# 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::PentSpiral;
use 5.004;
use strict;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 122;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'round_nearest';
# uncomment this to run the ### lines
#use Smart::Comments;
use constant parameter_info_array =>
[
Math::PlanePath::Base::Generic::parameter_info_nstart1(),
];
sub x_negative_at_n {
my ($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 => -2;
use constant dx_maximum => 2;
use constant dy_minimum => -1;
use constant dy_maximum => 1;
use constant _UNDOCUMENTED__dxdy_list => (2,0, # E by 2
1,1, # NE
-2,1, # WNW
-2,-1, # WSW
1,-1, # SE
);
use constant absdx_minimum => 1;
use constant dsumxy_minimum => -3; # SW -2,-1
use constant dsumxy_maximum => 2; # dX=+2 and NE diag
use constant ddiffxy_minimum => -3; # NW dX=-2,dY=+1
use constant ddiffxy_maximum => 2;
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 $self;
}
# base South-West diagonal
# d = [ 1, 2, 3, 4 ]
# n = [ 0, 4, 13, 27 ]
# N = (5/2 d^2 - 7/2 d + 1)
# = (5/2*$d**2 - 7/2*$d + 1)
# = ((5/2*$d - 7/2)*$d + 1)
# d = 7/10 + sqrt(2/5 * $n + 9/100)
# = (sqrt(40*$n + 9) + 7) / 10
#
# split Y axis
# d = [ 1, 2, 3 ]
# n = [ 2, 9, 21 ]
# N = ((5/2*$d - 1/2)*$d)
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 ($n < 1) {
# bottom horizontal
return (2*$n+$d-1, -$d+1);
} else {
# lower right diagonal ...
return ($n+$d, $n-$d);
}
} else {
if ($n <= $d) {
### top 2,1 slope left and right diagonals ...
return (-2*$n,
-abs($n) + $d);
} else {
### lower left diagonal ...
return ($n - 3*$d,
-$n + $d);
}
}
}
sub xy_to_n {
my ($self, $x, $y) = @_;
$x = round_nearest ($x);
$y = round_nearest ($y);
# nothing on odd points
# when y>=0 any odd x is not covered
# when y<0 the uncovered alternates, x even on y=-1, x odd on y=-2, x even
# y=-3 etc
if (($x%2) ^ ($y < 0 ? $y%2 : 0)) {
return undef;
}
if ($y >= 0) {
### top left and right slopes
# vertical at x=0
# d = [ 1, 2, 3 ]
# n = [ 3, 10, 22 ]
# n = (5/2*$d**2 + -1/2*$d + 1)
#
### assert: ($x%2)==0
$x /= 2;
my $d = abs($x) + $y;
return (5*$d - 1)*$d/2 - $x + $self->{'n_start'};
}
if ($x < $y) {
### lower left slope
# horizontal leftwards at y=0
# d = [ 1, 2, 3 ]
# n = [ 4, 12, 25 ]
# n = (5/2*$d**2 + 1/2*$d + 1)
# = (2.5*$d + 0.5)*$d + 1
my $d = -($x+$y)/2;
return (5*$d + 1)*$d/2 - $y + $self->{'n_start'};
}
if ($x > -$y) {
### lower right slope
# horizontal rightwards at y=0
# d = [ 1, 2, 3, ]
# n = [ 2, 8, 19,]
# n = (5/2*$d**2 + -3/2*$d + 1)
# = (2.5*$d - 1.5)*$d + 1
my $d = ($x-$y)/2;
return (5*$d - 3)*$d/2 + $y + $self->{'n_start'};
}
### bottom horizontal
# vertical downwards at x=0 is
# y = [ -1, -2, -3 ]
# n = [ 5.5, 15, 29.5 ]
# n = (5/2*$y**2 + -2*$y + 1)
# = (2.5*$y - 2)*$y + 1
# so
# N = (2.5*$y - 2)*$y + 1 + $x/2
# = ((5*$y - 4)*$y + $x)/2 + 1
#
return ((5*$y-4)*$y + $x)/2 + $self->{'n_start'};
}
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### PentSpiral 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 $this_d = 1 + ($y >= 0 ? abs($x) + $y
: $x < $y ? -($x+$y)/2
: $x > -$y ? ($x-$y)/2
: -$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::PentSpiral -- integer points in a pentagonal shape
=head1 SYNOPSIS
use Math::PlanePath::PentSpiral;
my $path = Math::PlanePath::PentSpiral->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path makes a pentagonal (five-sided) spiral with points spread out to
fit on a square grid.
22 3
23 10 21 2
24 11 3 9 20 1
25 12 4 1 2 8 19 <- Y=0
26 13 5 6 7 18 ... -1
27 14 15 16 17 33 -2
28 29 30 31 32 -2
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
Each horizontal gap is 2, so for instance n=1 is at x=0,y=0 then n=2 is at
x=2,y=0. The lower diagonals are 1 across and 1 down, so n=17 is at
x=4,y=-2 and n=18 is x=5,y=-1. But the upper angles go 2 across and 1 up,
so n=20 is x=4,y=1 then n=21 is x=2,y=2.
The effect is to make the sides equal length, except for a kink at the lower
right corner. Only every second square in the plane is used. In the top
half (y>=0) those points line up, in the lower half (y<0) they're offset on
alternate rows.
=head2 N Start
The default is to number points starting N=1 as shown above. An optional
C<n_start> can give a different start, in the same pattern. For example to
start at 0,
=cut
# math-image --path=PentSpiral,n_start=0 --expression='i<=57?i:0' --output=numbers --size=120x11
=pod
n_start => 0 38
39 21 37
...
40 22 9 20 36 57
41 23 10 2 8 19 35 56
42 24 11 3 0 1 7 18 34 55
43 25 12 4 5 6 17 33 54
44 26 13 14 15 16 32 53
45 27 28 29 30 31 52
46 47 48 49 50 51
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::PentSpiral-E<gt>new ()>
=item C<$path = Math::PlanePath::PentSpiral-E<gt>new (n_start =E<gt> $n)>
Create and return a new pentagon spiral object.
=item C<$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 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,
21 loop d=3
-- --
22 20
-- --
23 19
-- --
24 0 18
\ /
25 . 17
\ /
26 13----14----15----16
\
.
The SW diagonal is N=0,4,13,27,46,etc which is
N = (5d-7)*d/2 + 1 # starting d=1 first loop
This can be inverted to get d from N
d = floor( (sqrt(40*N + 9) + 7) / 10 )
Each side is length d, except the lower right diagonal slope which is d-1.
For the very first loop that lower right is length 0.
=head1 OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this
path include
=over
L<http://oeis.org/A140066> (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
A005891 N on South-East diagonal, centred pentagonals
A134238 N on South-West diagonal
n_start=0
A000566 N on X axis, heptagonal numbers
A005476 N on Y axis
A028895 N on South-East diagonal
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::PentSpiralSkewed>,
L<Math::PlanePath::HexSpiral>
=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