# Copyright 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::DiagonalsAlternating;
use 5.004;
use strict;
use vars '$VERSION', '@ISA';
$VERSION = 120;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'round_nearest';
# uncomment this to run the ### lines
# use Smart::Comments;
use constant parameter_info_array =>
[ Math::PlanePath::Base::Generic::parameter_info_nstart1() ];
use constant class_x_negative => 0;
use constant class_y_negative => 0;
use constant dx_minimum => -1;
use constant dx_maximum => 1;
use constant dy_minimum => -1;
use constant dy_maximum => 1;
use constant 1.02 _UNDOCUMENTED__dxdy_list => (1,0, # E at N=3 in default n_start=1
0,1, # N at N=1
-1,1, # NW at N=4
1,-1, # SE at N=2
);
sub _UNDOCUMENTED__dxdy_list_at_n {
my ($self) = @_;
return $self->n_start + 3;
}
use constant dsumxy_minimum => 0; # advancing diagonals
use constant dsumxy_maximum => 1;
use constant ddiffxy_minimum => -2; # NW diagonal
use constant ddiffxy_maximum => 2; # SE diagonal
use constant dir_maximum_dxdy => (1,-1); # South-East
#------------------------------------------------------------------------------
sub new {
my $self = shift->SUPER::new(@_);
if (! defined $self->{'n_start'}) {
$self->{'n_start'} = $self->default_n_start;
}
$self->{'x_start'} ||= 0;
$self->{'y_start'} ||= 0;
return $self;
}
# \
# 15 26
# |\ \
# 14 16 25
# \ \
# . . 17 24
# \ \
# . . . 18 23
# \ \
# . . . . 19 22
# \ \
# . . . . . 20-21
# Basis N=0
# d = [ 0, 1, 2, 3 ]
# n = [ 0, 5, 14, 27 ]
# N = (2 d^2 + 3 d)
# = (2*$d**2 + 3*$d)
# = ((2*$d + 3)*$d)
# d = -3/4 + sqrt(1/2 * $n + 9/16)
# = (sqrt(8*$n + 9) - 3) / 4
#
# Midpoint
# d = [ 0, 1, 2, 3 ]
# n = [ 2, 9, 20, 35 ]
# N = (2 d^2 + 5 d + 2)
# = (2*$d**2 + 5*$d + 2)
# = ((2*$d + 5)*$d + 2)
sub n_to_xy {
my ($self, $n) = @_;
### DiagonalsAlternating n_to_xy(): "$n ".(ref $n || '')
# adjust to N=0 at origin X=0,Y=0
$n = $n - $self->{'n_start'};
if ($n < 0) { return; }
my $d = int( (sqrt(8*int($n)+9) - 3)/4 );
$n -= (2*$d + 5)*$d + 3;
### remainder: $n
my ($x,$y);
if ($n >= -1) {
if ($n < 0) {
### horizontal X axis ...
$x = $n + 2*$d+2;
$y = 0;
} else {
### diagonal upwards ...
$x = -$n + 2*$d+2;
$y = $n;
}
} else {
$n += 2*$d+2; # -1 <= $n < ...
### added n: $n
### assert: ! ($n < -1)
if ($n < 0) {
### vertical Y axis ...
$x = 0;
$y = $n + 2*$d+1;
} else {
### diagonal downwards ...
$x = $n;
$y = -$n + 2*$d+1;
}
}
return ($x + $self->{'x_start'},
$y + $self->{'y_start'});
}
sub xy_to_n {
my ($self, $x, $y) = @_;
### xy_to_n(): $x, $y
$x = $x - $self->{'x_start'}; # "-" operator to provoke warning if x==undef
$y = $y - $self->{'y_start'};
$x = round_nearest ($x);
$y = round_nearest ($y);
if ($x < 0 || $y < 0) {
return undef; # outside first quadrant
}
my $d = $x + $y;
