# 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/>.
# ENHANCE-ME: n_to_xy() might be done with some rotates etc around its
# symmetry instead of 8 or 16 cases.
#
package Math::PlanePath::OctagramSpiral;
use 5.004;
use strict;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 121;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'round_nearest';
# uncomment this to run the ### lines
#use Smart::Comments;
use constant xy_is_visited => 1;
use constant dx_minimum => -1;
use constant dx_maximum => 1;
use constant dy_minimum => -1;
use constant dy_maximum => 1;
*_UNDOCUMENTED__dxdy_list = \&Math::PlanePath::_UNDOCUMENTED__dxdy_list_eight;
sub x_negative_at_n {
my ($self) = @_;
return $self->n_start + 6;
}
sub y_negative_at_n {
my ($self) = @_;
return $self->n_start + 10;
}
sub _UNDOCUMENTED__dxdy_list_at_n {
my ($self) = @_;
return $self->n_start + 8;
}
use constant dsumxy_minimum => -2; # diagonals
use constant dsumxy_maximum => 2;
use constant ddiffxy_minimum => -2;
use constant ddiffxy_maximum => 2;
use constant dir_maximum_dxdy => (1,-1); # South-East
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 $self;
}
sub n_to_xy {
my ($self, $n) = @_;
#### OctagramSpiral n_to_xy: $n
# adjust to N=1 at origin X=0,Y=0
$n = $n - $self->{'n_start'} + 1;
if ($n <= 2) {
if ($n < 1) {
return;
} else {
return ($n-1, 0);
}
}
# sqrt() done in integers to avoid limited precision from Math::BigRat sqrt()
#
my $d = int ((sqrt(int(32*$n) + 17) + 7) / 16);
#### d frac: ((sqrt(int(32*$n) + 17) + 7) / 16)
#### $d
#### base: ((8*$d - 7)*$d + 1)
$n -= ((8*$d - 7)*$d + 1);
#### remainder: $n
if ($n < $d) {
return ($d + $n, $n);
}
$n -= 2*$d;
if ($n < $d) {
if ($n < 0) {
return (- $n + $d,
$d);
} else {
return ($d,
$n + $d);
}
}
$n -= 2*$d;
if ($n < $d) {
return (-$n,
abs($n) + $d);
}
$n -= 2*$d;
if ($n < $d) {
if ($n < 0) {
return (-$d,
-$n + $d);
} else {
return (-$n - $d,
$d);
}
}
$n -= 2*$d;
if ($n < $d) {
return (-$d-abs($n), -$n);
}
$n -= 2*$d;
if ($n < $d) {
if ($n < 0) {
return (-$d + $n,
-$d);
} else {
return (-$d,
-$d - $n);
}
}
$n -= 2*$d;
if ($n < $d) {
return ($n,
- abs($n) - $d);
}
$n -= 2*$d;
if ($n < $d+1) {
if ($n < 0) {
return ($d,
$n - $d);
} else {
return ($n + $d,
-$d);
}
}
# $n >= $d+1 through to 2*$d+1
return (-$n + 3*$d+2, $n - 2*$d-1);
}
sub xy_to_n {
my ($self, $x, $y) = @_;
$x = round_nearest ($x);
$y = round_nearest ($y);
### xy_to_n: "x=$x, y=$y"
my $n;
if ($x > 0 && $y < 0 && -2*$y < $x) {
### longer bottom right horiz
$x--;
$n = 1;
} else {
$n = 0;
}
my $d_offset = 0;
if ($y < 0) {
$y = -$y;
$x = -$x;
$d_offset = 8;
### rotate 180 back: "$x, $y"
}
if ($x < 0) {
($x, $y) = ($y, -$x);
$d_offset += 4;
### rotate 90 back: "$x, $y"
}
if ($y > $x) {
($x, $y) = ($y, $y-$x);
$d_offset += 2;
### rotate 45 back: "$x, $y"
}
my $d;
if (2*$y < $x) {
### diag up
$d = $x - $y;
$n += $y;
} else {
### horiz back
$d = $y;
$n -= $x;
$d_offset += 3;
}
### final
### $d
### $d_offset
### $n
# horiz base 2,19,54,...
return $n + (8*$d - 7 + $d_offset)*$d + $self->{'n_start'};
}
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
my $d = max (1, map {abs(round_nearest($_))} $x1,$y1,$x2,$y2);
