_start'} + (7*$d - 5)*$d/2);
}
1;
__END__
=for stopwords PlanePath Ryde Math-PlanePath OEIS
=head1 NAME
Math::PlanePath::HeptSpiralSkewed -- integer points around a skewed seven sided spiral
=head1 SYNOPSIS
use Math::PlanePath::HeptSpiralSkewed;
my $path = Math::PlanePath::HeptSpiralSkewed->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path makes a seven-sided spiral by cutting one corner of a square
=cut
# math-image --path=HeptSpiralSkewed --expression='i<=44?i:0' --output=numbers_dash --size=60x18
=pod
31-30-29-28 3
| \
32 14-13-12 27 2
| | \ \
33 15 4--3 11 26 1
| | | \ \ \
34 16 5 1--2 10 25 <- Y=0
| | | | |
35 17 6--7--8--9 24 -1
| | |
36 18-19-20-21-22-23 -2
|
37-38-39-40-41-... -3
^
-3 -2 -1 X=0 1 2 3
The path is as if around a heptagon, with the left and bottom here as two
sides of the heptagon straightened out, and the flat top here skewed across
to fit a square grid.
=head2 N Start
The default is to number points starting N=1 as shown above. An optional
C<n_start> can give a different start, in the same pattern. For example to
start at 0,
=cut
# math-image --path=HeptSpiralSkewed,n_start=0 --expression='i<=40?i:0' --output=numbers --size=60x11
=pod
30 29 28 27 n_start => 0
31 13 12 11 26
32 14 3 2 10 25
33 15 4 0 1 9 24
34 16 5 6 7 8 23
35 17 18 19 20 21 22
36 37 38 39 40 ...
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::HeptSpiralSkewed-E<gt>new ()>
=item C<$path = Math::PlanePath::HeptSpiralSkewed-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 N
in the path as centred in a square of side 1, so the entire plane is
covered.
=back
=head1 FORMULAS
=head2 N to X,Y
It's convenient to work in terms of Nstart=0 and to take each loop as
beginning on the South-West diagonal,
=cut
# math-image --path=HeptSpiralSkewed,n_start=0 --expression='i<=37?i:0' --output=numbers_dash --size=25x16
=pod
top length = d
30-29-28-27
| \
31 26 diagonal length = d
left | \
length 32 25
= 2*d | \
33 0 24
| | right
34 . 23 length = d-1
| |
35 17-18-19-20-21-22
|
. bottom length = 2*d-1
The SW diagonal is N=0,5,17,36,etc which is
N = (7d-11)*d/2 + 2 # starting d=1 first loop
This can be inverted to get d from N
d = floor( (sqrt(56*N+9)+11)/14 )
The side lengths are as shown above. The first loop is d=1 and for it the
"right" vertical length is zero, so no such side on that first loop 0 E<lt>=
N < 5.
=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
A140065 N on Y axis
n_start=0
A001106 N on X axis, 9-gonal numbers
A218471 N on Y axis
A022265 N on X negative axis
A179986 N on Y negative axis, second 9-gonals
A195023 N on X=Y diagonal
A022264 N on North-West diagonal
A186029 N on South-West diagonal
A024966 N on South-East diagonal
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::SquareSpiral>,
L<Math::PlanePath::PentSpiralSkewed>,
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