# odd, downwards ...
# d= [ 1,3,5 ]
# N= [ 2,7,16 ]
# N = ((1/2*$d + 1/2)*$d + 1)
#
# even, upwards
# d= [ 0,2,4 ]
# N= [ 1,4,11 ]
# N = ((1/2*$d + 1/2)*$d + 1)
# = ($d + 1)*$d/2 + 1
my $n = ($d + 1)*$d/2 + $self->{'n_start'};
if ($d % 2) {
return $n + $x;
} else {
return $n + $y;
}
}
use Math::PlanePath::Diagonals;
*rect_to_n_range = \&Math::PlanePath::Diagonals::rect_to_n_range;
1;
__END__
=for stopwords PlanePath Ryde Math-PlanePath OEIS
=head1 NAME
Math::PlanePath::DiagonalsAlternating -- points in diagonal stripes of alternating directions
=head1 SYNOPSIS
use Math::PlanePath::DiagonalsAlternating;
my $path = Math::PlanePath::DiagonalsAlternating->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path follows successive diagonals going from the Y axis down to the X
axis and then back again,
=cut
# math-image --path=DiagonalsAlternating --expression='i<=31?i:0' --output=numbers
=pod
7 | 29
6 | 28 30
5 | 16 27 31
4 | 15 17 26 ...
3 | 7 14 18 25
2 | 6 8 13 19 24
1 | 2 5 9 12 20 23
Y=0 | 1 3 4 10 11 21 22
+----------------------------
X=0 1 2 3 4 5 6
X<Triangular numbers>The triangular numbers 1,3,6,10,etc k*(k+1)/2 are the
start of each run up or down alternately on the X axis and Y axis.
X<Hexagonal numbers>N=1,6,15,28,etc on the Y axis (Y even) are the hexagonal
numbers j*(2j-1). N=3,10,21,36,etc on the X axis (X odd) are the hexagonal
numbers of the second kind j*(2j+1).
=head2 N Start
The default is to number points starting N=1 as shown above. An optional
C<n_start> can give a different start, in the same pattern. For example to
start at 0,
=cut
# math-image --path=DiagonalsAlternating,n_start=0 --expression='i<=14?i:0' --output=numbers --size=35x5
=pod
n_start => 0
4 | 14
3 | 6 13
2 | 5 7 12
1 | 1 4 8 11
Y=0 | 0 2 3 9 10
+-----------------
X=0 1 2 3 4
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::DiagonalsAlternating-E<gt>new ()>
=item C<$path = Math::PlanePath::DiagonalsAlternating-E<gt>new (n_start =E<gt> $n)>
Create and return a new path object.
=item C<($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 E<lt> 1> the return is an empty list, it being considered the path
begins at 1.
=back
=head1 FORMULAS
=head2 Rectangle to N Range
Within each row increasing X is increasing N, and in each column increasing
Y is increasing N. So in a rectangle the lower left corner is the minimum N
and the upper right is the maximum N.
| N max
| ----------+
| | ^ |
| | | |
| | ----> |
| +----------
| N min
+-------------------
=head1 OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this
path include
=over
L<http://oeis.org/A131179> (etc)
=back
n_start=1
A131179 N on X axis (extra initial 0)
A128918 N on Y axis (extra initial 1)
A001844 N on X=Y diagonal
A038722 permutation N at transpose Y,X
n_start=0
A003056 X+Y
A004247 X*Y
A049581 abs(X-Y)
A048147 X^2+Y^2
A004198 X bit-and Y
A003986 X bit-or Y
A003987 X bit-xor Y
A004197 min(X,Y)
A003984 max(X,Y)
A101080 HammingDist(X,Y)
A023531 dSum = dX+dY, being 1 at N=triangular+1 (and 0)
A046092 N on X=Y diagonal
A061579 permutation N at transpose Y,X
A056011 permutation N at points by Diagonals,direction=up order
A056023 permutation N at points by Diagonals,direction=down
runs alternately up and down, both are self-inverse
The coordinates such as A003056 X+Y are the same here as in the Diagonals
path. C<DiagonalsAlternating> transposes X,Y -E<gt> Y,X in every second
diagonal but forms such as X+Y are unchanged by swapping to Y+X.
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::Diagonals>,
L<Math::PlanePath::DiagonalsOctant>
=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