# Copyright 2010, 2011, 2012, 2013, 2014 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::TriangleSpiral;
use 5.004;
use strict;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 116;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'round_nearest';
# uncomment this to run the ### lines
#use Smart::Comments;
*xy_is_visited = \&Math::PlanePath::Base::Generic::xy_is_even;
use constant parameter_info_array =>
[ Math::PlanePath::Base::Generic::parameter_info_nstart1() ];
sub x_negative_at_n {
my ($self) = @_;
return $self->n_start + 4;
}
sub y_negative_at_n {
my ($self) = @_;
return $self->n_start + 6;
}
sub _UNDOCUMENTED__dxdy_list_at_n {
my ($self) = @_;
return $self->n_start + 3;
}
use constant dx_minimum => -1;
use constant dx_maximum => 2;
use constant dy_minimum => -1;
use constant dy_maximum => 1;
use constant _UNDOCUMENTED__dxdy_list => (2,0, # E
-1,1, # NW
-1,-1); # SW
use constant absdx_minimum => 1;
use constant dsumxy_minimum => -2; # SW diagonal
use constant dsumxy_maximum => 2; # dX=+2 horiz
use constant ddiffxy_minimum => -2; # NW diagonal
use constant ddiffxy_maximum => 2; # dX=+2 horiz
use constant dir_maximum_dxdy => (-1,-1); # at most South-West diagonal
#------------------------------------------------------------------------------
sub new {
my $self = shift->SUPER::new (@_);
if (! defined $self->{'n_start'}) {
$self->{'n_start'} = $self->default_n_start;
}
return $self;
}
# base at bottom right corner
# d = [ 1, 2, 3 ]
# n = [ 2, 11, 29 ]
# $d = 1/2 + sqrt(2/9 * $n + -7/36)
# = 1/2 + sqrt(8/36 * $n + -7/36)
# = 0.5 + sqrt(8*$n + -7)/6
# = (1 + 2*sqrt(8*$n + -7)/6) / 2
# = (1 + sqrt(8*$n + -7)/3) / 2
# = (3 + sqrt(8*$n - 7)) / 6
#
# $n = (9/2*$d**2 + -9/2*$d + 2)
# = (4.5*$d - 4.5)*$d + 2
#
# top of pyramid
# d = [ 1, 2, 3 ]
# n = [ 4, 16, 37 ]
# $n = (9/2*$d**2 + -3/2*$d + 1)
# so remainder from there
# rem = $n - (9/2*$d**2 + -3/2*$d + 1)
# = $n - (4.5*$d*$d - 1.5*$d + 1)
# = $n - ((4.5*$d - 1.5)*$d + 1)
#
#
sub n_to_xy {
my ($self, $n) = @_;
#### TriangleSpiral n_to_xy: $n
$n = $n - $self->{'n_start'}; # starting $n==0, warn if $n==undef
if ($n < 0) { return; }
my $d = int ((3 + sqrt(8*$n+1)) / 6);
#### $d
$n -= (9*$d - 3)*$d/2;
#### remainder: $n
if ($n <= 3*$d) {
### sides, remainder pos/neg from top
return (-$n,
2*$d - abs($n));
} else {
### rightwards from bottom left
### remainder: $n - 3*$d
# corner is x=-3*$d
# so -3*$d + 2*($n - 3*$d)
# = -3*$d + 2*$n - 6*$d
# = -9*$d + 2*$n
# = 2*$n - 9*$d
return (2*$n - 9*$d,
-$d);
}
}
sub xy_to_n {
my ($self, $x, $y) = @_;
$x = round_nearest ($x);
$y = round_nearest ($y);
### xy_to_n(): "$x,$y"
if (($x ^ $y) & 1) {
return undef; # nothing on odd points
}
if ($y < 0 && 3*$y <= $x && $x <= -3*$y) {
### bottom horizontal
# negative y, at vertical x=0
# [ -1, -2, -3, -4, -5, -6 ]
# [ 8.5, 25, 50.5, 85, 128.5, 181 ]
# $n = (9/2*$y**2 + -3*$y + 1)
# = (4.5*$y*$y + -3*$y + 1)
# = ((4.5*$y -3)*$y + 1)
# from which $x/2
#
return ((9*$y - 6)*$y/2) + $x/2 + $self->{'n_start'};
} else {
### sides diagonal
#
# positive y, x=0 centres
# [ 2, 4, 6, 8 ]
# [ 4, 16, 37, 67 ]
# n = (9/8*$d**2 + -3/4*$d + 1)
# = (9/8*$d + -3/4)*$d + 1
# = (9*$d + - 6)*$d/8 + 1
# from which -$x offset
#
my $d = abs($x) + $y;
return ((9*$d - 6)*$d/8) - $x + $self->{'n_start'};
}
}
# not exact
sub rect_to_n_range {
my ($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 + ($y < 0 && 3*$y <= $x && $x <= -3*$y
? -$y # bottom horizontal
: int ((abs($x) + $y) / 2))); # sides
}
}
return ($self->{'n_start'},
(9*$d - 9)*$d/2 + $self->{'n_start'});
}
1;
__END__
=for stopwords Ryde Math-PlanePath hendecagonal 11-gonal (s+2)-gonal OEIS hendecagonals
=head1 NAME
Math::PlanePath::TriangleSpiral -- integer points drawn around an equilateral triangle
=head1 SYNOPSIS
use Math::PlanePath::TriangleSpiral;
my $path = Math::PlanePath::TriangleSpiral->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path makes a spiral shaped as an equilateral triangle (each side the
same length).
