# Copyright 2010, 2011, 2012, 2013 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 .
package Math::PlanePath::PyramidSpiral;
use 5.004;
use strict;
#use List::Util 'min';
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 114;
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;
use constant dx_minimum => -1;
use constant dx_maximum => 1;
use constant dy_minimum => -1;
use constant dy_maximum => 1;
use constant absdx_minimum => 1;
use constant dsumxy_minimum => -2; # SW diagonal
use constant dsumxy_maximum => 1;
use constant ddiffxy_minimum => -2; # NW diagonal
use constant ddiffxy_maximum => 1;
use constant dir_maximum_dxdy => (-1,-1); # South-West
#------------------------------------------------------------------------------
sub new {
my $self = shift->SUPER::new (@_);
if (! defined $self->{'n_start'}) {
$self->{'n_start'} = $self->default_n_start;
}
return $self;
}
# bottom left corner, N=0 basis
# [ 0, 1, 2, 3 ],
# [ 0, 4, 16, 36 ]
# N = (4 d^2)
# = (4*$d**2)
# = (4*$d**2)
# d = 0 + sqrt(1/4 * $n + 0)
# = sqrt($n)/2
#
sub n_to_xy {
my ($self, $n) = @_;
#### PyramidSpiral n_to_xy: $n
$n = $n - $self->{'n_start'}; # start N=0, and warn if $n==undef
if ($n < 0) { return; }
my $d = int(sqrt(int($n))/2);
$n -= (4*$d+2)*$d; # to $n==0 on Y negative axis
if ($n <= 2*$d+1) {
### bottom horizontal ...
return ($n, -$d);
}
$n -= 4*$d+2;
### sides, remainder pos/neg from top ...
return (-$n,
- abs($n) + $d + 1);
}
# negative y, x=0 centres
# [ 1, 2, 3 ]
# [ 7, 21, 43 ]
# n = (4*$y*$y + 2*abs($y) + 1)
#
# positive y, x=0 centres
# [ 1, 2, 3 ]
# [ 3, 13, 31 ]
# n = (4*$d*$d + -2*$d + 1)
#
sub xy_to_n {
my ($self, $x, $y) = @_;
$x = round_nearest ($x);
$y = round_nearest ($y);
### xy_to_n(): "$x,$y"
if ($y < 0 && abs($x) <= 2*-$y) {
### bottom horizontal
return 4*$y*$y - 2*$y + $x + $self->{'n_start'};
}
### sides diagonal
my $k = 2 * (abs($x) + $y);
return $k*($k-1) - $x + $self->{'n_start'};
}
# Each row N increases away from some midpoint.
# Each column N increase away from some midpoint.
# So maximum must be at one of the corners.
#
# maybe:
# Minimum in row is at X=0 when Y>=0
# or at X=2*Y slope when Y<0
# Minimum in column is at Y=floor(X/2) when X<=0
# or at Y=-floor(X/2) when X>=0
#
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
# ENHANCE-ME: find exact minimum
return ($self->n_start,
max ($self->xy_to_n($x1,$y1),
$self->xy_to_n($x1,$y2),
$self->xy_to_n($x2,$y1),
$self->xy_to_n($x2,$y2)));
}
1;
__END__
=for stopwords pronic PlanePath Ryde Math-PlanePath OEIS
=head1 NAME
Math::PlanePath::PyramidSpiral -- integer points drawn around a pyramid
=head1 SYNOPSIS
use Math::PlanePath::PyramidSpiral;
my $path = Math::PlanePath::PyramidSpiral->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path makes a pyramid shaped spiral,
=cut
# math-image --path=PyramidSpiral --all --output=numbers_dash
=pod
31 3
/ \
32 13 30 2
/ / \ \
33 14 3 12 29 1
/ / / \ \ \
34 15 4 1--2 11 28 ... <- Y=0
/ / / \ \ \
35 16 5--6--7--8--9-10 27 52 -1
/ / \ \
36 17-18-19-20-21-22-23-24-25-26 51 -2
/ \
37-38-39-40-41-42-43-44-45-46-47-48-49-50 -3
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
XThe perfect squares 1,4,9,16 fall one before the bottom
left corner of each loop, and the Xpronic numbers
2,6,12,20,30,etc are the vertical upwards from X=1,Y=0.
=head2 Square Spiral
This spiral goes around at the same rate as the C. It's as if
two corners are cut off (like the C) and two others extended
(like the C). The net effect is the same looping rate but
the points pushed around a bit.
Taking points up to a perfect square shows the similarity. The two
triangular cut-off corners marked by "."s are matched by the two triangular
extensions.
+--------------------+ 7x7 square
| . . . 31 . . .|
| . . 32 13 30 . .|
| . 33 14 3 12 29 .|
|34 15 4 1 2 11 28|
35|16 5 6 7 8 9 10|27
36 17|18 19 20 21 22 23 24|25 26
37 38 39|40 41 42 43 44 45 46|47 48 49
+--------------------+
=head2 N Start
The default is to number points starting N=1 as shown above. An optional
C can give a different start, with the same shape etc. For example
to start at 0,
=cut
# math-image --path=PyramidSpiral,n_start=0 --all --output=numbers_dash --size=35x16
=pod
12 n_start => 0
/ \
13 2 11
/ / \ \
14 3 0--1 10
/ / \
15 4--5--6--7--8--9
/
16-17-18-19-20-21-22-...
=head1 FUNCTIONS
See L for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::PyramidSpiral-Enew ()>
=item C<$path = Math::PlanePath::PyramidSpiral-Enew (n_start =E $n)>
Create and return a new pyramid spiral object.
=item C<$n = $path-Exy_to_n ($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 path is in Sloane's Online Encyclopedia of Integer Sequences as
=over
L (etc)
=back
n_start=1 (the default)
A053615 abs(X), distance to next pronic, but starts n=0
A054552 N on X axis, 4n^2 - 3n + 1
A033951 N on South-East diagonal, 4n^2 + 3n + 1
A214250 sum N of eight surrounding cells
A217013 permutation N of points in SquareSpiral order
rotated +90 degrees
A217294 inverse
In the two permutations the pyramid spiral is conceived as starting to the
left and the square spiral starting upwards. The paths here start in the
same direction (both to the right), hence rotate 90 to adjust the
orientation.
n_start=0
A001107 N on X axis, decagonal numbers
A002939 N on Y axis
A033991 N on X negative axis
A002943 N on Y negative axis
A007742 N on diagonal South-West
A033954 N on diagonal South-East, decagonal second kind
n_start=2
A185669 N on diagonal South-East
=head1 SEE ALSO
L,
L,
L,
L,
L
=head1 HOME PAGE
L
=head1 LICENSE
Copyright 2010, 2011, 2012, 2013 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 .
=cut