### $d
# ENHANCE-ME: find actual minimum if rect doesn't cover 0,0
return ($self->{'n_start'},
# bottom-right inner corner 16,47,94,...
$self->{'n_start'} + (8*$d + 7)*$d);
}
1;
__END__
# 29 25 4
# | \ / |
# 30 28 26 24 ...-56--55 3
# | \ / | /
# 33--32--31 7 27 5 23--22--21 54 2
# \ | \ / | / /
# 34 9-- 8 6 4-- 3 20 53 1
# \ \ / / /
# 35 10 1---2 19 52 <- Y=0
# / / \ \
# 36 11--12 14 16--17--18 51 -1
# / | / \ | \
# 37--38--39 13 43 15 47--48--49--50 -2
# | / \ |
# 40 42 44 46 -3
# | / \ |
# 41 45 -4
#
# ^
# -4 -3 -2 -1 X=0 1 2 3 4 5 ...
#
#
#
#
#
#
#
#
#
# 28 24 4
# | \ / |
# 29 27 25 23 ...-54--53 3
# | \ / | /
# 32--31--30 7 26 5 22--21--20 52 2
# \ | \ / | / /
# 33 9-- 8 6 4-- 3 19 51 1
# \ \ / / /
# 34 10 1---2 18 50 <- Y=0
# / / | |
# 35 11--12 14 16--17 49 -1
# / | / \ | \
# 36--37--38 13 42 15 46--47--48 -2
# | / \ |
# 39 41 43 45 -3
# | / \ |
# 40 44 -4
#
# ^
# -4 -3 -2 -1 X=0 1 2 3 4 5 ...
=for stopwords Ryde Math-PlanePath octagram 18-gonal OEIS
=head1 NAME
Math::PlanePath::OctagramSpiral -- integer points drawn around an octagram
=head1 SYNOPSIS
use Math::PlanePath::OctagramSpiral;
my $path = Math::PlanePath::OctagramSpiral->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path makes a spiral around an octagram (8-pointed star),
29 25 4
| \ / |
30 28 26 24 ...56-55 3
| \ / | /
33-32-31 7 27 5 23-22-21 54 2
\ |\ / | / /
34 9- 8 6 4- 3 20 53 1
\ \ / / /
35 10 1--2 19 52 <- Y=0
/ / \ \
36 11-12 14 16-17-18 51 -1
/ |/ \ | \
37-38-39 13 43 15 47-48-49-50 -2
| / \ |
40 42 44 46 -3
|/ \ |
41 45 -4
^
-4 -3 -2 -1 X=0 1 2 3 4 5 ...
Each loop is 16 longer than the previous. The 18-gonal numbers
18,51,100,etc fall on the horizontal at Y=-1.
The inner corners like 23, 31, 39, 47 are similar to the C<SquareSpiral>
path, but instead of going directly between them the octagram takes a detour
out to make the points of the star. Those excursions make each loops 8
longer (1 per excursion), hence a step of 16 here as compared to 8 for the
C<SquareSpiral>.
=head2 N Start
The default is to number points starting N=1 as shown above. An optional
C<n_start> can give a different start, in the same pattern. For example to
start at 0,
=cut
# math-image --path=OctagramSpiral,n_start=0 --expression='i<=55?i:0' --output=numbers --size=80x13
=pod
n_start => 0
28 24
29 27 25 23 ... 55 54
32 31 30 6 26 4 22 21 20 53
33 8 7 5 3 2 19 52
34 9 0 1 18 51
35 10 11 13 15 16 17 50
36 37 38 12 42 14 46 47 48 49
39 41 43 45
40 44
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::OctagramSpiral-E<gt>new ()>
Create and return a new octagram 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 FORMULAS
=head2 X,Y to N
The symmetry of the octagram can be used by rotating a given X,Y back to the
first star excursion such as N=19 to N=23. If Y is negative then rotate
back by 180 degrees, then if X is negative rotate back by 90, and if Y>=X
then by a further 45 degrees. Each such rotation, if needed, is counted as
a multiple of the side-length to be added to the final N. For example at
N=19 the side length is 2. Rotating by 180 degrees is 8 side lengths, by 90
degrees 4 sides, and by 45 degrees is 2 sides.
=head1 OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this
path include
=over
L<http://oeis.org/A125201> (etc)
=back
n_start=1 (the default)
A125201 N on X axis, from X=1 onwards, 18-gonals + 1
A194268 N on diagonal South-East
n_start=0
A051870 N on X axis, 18-gonal numbers
A139273 N on Y axis
A139275 N on X negative axis
A139277 N on Y negative axis
A139272 N on diagonal X=Y
A139274 N on diagonal North-West
A139276 N on diagonal South-West
A139278 N on diagonal South-East, second 18-gonals
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::SquareSpiral>,
L<Math::PlanePath::PyramidSpiral>
=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