16 4
/ \
17 15 3
/ \
18 4 14 ... 2
/ / \ \ \
19 5 3 13 32 1
/ / \ \ \
20 6 1-----2 12 31 <- Y=0
/ / \ \
21 7-----8-----9----10----11 30 -1
/ \
22----23----24----25----26----27----28----29 -2
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8
Cells are spread horizontally to fit on a square grid as per
L. The horizontal gaps are 2, so for
instance n=1 is at x=0,y=0 then n=2 is at x=2,y=0. The diagonals are 1
across and 1 up or down, so n=3 is at x=1,y=1. Each alternate row is offset
from the one above or below.
This grid is the same as the C and the path is like that spiral
except instead of a flat top and SE,SW sides it extends to triangular peaks.
The result is a longer loop and each successive loop is step=9 longer than
the previous (whereas the C is step=6 more).
XThe triangular numbers 1, 3, 6, 10, 15, 21, 28, 36 etc,
k*(k+1)/2, fall one before the successive corners of the triangle, so when
plotted make three lines going vertically and angled down left and right.
The 11-gonal "hendecagonal" numbers 11, 30, 58, etc, k*(9k-7)/2 fall on a
straight line horizontally to the right. (As per the general rule that a
step "s" lines up the (s+2)-gonal numbers.)
=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=TriangleSpiral,n_start=0 --expression='i<=31?i:0' --output=numbers_dash
=pod
n_start => 0 15
/ \
16 14
/ \
17 3 13
/ / \ \
18 4 2 12 ...
/ / \ \ \
19 5 0-----1 11 30
/ / \ \
20 6-----7-----8-----9----10 29
/ \
21----22----23----24----25----26----27----28
With this adjustment the X axis N=0,1,11,30,etc is the hendecagonal numbers
(9k-7)*k/2. And N=0,8,25,etc diagonally South-East is the hendecagonals of
the second kind which is (9k-7)*k/2 for k negative.
=head1 FUNCTIONS
See L for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::TriangleSpiral-Enew ()>
=item C<$path = Math::PlanePath::TriangleSpiral-Enew (n_start =E $n)>
Create and return a new triangle spiral object.
=item C<($x,$y) = $path-En_to_xy ($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 = $path-Exy_to_n ($x,$y)>
Return the point number for coordinates C<$x,$y>. C<$x> and C<$y> are
each rounded to the nearest integer, which has the effect of treating each
C<$n> in the path as a square of side 1.
Only every second square in the plane has an N. If C<$x,$y> is a
position without an N then the return is C.
=back
=head1 OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to
this path include
=over
L (etc)
=back
n_start=1 (default)
A010054 turn 1=left,0=straight, extra initial 1
A117625 N on X axis
A081272 N on Y axis
A006137 N on X negative axis
A064226 N on X=Y leading diagonal, but without initial value=1
A064225 N on X=Y negative South-West diagonal
A081267 N on X=-Y negative South-East diagonal
A081589 N on ENE slope dX=3,dY=1
A038764 N on WSW slope dX=-3,dY=-1
A060544 N on ESE slope dX=3,dY=-1 diagonal
A063177 total sum previous row or diagonal
n_start=0
A051682 N on X axis (11-gonal numbers)
A062741 N on Y axis
A062708 N on X=Y leading diagonal
A081268 N on X=Y+2 diagonal (right of leading diagonal)
A062728 N on South-East diagonal (11-gonal second kind)
A062725 N on South-West diagonal
A081275 N on ENE slope from X=2,Y=0 then dX=+3,dY=+1
A081266 N on WSW slope dX=-3,dY=-1
A081271 N on X=2 vertical
n_start=-1
A023531 N position of turns (to the left)
1 at N=k*(k+3)/2
A023531 is C to match its "offset=0" for the first turn, being
the second point of the path. A010054 which is 1 at triangular numbers
k*(k+1)/2 is the same except for an extra initial 1.
=head1 SEE ALSO
L,
L,
L
=head1 HOME PAGE
L
=head1 LICENSE
Copyright 2010, 2011, 2012, 2013, 2014 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